Smart antennas for next generation wireless systems eurasip

EURASIP Journal on Wireless Communications and Networking

Smart Antennas for Next Generation Wireless Systems

Guest Editors: Angeliki Alexiou, Monica Navarro, and Robert W. Heath Jr.

Smart Antennas for NextGeneration Wireless Systems

EURASIP Journal onWireless Communications and Networking

Smart Antennas for NextGeneration Wireless Systems

Guest Editors: Angeliki Alexiou, Monica Navarro,and Robert W. Heath Jr.

Copyright © 2007 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in volume 2007 of “EURASIP Journal on Wireless Communications and Networking.” All articles areopen access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Editor-in-ChiefLuc Vandendorpe, Universite Catholique de Louvain, Belgium

Associate Editors

Thushara Abhayapala, AustraliaMohamed H. Ahmed, CanadaFarid Ahmed, USAAlagan Anpalagan, CanadaAnthony Boucouvalas, GreeceLin Cai, CanadaBiao Chen, USAYuh-Shyan Chen, TaiwanPascal Chevalier, FranceChia-Chin Chong, South KoreaHuaiyu Dai, USASoura Dasgupta, USAIbrahim Develi, TurkeyPetar M. Djuric, USAMischa Dohler, FranceAbraham O. Fapojuwo, CanadaMichael Gastpar, USAAlex Gershman, GermanyWolfgang Gerstacker, Germany

David Gesbert, FranceZ. Ghassemlooy, UKChristian Hartmann, GermanyStefan Kaiser, GermanyG. K. Karagiannidis, GreeceChi Chung Ko, SingaporeVisa Koivunen, FinlandRichard Kozick, USABhaskar Krishnamachari, USAS. Lambotharan, UKVincent Lau, Hong KongDavid I. Laurenson, UKTho Le-Ngoc, CanadaYonghui Li, AustraliaWei Li, USATongtong Li, USAZhiqiang Liu, USASteve McLaughlin, UKSudip Misra, Canada

Marc Moonen, BelgiumEric Moulines, FranceSayandev Mukherjee, USAKameswara Rao Namuduri, USAAmiya Nayak, CanadaA. Pandharipande, South KoreaAthina Petropulu, USAPhillip Regalia, FranceA. Lee Swindlehurst, USASergios Theodoridis, GreeceGeorge S. Tombras, GreeceLang Tong, USAAthanasios V. Vasilakos, GreeceWeidong Xiang, USAYang Xiao, USAXueshi Yang, USALawrence Yeung, Hong KongDongmei Zhao, CanadaWeihua Zhuang, Canada

Contents

Smart Antennas for Next Generation Wireless Systems, Angeliki Alexiou, Monica Navarro,and Robert W. Heath Jr.Volume 2007, Article ID 20427, 2 pages

Tower-Top Antenna Array Calibration Scheme for Next Generation Networks, Justine McCormack,Tim Cooper, and Ronan FarrellVolume 2007, Article ID 41941, 12 pages

Optimal Design of Uniform Rectangular Antenna Arrays for Strong Line-of-Sight MIMO Channels,Frode Bøhagen, Pal Orten, and Geir ØienVolume 2007, Article ID 45084, 10 pages

Joint Estimation of Mutual Coupling, Element Factor, and Phase Center in Antenna Arrays,Marc Mowler, Bjorn Lindmark, Erik G. Larsson, and Bjorn OtterstenVolume 2007, Article ID 30684, 9 pages

Diversity Characterization of Optimized Two-Antenna Systems for UMTS Handsets, A. Diallo,P. Le Thuc, C. Luxey, R. Staraj, G. Kossiavas, M. Franzen, and P.-S. KildalVolume 2007, Article ID 37574, 9 pages

Optimal Design of Nonuniform Linear Arrays in Cellular Systems by Out-of-Cell InterferenceMinimization, S. Savazzi, O. Simeone, and U. SpagnoliniVolume 2007, Article ID 93421, 9 pages

Capacity Performance of Adaptive Receive Antenna Subarray Formation for MIMO Systems,Panagiotis Theofilakos and Athanasios G. KanatasVolume 2007, Article ID 56471, 12 pages

Capacity of MIMO-OFDM with Pilot-Aided Channel Estimation, Ivan Cosovic and Gunther AuerVolume 2007, Article ID 32460, 12 pages

Distributed Antenna Channels with Regenerative Relaying: Relay Selection and AsymptoticCapacity, Aitor del Coso and Christian IbarsVolume 2007, Article ID 21093, 12 pages

A Simplified Constant Modulus Algorithm for Blind Recovery of MIMO QAM and PSK Signals:A Criterion with Convergence Analysis, Aissa Ikhlef and Daniel Le GuennecVolume 2007, Article ID 90401, 13 pages

Employing Coordinated Transmit and Receive Beamforming in Clustering Double-DirectionalRadio Channel, Chen Sun, Makoto Taromaru, and Takashi OhiraVolume 2007, Article ID 57175, 10 pages

Inter- and Intrasite Correlations of Large-Scale Parameters from Macrocellular Measurements at1800 MHz, Niklas Jalden, Per Zetterberg, Bjorn Ottersten, and Laura GarciaVolume 2007, Article ID 25757, 12 pages

Multiple-Antenna Interference Cancellation for WLAN with MAC Interference Avoidance inOpen Access Networks, Alexandr M. Kuzminskiy and Hamid Reza KarimiVolume 2007, Article ID 51358, 11 pages

Transmit Diversity at the Cell Border Using Smart Base Stations, Simon Plass, Ronald Raulefs,and Armin DammannVolume 2007, Article ID 60654, 11 pages

SmartMIMO: An Energy-Aware Adaptive MIMO-OFDM Radio Link Control for Next-GenerationWireless Local Area Networks, Bruno Bougard, Gregory Lenoir, Antoine Dejonghe, Liesbet Van der Perre,Francky Catthoor, and Wim DehaeneVolume 2007, Article ID 98186, 15 pages

Cross-Layer Admission Control Policy for CDMA Beamforming Systems, Wei Sheng andSteven D. BlosteinVolume 2007, Article ID 14562, 15 pages

Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2007, Article ID 20427, 2 pagesdoi:10.1155/2007/20427

EditorialSmart Antennas for Next Generation Wireless Systems

Angeliki Alexiou,1 Monica Navarro,2 and Robert W. Heath Jr.3

1 Bell Laboratories, Lucent Technologies, Suffolk CO10 1LN, UK2 Centre Tecnologic de Telecomunicacions de Catalunya (CTTC), 08860 Barcelona, Spain3 Department of Electrical and Computer Engineering, The University of Texas at Austin, TX 78712, USA

Received 31 December 2007; Accepted 31 December 2007

Copyright © 2007 Angeliki Alexiou et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The adoption of multiple antenna techniques in future wire-less systems is expected to have a significant impact onthe efficient use of the spectrum, the minimisation of thecost of establishing new wireless networks, the enhancementof the quality of service, and the realisation of reconfig-urable, robust, and transparent operation across multitech-nology wireless networks. Although a considerable amountof research effort has been dedicated to the investigation ofMIMO systems performance, results, conclusions, and ideason the critical implementation aspects of smart antennas infuture wireless systems remain fragmental.

The objective of this special issue is to address these crit-ical aspects and present the most recent developments inthe areas of antenna array design, implementation, measure-ments, and MIMO channel modelling, robust signal process-ing for multiple antenna systems and interference-aware sys-tem level optimisation.

In the area of antenna array design, the paper by T.Cooper et al. presents a tower-top smart antenna calibra-tion scheme, designed for high-reliability tower-top opera-tion and based upon an array of coupled reference elementswhich sense the array’s output. The theoretical limits of theaccuracy of this calibration approach are assessed and the ex-pected performance is evaluated by means of initial proto-typing of a precision coupler circuit for a 2 × 2 array.

The design of uniform rectangular arrays for MIMOcommunication systems with strong line-of-sight compo-nents is studied in the paper by F. Bøhagen et al., basedon an orthogonality requirement inspired by the mutual in-formation. Because the line-of-sight channel is more sensi-tive to antenna geometry and orientation, a new geometricalmodel is proposed in this paper that includes the transmitand receive orientation through a reference coordinate sys-tem, along with a spherical wave propagation model to pro-

vide more accurate propagation predictions for the line-of-sight channel. It is shown that the separation distance be-tween antennas can be optimized in several cases and thatthese configurations are robust in Ricean channels with dif-ferent K-factors.

The paper by M. Mowler et al. considers estimation ofthe mutual coupling matrix for an adaptive antenna arrayfrom calibration measurements. This paper explicitly incor-porates the lack of information about the phase centre andthe element factor of the array by including them in an it-erative joint optimization. In particular, the element factorof the array is approximated through a basis expansion—thecoefficients of the expansion are estimated by the algorithm.Analysis, simulations, and experimental results demonstratethe efficacy of the proposed estimator.

Diversity characterisation of two-antenna systems forUMTS terminals by means of measurements performed con-currently with the help of a reverberation chamber and aWheeler Cap setup is addressed in the paper by A. Diallo et al.It is shown that even if the envelope correlation coefficientsof these systems are very low, having antennas with high iso-lation improves the total efficiency by increasing the effectivediversity gain.

In the paper by S. Savazzi et al., the authors address thedesign problem of linear antenna array optimization to en-hance the overall throughput of an interference-limited sys-tem. They focus on the design of linear antenna arrays withnonuniform spacing between antenna array elements, whichexplicitly takes into account the cellular layout and the prop-agation model, and show the potential gains with respect tothe conventional half wavelength systems. For such purposestwo optimisation criteria are considered: one based on theminimization of the average interference power at the out-put of a conventional beamformer, for which a closed-form

2 EURASIP Journal on Wireless Communications and Networking

solution is derived and from which the justification for un-equal spacings is inferred; a second targets the maximizationof the ergodic capacity and resorts to numerical results.

Addressing the robust MIMO signal processing aspect, thepaper by P. Theofilakos and A. Kanatas considers the use ofsubarrays as a way to improve performance in MIMO com-munication systems. In this approach, each radio frequencychain is coupled to the antenna through a beamforming vec-tor on a subarray of antenna elements. Bounds on capacityfor Rayleigh fading channels highlight the benefits of subar-ray formation while low-complexity algorithms for groupingantennas into subarrays illustrate how to realize this conceptin practical systems.

The paper by I. Cosovic and G. Auer presents an analyti-cal framework for the assessment of the pilot grid for MIMO-OFDM in terms of overhead and power allocation. The op-timum pilot grid is identified based on the criterion of thecapacity for OFDM operating in time-variant frequency se-lective channels. A semianalytical procedure is also proposedto maximize the capacity with respect to the considered esti-mator for realizable and suboptimal estimation schemes.

The paper by A. Del Coso and C. Ibars analyses a point-to-point multiple relay Gaussian channel that uses a decode-and-forward relaying strategy under a half duplex constraint.It derives the instantaneous achievable rate under perfectCSI assumption and obtains the relay selection algorithmand power allocation strategy within the two consecutivetime slots that maximises the achievable rate. Furthermorethe study provides upper and lower bounds on the ergodicachievable rate, and derives the asymptotic behaviour interms of the number of relays, showing that for any randomdistribution of relays the ergodic capacity of the multiple re-lay channel under AWGN grows asymptotically with the log-arithm of the Lambert function of the total number of relays.

In the paper by A. Ikhlef and D. Le Guennec, the problemof signal detection in MIMO systems is addressed focusingon blind reduced complexity schemes. In particular, the au-thors propose the use of blind source separation techniques,which avoid the use of training sequences, for blind recoveryof QAM and PSK signals in MIMO channels. The proposedlow-complexity algorithm is a simplified version of the CMAalgorithm that operates over a single signal dimension, thatis, either on the real or imaginary part and of which conver-gence is also proved in the paper.

In the area of system level optimisation, the paper by C.Sun et al. considers the application of switched directionalbeams at the transmitter and receiver of a MIMO commu-nication link. The beams provide a capacity gain by focus-ing on different dominant wave clusters in the environment;switching between beams gives additional diversity benefits.Electronically steerable parasitic array radiators are suggestedas a means to implement the beamforming in the RF. Per-formance is particularly enhanced at low SNRs compared toa conventional MIMO system that requires an RF chain foreach antenna.

In the paper by N. Jalden et al., the inter- and intrasitecorrelation properties of shadow fading and power-weightedangular spread at both the mobile station and the base sta-tion are studied, for different interbase station distances, uti-

lizing narrow band multisite MIMO measurements in the1800 MHz band.

A. M. Kuzminskiy and H. R. Karimi show in their pa-per the potential increase in throughput when multiantennainterference cancellation techniques are considered to com-plement the multiple access control protocol. The work eval-uates the gains that multiantenna interference cancellationschemes provide in the context of WLAN systems which im-plement the CSMA/CA MAC protocol.

Transmit diversity techniques and the resulting gains atthe cell border in a cellular MC-CDMA environment usingsmart base stations are addressed in the paper by S. Plass etal. Cellular cyclic delay diversity (C-CDD) and cellular Alam-outi technique (CAT) are proposed, that improves the per-formance at the cell borders by enhancing macrodiversityand reducing the overall intercell interference.

The paper by B. Bougard et al. investigates the transceiverenergy efficiency of multiantenna broadband transmissionschemes and evaluates such transceiver power consumptionfor an adaptive system. In particular, the paper evaluates thetradeoff between the net throughput at the MAC layer ver-sus the average power consumption, that an adaptive systemswitching between a space-division multiplexing, space-timecoding or single antenna transmission achieves. Authors pro-vide a model that aims to capture channel state informationin a compact way, and from which a simple policy-basedadaptation scheme can be implemented.

In the last paper, W. Sheng and S. D. Blostein formulatethe problem of admission control for a CDMA beamform-ing system as a cross-layer design problem. In the proposedframework, the parameters of a truncated automatic retrans-mission algorithm and a packet level admission control pol-icy are jointly optimized to maximize throughput subject toquality-of-service requirements. Numerical examples showthat throughput can be increased substantially in the lowpacket error rate regime.

The theme of this special session was inspired by the jointresearch collaboration in the area of smart antennas withinthe ACE project, a Network of Excellence under the FP6 Eu-ropean Commission’s Information Society Technologies Ini-tiative. The objective of this issue is to share some insight andencourage more research on the critical implementation as-pects for the adoption of smart antennas in future wirelesssystems.

We would like to thank the authors, the reviewers andHindawi staff for their efforts in the preparation of this spe-cial issue.

Angeliki AlexiouMonica Navarro

Robert W. Heath Jr.


Research ArticleTower-Top Antenna Array Calibration Scheme forNext Generation Networks

Justine McCormack, Tim Cooper, and Ronan Farrell

Centre for Telecommunications Value-Chain Research, Institute of Microelectronics and Wireless Systems,National University of Ireland, Kildare, Ireland

Received 1 November 2006; Accepted 31 July 2007

Recommended by A. Alexiou

Recently, there has been increased interest in moving the RF electronics in basestations from the bottom of the tower to the top,yielding improved power efficiencies and reductions in infrastructural costs. Tower-top systems have faced resistance in the pastdue to such issues as increased weight, size, and poor potential reliability. However, modern advances in reducing the size andcomplexity of RF subsystems have made the tower-top model more viable. Tower-top relocation, however, faces many significantengineering challenges. Two such challenges are the calibration of the tower-top array and ensuring adequate reliability. We presenta tower-top smart antenna calibration scheme designed for high-reliability tower-top operation. Our calibration scheme is basedupon an array of coupled reference elements which sense the array’s output. We outline the theoretical limits of the accuracyof this calibration, using simple feedback-based calibration algorithms, and present their predicted performance based on initialprototyping of a precision coupler circuit for a 2 × 2 array. As the basis for future study a more sophisticated algorithm for arraycalibration is also presented whose performance improves with array size.

Copyright © 2007 Justine McCormack et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Antennas arrays have been commercially deployed in recentyears in a range of applications such as mobile telephony, inorder to provide directivity of coverage and increase systemcapacity. To achieve this, the gain and phase relationship be-tween the elements of the antenna array must be known. Im-balances in these relationships can arise from thermal effects,antenna mutual coupling, component aging, and finite man-ufacturing tolerance [1]. To overcome these issues, calibra-tion is required [2, 3]. Traditionally, calibration would havebeen undertaken at the manufacturer, address static effectsarising from the manufacturing tolerances. However, imbal-ances due to dynamic effects require continual or dynamiccalibration.

Array calibration of cellular systems has been the subjectof much interest over the last decade (e.g., [4–6]), and al-though many calibration processes already exist, the issue ofarray calibration has, until now, been studied in a “tower-bottom” smart antenna context (e.g., tsunami(II) [2]). In-dustry acceptance of smart antennas has been slow, princi-pally due to their expense, complexity, and stringent relia-

bility requirements. Therefore, alternative technologies havebeen used to increase network performance, such as cell split-ting and tower-bottom hardware upgrades [7, 8].

To address the key impediments to industry acceptanceof complexity and expense, we have been studying the fea-sibility of a self-contained, self-calibrating “tower-top” basetransceiver station (BTS). This system sees the RF and mixedsignal components of the base station relocated next to theantennas. This provides potential capital and operationalsavings from the perspective of the network operator due tothe elimination of the feeder cables and machined duplexerfilter. Furthermore, the self-contained calibration electron-ics simplify the issue of phasing the tower-top array from theperspective of the network provider.

Recent base station architectures have seen some depar-ture from the conventional tower-bottom BTS and tower-top antenna model. First, amongst these was the deploy-ment of tower-top duplexer low-noise amplifiers (TT-LNA),demonstrating a tacit willingness on the part of the net-work operator to relocate equipment to the tower-top ifperformance gains proved adequate and sufficient reliabilitycould be achieved [9]. This willingness can be seen with the


TRx

TRx

TRx

TRx

DA/AD

DA/AD

DA/AD

DA/AD

Ctrl

Tower top

Tower bottom

BasebandBTS

Figure 1: The hardware division between tower top and bottom forthe tower-top BTS.

exploration of novel basestation architectures, with examplessuch as reduced RF feeder structures utilising novel switchingmethodologies [10, 11], and the development of basestationhotelling with remote RF heads [12]. Such approaches aimto reduce capital infrastructure costs, and also site rental oracquisition costs [13].

In this paper, we present our progress toward a reliable,self-contained, low-cost calibration system for a tower-topcellular BTS. The paper initially presents a novel schemefor the calibration of an arbitray-sized rectilinear array us-ing a structure of interlaced reference elements. This is fol-lowed in Section 3 by a theoretical analysis of this schemeand predicted performance. Section 4 presents a descriptionof a prototype implementation with a comparison betweenexperimental and predicted performance. Section 5 presentssome alternative calibration approaches utilising the samephysical structure.

2. RECTILINEAR ARRAY CALIBRATION

2.1. Array calibration

To yield a cost-effective solution for the cellular BTS mar-ket, we have been studying the tower-top transceiver config-uration shown in Figure 1. This configuration has numerousadvantages over the tower-bottom system but, most notably,considerably lower hardware cost than a conventional tower-bottom BTS may be achieved [14].

We define two varieties of array calibration. The first,radiative calibration, employs free space as the calibrationpath between antennas. The second, where calibration is per-formed by means of a wired or transmission line path andany radiation from the array in the process of calibrationis ancillary, is refered to as “nonradiative” calibration. Thesetup of Figure 2 is typically of a nonradiative calibrationprocess [2]. This process is based upon a closed feedbackloop between the radiative elements of the array and a sensor.This sensor provides error information on the array outputand generates an error signal. This error signal is fed back tocorrectively weight the array element’s input (transmit cal-

TRx

TRx

TRx

TRx

DA/AD

DA/AD

DA/AD

DA/AD

Ctrl I/O

Sense

Figure 2: A simplified block schematic diagram of a typical arraycalibration system.

ibration) or output (receive calibration). It is important toobserve that this method of calibration does not correct forerrors induced by antenna mutual coupling. Note that in ourcalibration scheme, a twofold approach will be taken to com-pensate for mutual coupling. The first is to minimise mu-tual coupling by screening neighbouring antennas—and per-haps using electromagnetic (EM) bandgap materials to re-duce surface wave propagation to distant antennas in largearrays. The second is the use of EM modelling-based mitiga-tion such as that demonstrated by Dandekar et al. [6]. Fur-ther discussion of mutual coupling compensation is beyondthe scope of this paper.

While wideband calibration is of increasing interest, it re-mains difficult to implement. On the other hand, narrow-band calibration schemes are more likely to be practicallyimplemented [1]. The calibration approach presented hereis directed towards narrowband calibration. However, themethodology supports wideband calibration through sam-pling at different frequencies.

2.2. Calibration of a 2 × 2 array

Our calibration process employs the same nonradiative cal-ibration principle as shown in Figure 2. The basic build-ing block, however, upon which our calibration system isbased is shown in Figure 3. This features four radiative arraytransceiver elements, each of which is coupled by transmis-sion line to a central, nonradiative reference element.

In the case of transmit calibration (although by reci-procity receive calibration is also possible), the transmit sig-nal is sent as a digital baseband signal to the tower-top andis split (individually addressed) to each transmitter for SISO(MIMO) operation. This functionality is subsumed into thecontrol (Ctrl) unit of Figure 3.

Remaining with our transmit calibration example, thereference element sequentially receives the signals in turnfrom the feed point of each of the radiative array elements.This enables the measurement of their phase and amplituderelative to some reference signal. This information on the

Justine McCormack et al. 3

ZTRx TRx

DA/AD DA/AD

TRx TRx

DA/AD DA/ADCtrl

I/O

Sense

Figure 3: A central, nonradiative reference sensor element coupledto four radiative array transceiver elements.

TRx TRx TRx

Ref Ref

TRx TRx TRx

Figure 4: A pair of reference elements, used to calibrate a 2×3 array.

relative phase and amplitude imbalance between the feedpoints of each of the transceivers is used to create an errorsignal. This error signal is fed back and used to weight the in-put signal to the transceiver element—effecting calibration.Repeating this procedure for the two remaining elements cal-ibrates our simple 2×2 array. This baseband feedback systemis to be implemented in the digital domain, at the tower-top.The functionality of this system and the attendant comput-ing power, energy, and cost requirements of this system arecurrently under investigation.

2.3. Calibration of an n × n array

By repeating this basic 2 × 2 pattern with a central referenceelement, it becomes possible to calibrate larger arrays [15].Figure 4 shows the extension of this basic calibration princi-ple to a 2× 3 array.

X + ΔTx1 ΔC1 ΔC2 X + ΔTx1 + ΔC1− ΔC2

RefΔTx1 Tx Tx ΔTx2

X q[ ] Y

−+ +

Err

Figure 5: Propagation of error between calibrating elements.

To calibrate a large, n × n, antenna array, it is easy to seehow this tessellation of array transceivers and reference ele-ments could be extended arbitrarily to make any rectilineararray geometry.

From the perspective of a conventional array, this has theeffect of interleaving a second array of reference sensor el-ements between the lines of radiative transceiver elements,herein referred to as “interlinear” reference elements, to per-form calibration. Each reference is coupled to four adjacentradiative antenna elements via the six-port transmission linestructure as before. Importantly, because there are referenceelements shared by multiple radiative transceiver elements, asequence must be imposed on the calibration process. Thus,each transceiver must be calibrated relative to those alreadycharacterised.

Cursorily, this increase in hardware at the tower-top dueto our interlinear reference elements has the deleterious ef-fect of increasing the cost, weight, and power inefficiency ofthe radio system. The reference element hardware overhead,however, produces three important benefits in a tower-topsystem: (i) many shared reference elements will enhance thereliability of the calibration scheme—a critical parameter fora tower-top array; (ii) the array design is inherently scalableto large, arbitrary shape, planar array geometries; (iii) as wewill show later in this paper, whilst these reference nodes arefunctional, the multiple calibration paths between them maypotentially be used to improve the calibration accuracy of thearray. For now, however, we consider basic calibration basedon a closed loop feedback mechanism.

3. RECTILINEAR CALIBRATION—THEORYOF OPERATION

3.1. Basic calibration

Figure 5 shows a portion of an n × n array where two ofthe radiative elements of our array are coupled to a centralreference transceiver. As detailed in Section 2.2, the calibra-tion begins by comparing the output of transceiver 1 withtransceiver 2, via the coupled interlinear reference element.Assuming phase only calibration of a SISO system, at a singlefrequency and with perfect impedance matching, each of thearbitrary phase errors incured on the signals, that are sentthrough the calibration system, may be considered additive


constants (Δi, where i is the system element in question).Where there is no variation between the coupled paths andthe accuracy of the phase measurement process is arbitrarilyhigh, then, as can be seen in Figure 5, the calibration processis essentially perfect.

However, due to finite measurement accuracy and coup-ler balance, errors propagate through the calibration scheme.Initial sensitivity analysis [16] showed that when the reso-lution of the measurement accuracy, q[ ], is greater than orequal to 14 bits (such as that attainable using modern DDS,e.g., AD9954 [17] for phase control), the dominant source oferror is the coupler imbalance.

From Figure 5 it is clear that an error, equal in magnitudeto the pair of coupler imbalances that the calibration signalencounters, is passed on to the feed point of each calibratedtransceiver. If this second transceiver is then used in subse-quent calibration operations, this error is passed on. Clearly,this cumulative calibration error is proportional to the num-ber of the calibration couplers in a given calibration path. Forsimple calibration algorithms such as that shown in Figure 5,the array geometry and calibration path limit the accuracywith which the array may be calibrated.

3.2. Theoretical calibration accuracy

3.2.1. Linear array

Figure 6(a) shows the hypothetical calibration path taken inphasing a linear array of antennas. Each square represents aradiative array element. Each number denotes the number ofcoupled calibration paths accrued in the calibration of thatelement, relative to the first element numbered 0 (here thecentremost). If we choose to model the phase and ampli-tude imbalance of the coupler (σck ) as identically distributedGaussian, independent random variables, then the accuracyof calibration for the linear array of N elements relative tothe centre element, σak , will be given by the following:

even N :

σ2ak =

2σc2k

N − 1

N/2∑

i=1

2i, (1)

odd N :

σ2ak =

2σc2k

N − 1

([N/2∑

i=1

2i

]+ 1

), (2)

where the subscript k = A or φ for amplitude or phase error.With this calibration topology, linear arrays are the hardestto accurately phase as they encounter the highest cumulativeerror. This can be mitigated in part (as shown here) by start-ing the calibration at the centre of the array.

3.2.2. Square array

Based on this observation, a superior array geometry forthis calibration scheme is a square. Two example square ar-rays calibration methods are shown in Figures 6(b) and 6(c).The former initiates calibration relative to the top-left hand

· · · 8 6 4 2 0 2 4 6 8 · · ·

(a)

0 2 4 6 8

2 2 4 6 8

4 4 4 6 8

6 6 6 6 8

8 8 8 8 8

· · ·

. . ....

(b)

4 4 4 4 4

4 2 2 2 4

4 2 0 2 4

4 2 2 2 4

4 4 4 4 4

· · · · · ·

...

...

(c)

Figure 6: Calibration paths through (a) the linear array. Also thesquare array starting from (b) the top left and (c) the centre of thearray.

transceiver element. The calibration path then propagatesdown through to the rest of the array taking the shortest pathpossible. Based upon the preceding analysis, the predictedcalibration accuracy due to coupler imbalance of an n × narray is given by

σ2ak =

2σ2ck

N − 1

n∑

i=1

(2i− 1)(i− 1) (3)

with coupler error variance σ2ck , centred around a mean equal

to the value of the first element.Figure 6(c) shows the optimal calibration path for a

square array, starting at the centre and then radiating to theperiphery of the array by the shortest path possible. Theclosed form expressions for predicting the overall calibrationaccuracy of the array relative to element 0 are most conve-niently expressed for the odd and even n, where n2 = N :

even n:

σ2ak =

2σ2ck

N − 1

([n/2−1∑

i=1

(8i)(2i)

])+

2n− 1N − 1

nσ2ck , (4)


11

10

9

8

7

6

5

4

rms

phas

eer

ror

(deg

rees

)

0 20 40 60 80 100

Number of elements, N

Top leftCentre

Figure 7: Comparison of the theoretical phase accuracy predictedby the closed form expressions for the square array calibrationschemes, with σcφ = 3◦.

Tx

Cal

Ref

Cal

Tx

Figure 8: Block schematic diagram of the array calibration simula-tion used to test the accuracy of the theoretical predictions.

odd n:

σ2ak =

2σ2ck

N − 1

n/2−1/2∑

i=1

(8i)(2i). (5)

A graph of the relative performance of each of these twocalibration paths as a function of array size (for square arraysonly) is shown in Figure 7. This shows, as predicted, that thephasing error increases with array size. The effect of this erroraccumulation is reduced when the number of coupler errorsaccrued in that calibration is lower—that is, when the cali-bration path is shorter. Hence, the performance of the centrecalibrated array is superior and does not degrade as severelyas the top-left calibrated array for large array sizes.

As array sizes increase, the calibration path lengths willinherently increase. This will mean that the outer elementswill tend to have a greater error compared to those near thereference element. While this will have impact on the ar-ray performance, for example, in beamforming, it is difficultto quantify. However, in a large array the impact of a smallnumber of elements with relatively large errors is reduced.

Table 1

Component (i) μiA σiA μiφ σiφ

Tx S21 50 dB 3 dB 10◦ 20◦

Ref S21 60 dB 3 dB 85◦ 20◦

Cal S21 −40 dB 0.1 dB 95◦ 3◦

8

7.5

7

6.5

6

5.5

5

4.5

4

rms

phas

eer

ror

(deg

rees

)0 20 40 60 80 100


TheorySimulation

Figure 9: The overall array calibration accuracy predicted by (4)and the calibration simulation for σcφ = 3◦.

3.3. Simulation

3.3.1. Calibration simulation system

To determine the accuracy of our theoretical predictions onarray calibration, a simulation comprising the system shownin Figure 8 was implemented. This simulation was based onthe S-parameters of each block of the system, again assumingperfect impedance matching and infinite measurement reso-lution. Attributed to each block of this schematic was a meanperformance (μik ) and a normally distributed rms error (σik ),which are shown in Table 1.

3.3.2. Results

For each of the square array sizes, the results of 10 000 simu-lations were complied to obtain a statistically significant sam-ple of results. For brevity and clarity, only the phase resultsfor the centre-referenced calibration are shown, althoughcomparable accuracy was also attained for both the ampli-tude output and the “top-left” algorithm. Figure 9 showsthe phase accuracy of the centre-referenced calibration algo-rithm. Here we can see good agreement between theory andsimulation. The reason for the fluctuation in both the theo-retical and simulated values is because of the difference be-tween the even and odd n predictions for the array accuracy.This difference arises because even n arrays do not have acentre element, thus the periphery of the array farthest fromthe nominated centre element incurs slightly higher error.


F E

A B

C D

Figure 10: Schematic representation of the six-port, precision di-rectional coupler.

3.3.3. Practical calibration accuracy

These calibration schemes are only useful if they can calibratethe array to within the limits useful for adaptive beamform-ing. The principle criterion on which this usefulness is basedis on meeting the specifications of 1 dB peak amplitude er-ror and 5◦ rms phase error [16]. The preceding analysis hasshown that, in the absence of measurement error,

limσc→0

σa −→ 0, (6)

where σa is the rms error of the overall array calibration er-ror. Because of this, limiting the dominant source of phaseand amplitude imbalance, that of the array feed-point cou-pler structure, will directly improve the accuracy of the arraycalibration.

4. THE CALIBRATION COUPLER

4.1. 2 × 2 array calibration coupler

The phase and amplitude balance of the six-port couplerstructure at the feed point of every transceiver and refer-ence element in Figure 4 is crucial to the performance of ourcalibration scheme. This six-port coupler structure is shownschematically in Figure 10. In the case of the reference ele-ment, the output (port B) is terminated in a matched load(antenna) and the input connected to the reference elementhardware (port A). Ports C−F of the coupler feed adjacenttransceiver or reference elements. Similarly, for the radiativetransceiver element, port B is connected to the antenna ele-ment and port A the transceiver RF hardware. For the indi-vidual coupler shown in Figure 10 using conventional low-cost, stripline, board fabrication techniques, phase balanceof 0.2 dB and 0.9◦ is possible [18]. By interconnecting five ofthese couplers, then the basic 2 × 2 array plus single refer-ence sensor element building block of our scheme is formed.It is this pair of precision six-port directional couplers whosecombined error will form the individual calibration paths be-tween transceiver and reference element.

A schematic representation of the 2 × 2 array coupler isshown in Figure 11. This forms the feed-point coupler struc-ture of Figure 4, with the central coupler (port 1) connectedto the reference element and the load (port 2). Each periph-eral couplers is connected to a radiative transceiver element

6 6′ 5 5′

Z Y

X1 2

X

Z Y

3 3′ 4 4′

Figure 11: Five precision couplers configured for 2 × 2 array cali-bration.

(ports 3–6). By tiling identical couplers at half integer wave-length spacing, our objective was to produce a coupler net-work with very high phase and amplitude balance.

4.2. Theoretical coupler performance

The simulation results for our coupler design, using ADSmomentum, are shown in Figure 12 [19]. Insertion loss atthe design frequency of 2.46 GHz is predicted as 0.7 dB. Theintertransceiver isolation is high—a minimum of 70.4 dB be-tween transceivers. In the design of the coupler structure, atradeoff exists between insertion loss and transceiver isola-tion. By reducing the coupling factor between the antennafeeder transmission line and the coupled calibration path(marked X on Figure 11), higher efficiency may be attained.However, weaker calibration coupling than −40 dBm is un-desirable from the perspective of calibration reference ele-ment efficiency and measurement reliability. This necessi-tates stronger coupling between the calibration couplers—this stronger coupling in the second coupler stage (markedY or Z on Figure 11) will reduce transceiver isolation. It isfor this reason that −20 dB couplers are employed in all in-stances (X, Y, and Z).

The ADS simulation predicts that the calibration pathwill exhibit a coupling factor of−44.4 dB, slightly higher thandesired.

The phase and amplitude balance predicted by the sim-ulation is shown in Figures 13 and 14. This is lower thanreported for a single coupler. This is because the individ-ual coupler exhibits a natural bias toward high phase balancebetween the symmetrical pairs of coupled lines—ports D,Eand C,F of Figure 10. In placing the couplers as shown inFigure 11, the error in the coupled path sees the sum of an


0

−20

−40

−60

−80

−100

−120

−140

Am

plit

ude

(dB

)

1 1.5 2 2.5 3 3.5 4 4.5 5

Frequency (GHz)

S21S31

S34S36

Figure 12: The theoretically predicted response of the ideal 2 × 2coupler.

0.6

0.5

0.4

0.3

0.2

0.1

0

−0.1

−0.2

−0.3

Ph

ase

imba

lan

ce(d

egre

es)

1 1.5 2 2.5 3 3.5 4

Frequency (GHz)

Error 31–41Error 31–51Error 31–61

Figure 13: The predicted phase imbalance of an ideal 2×2 coupler.

A,D (X ,Z) type error and an A,C (X ,Y) type error. This hasthe overall effect of reducing error. Were there to be a diago-nal bias toward the distribution of error, then the error wouldaccumulate.

Also visible in these results is a greater phase and am-plitude balance between the symmetrically identical couplerpairs. For example, the phase and amplitude imbalance be-tween ports 3 and 6 is very high. This leads to efforts to in-crease symmetry in the design, particularly the grounding viascreens.

4.3. Measured coupler performance

Our design for Figure 11 was manufactured on a low-costFR-4 substrate using a stripline design produced in Eagle

0.06

0.04

0.02

0

−0.02

−0.04

−0.06

−0.08

Am

plit

ude

imba

lan

ce(d

B)

1 1.5 2 2.5 3 3.5 4

Frequency (GHz)


Figure 14: The predicted amplitude imbalance of an ideal 2 × 2coupler.

Figure 15: The PCB layout of the centre stripline controlledimpedance conductor layer.

[20]—see Figure 15. Additional grounding strips, connectedby blind vias to the top and bottom ground layers, are visi-ble which provide isolation between the individual couplers.A photograph of the finished 2× 2 coupler manufactured byECS circuits [21] is shown in Figure 16. Each of the couplerarms is terminated in low-quality surface mount 47Ω resis-tors.

The 2 × 2 coupler was then tested using an R&S ZVB20vector network analyser [22]. The results of this measure-ment with an input power of 0 dBm and 100 kHz of reso-lution bandwidth are shown in Figure 17. The coupler in-sertion loss is marginally higher than the theoretical pre-diction at 1.2 dB. This will affect the noise performanceof the receiver and the transmit efficiency and hence mustbe budgeted for in our tower-top transceiver design. The


Figure 16: A photograph of the transceiver side of the calibrationcoupler board. The opposite side connects to the antenna array andacts as the ground plane.

0

−20

−40

−60

−80

−100

−120

Am

plit

ude

(dB

)

1 1.5 2 2.5 3 3.5 4

Frequency (GHz)

S21S31

S34S36

Figure 17: The measured performance of the prototype 2× 2 cou-pler.

coupled calibration path exhibits the desired coupling fac-tor of −38.8 dB at our design frequency of 2.46 GHz. Thisstronger coupling, together with the finite loss tangent ofour FR4 substrate, explain the increased insertion loss. Themeasured inter-transceiver isolation was measured at a min-imum of−60.9 dB—thus the dominant source of (neighbor-ing) inter-element coupling is likely to be antenna mutualcoupling.

The other important characteristics of the coupler, itsphase and amplitude balance, are shown in Figures 18 and19 respectively. Phase balance is significantly poorer than in-dicated by the theoretical value. The maximum phase errorrecorded at our design frequency of 2.46 GHz for this cou-pler is 0.938◦—almost an order of magnitude worse than thepredicted imbalance shown in Figure 13.

15

10

5

0

−5

−10

−15

−20

Ph

ase

imba

lan

ce(d

egre

es)

1 1.5 2 2.5 3 3.5 4

Frequency (GHz)


Figure 18: The measured phase imbalance of the 2× 2 coupler.

3.5

3

2.5

2

1.5

1

0.5

0

−0.5

Am

plit

ude

imba

lan

ce(d

B)

1 1.5 2 2.5 3 3.5 4

Frequency (GHz)


Figure 19: The measured amplitude imbalance of the 2×2 coupler.

The amplitude balance results, Figure 19, are similarlyinferior to the ADS predictions (contrast with Figure 14).The greatest amplitude imbalance is between S31 and S61of 0.78 dB—compared with 0.18 dB in simulation. However,clearly visible in the amplitude response, and hidden in thephase error response, is the grouping of error characteristicsbetween the paths S31-S41 and S51-S61.

Because the coupler error did not cancel as predicted bythe ADS simulation, but is closer in performance to the seriesconnection of a pair of individual couplers, future simulationof the calibration coupler should include Monte Carlo analy-sis based upon fabrication tolerance to improve the accuracyof phase and amplitude balance predictions.

Clearly a single coupler board cannot be used to charac-terise all couplers. To improve the statistical relevance of our


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

P(A|9)

1 −0.5 0 0.5 1 1.5 2

Amplitude (dB)

2σ

DataPDF

Figure 20: The measured coupler amplitude imbalance fitteda Gaussian probability density function, σA = 0.4131 dB, μA =0.366 dB.

0.25

0.2

0.15

0.1

0.05

0

P(φ|9)

−4 −3 −2 −1 0 1 2 3 4

Phase (degrees)

2σ

DataPDF

Figure 21: The measured coupler phase imbalance fitted to a Gaus-sian probability density function σφ = 1.672◦, μφ = 0.371◦.

results, three 2 × 2 coupler boards were manufactured andthe phase and amplitude balance of each of them recorded atour design frequency of 2.46 GHz. These results are plottedagainst the Gaussian distribution to which the results werefitted for the amplitude and phase (Figures 20 and 21 cor-respondingly). Whilst not formed from a statistically signifi-cant sample (only nine points were available for each distri-bution), these results are perhaps representative of the cali-bration path imbalance in a small array. The mean and stan-dard deviation of the coupler amplitude imbalance distri-bution are μcA = 0.366 dB and σcA = 0.4131 dB. This erroris somewhat higher than predicted by our theoretical study.Work toward improved amplitude balance is ongoing. Thephase balance, with an rms error of 1.672◦, is of the orderanticipated given the performance of the individual coupler.

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

rms

ampl

itu

deer

ror

(dB

)

0 20 40 60 80 100


SimulationTheory

Figure 22: The theoretical prediction of overall array amplitude cal-ibration accuracy based upon the use of the coupler hardware ofSection 4.1.

3

2.8

2.6

2.4

2.2

2

1.8

1.6

1.4

rms

phas

eer

ror

(deg

rees

)

0 20 40 60 80 100


SimulationTheory

Figure 23: The theoretical prediction of overall array phase cali-bration accuracy based upon the use of the coupler hardware ofSection 4.1.

With this additional insight into the statistical distribu-tion of error for a single coupled calibration path, we maymake inferences about the overall array calibration accuracypossible with such a system.

4.4. Predicted array calibration performance

To investigate the utility, or otherwise, of our practical ar-ray calibration system, the coupler statistics derived fromour hardware measurements were fed into both the centre-referenced calibration algorithm simulation and the theoret-ical prediction of Section 3. The results of this simulation areshown in Figures 22 and 23.


TRx TRx TRx

Sense Sense

TRx TRx TRx

Sense Sense

TRx TRx TRx

Figure 24: The redundant coupled calibration paths which may beuseful in enhancing the quality of calibration.

The results from these figures show that the approachyields a highly accurate calibration, with rms phase errors fora typical 16-element array of less than 2◦ and a gain imbal-ance of less than 0.55 dB. As arrays increase in size, the er-rors do increase. For phase calibration, the increase is smalleven for very large arrays. Gain calibration is more sensitiveto size and a 96-element array would have a 0.85 dB rms er-ror. Ongoing work is focused upon improving the gain cali-bration performance for larger arrays. The following sectionis presenting some initial results for alternative calibrationschemes which utilise the additional information from theredundant calibration paths.

5. FUTURE WORK

5.1. Redundant coupler paths

In each of the calibration algorithms discussed thus far, onlya fraction of the available coupled calibration paths is em-ployed. Figure 24 shows the coupled paths which are redun-dant in the “top-left” calibration scheme of Figure 6(b). Thefocus of future work will be to exploit the extra informationwhich can be obtained from these redundant coupler paths.

5.2. Iterative technique

5.2.1. Operation

Given that we cannot measure the array output without in-curring error due to the imbalance of each coupler, we havedevised a heuristic method for enhancing the antenna arraycalibration accuracy. This method is designed to exploit theadditional, unused coupler paths and information about thegeneral distribution and component tolerance of the errorswithin the calibration system, to improve calibration accu-racy. One candidate technique is based loosely on the iter-ative algorithmic processes outlined in [23]. Our method isa heuristic, threshold-based algorithm and attempts to in-fer the actual error in each component of the calibrationsystem—allowing them to be compensated for.

TRx TRx

Ref

TRx TRxf (Tx, Ref, C)

(a)

Ref Ref

Tx

Ref Ref

f (C) f (C)

f (C) f (C)

(b)

Tx Tx Tx Tx Tx Tx

Tx Tx Tx Tx Tx Tx

Tx Tx Tx Tx Tx Tx

Tx Tx Tx Tx Tx Tx

(c)

Figure 25: The two main processes of our heuristic method: (a)reference characterisation and (b) transmitter characterisation. (c)The error dependency spreads from the neighbouring elementswith each iteration of the heuristic process.

Figure 25 illustrates the two main processes of our it-erative heuristic algorithm. The first stage, Figure 25(a), isthe measurement of each of the transmitters by the refer-ence elements connected to them. The output of these mea-surements, for each reference, then have the mean perfor-mance of each neighbouring measured blocks subtracted.This results in four error measurements (per reference ele-ment) that are a function of the proximate coupler, referenceand transmitter errors. Any error measurements which aregreater than one standard deviation from the mean trans-mitter and coupler output are discarded. The remaining er-ror measurements, without the outliers, are averaged and areused to estimate the reference element error.


The second phase, Figure 25(b), repeats the process de-scribed above, this time for each transmitter. Here the func-tionally equivalent step of measuring each transmitter by thefour neighbouring references is performed. Again, the meanperformance of each block in the signal path is calculated andsubtracted. However, during this phase the reference error istreated as a known quantity—using the inferred value fromthe previous measurement. Based on this assumption, the re-sultant error signal is a function of the coupler error and thecommon transceiver element alone.

By extrapolating the transmitter error, using the sameprocess as for the reference element, the coupler errorsmay be calculated and compensated for by weighting thetransceiver input. This process is repeated. In each subse-quent iteration, the dependency of the weighting error sig-nal is dependent upon successive concentric array elementsas illustrated in Figure 25(c).

The iterative process continues for much greater than niterations, until either subsequent corrective weightings arewithin a predefined accuracy, or until a time limit is reached.

Cognisant of the negative effect that the peripheral ele-ments of the array will have on the outcome of this calibra-tion scheme, these results are discarded. For the results pre-sented here, this corresponds to the connection of an addi-tional ring of peripheral reference elements to the array. Fu-ture work will focus on the combining algorithmic and con-ventional calibration techniques to negate the need for thisadditional hardware.

5.2.2. Provisional results

To test the performance of this calibration procedure, theresults are of 1000 simulations of a 10 × 10 array, eachperformed for 100 calibration iterations, was simulated us-ing the system settings of Section 4.4. The centre calibrationscheme gave an overall rms array calibration accuracy (σa) of0.857 dB and 2.91◦. The iterative calibration procedure givesa resultant phase accuracy of 1.32◦ and amplitude accuracyof 0.7148 dB. Figure 26 shows how the amplitude accuracy ofthe iterative calibration varies with each successive iteration.The horizontal line indicates the performance of the centre-referenced calibration. A characteristic of the algorithm is itsperiodic convergence. This trait, shared by simulated anneal-ing algorithms, prevents convergence to (false) local min-ima early in the calibration process. This, unfortunately, alsolimits the ultimate accuracy of the array calibration. For in-stance, the phase accuracy of this array (Figure 27) degradesby 0.1◦ to 1.32◦ from its minimum value, reached on the 37thiteration. Future work will focus on tuning the algorithm’sperformance, perhaps to attenuate this oscillation in later it-erations with a temperature parameter (T) and associated re-duction function f (T). Hybrid algorithms—targeting differ-ent calibration techniques at different sections of the array—are also currently under investigation.

6. CONCLUSION

In this paper, we have presented a new scheme for tower-toparray calibration, using a series of nonradiative, interlinear

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

rms

ampl

itu

deer

ror

(dB

)

0 20 40 60 80 100

Iterations

Figure 26: Resultant array amplitude feed-point calibration accu-racy (σaA) for a single N = 100 array, plotted versus the number ofcalibration iterations.

2.1

2

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

rms

phas

eer

ror

(deg

rees

)

0 20 40 60 80 100

Iterations

Figure 27: Resultant array phasing feed-point calibration accuracy(σaφ ) for a single N = 100 array, plotted versus the number of cali-bration iterations.

reference elements to sense the output of the array. The ac-curacy of this calibration scheme is a function of the arraysize, the calibration path taken in calibrating the array, andthe coupler performance. Where the measurement accuracyis unlimited, then the accuracy of this calibration is depen-dent upon the number of couplers in a given calibration path.

The basic building block of this calibration scheme is the2 × 2 array calibration coupler. We have shown that usinglow-cost fabrication techniques and low-quality FR-4 sub-strate, a broadband coupler network with rms phase balanceof 1.1175◦ and amplitude balance of 0.3295 dB is realisable.

Based upon this coupler hardware, we have shown thatphase calibration accurate enough for cellular smart antennaapplications is possible. Although amplitude accuracy is stilloutside our initial target, work is ongoing on improving the


precision coupler network and on the development of cali-bration algorithms to further reduce this requirement.

Finally, we presented examples of one such algorithm—whose performance, unlike that of the conventional feedbackalgorithms, improves with array size. Moreover, this calibra-tion algorithm, which is based upon exploiting randomnesswithin the array, outperforms conventional calibration forlarge arrays. Future work will focus on use of simulated an-nealing and hybrid calibration algorithms to increase calibra-tion accuracy.

ACKNOWLEDGMENT

The authors would like to thank Science Foundation Irelandfor their generous funding of this project through the Centrefor Telecommunications Value-Chain Research (CTVR).

REFERENCES

[1] N. Tyler, B. Allen, and H. Aghvami, “Adaptive antennas: thecalibration problem,” IEEE Communications Magazine, vol.42, no. 12, pp. 114–122, 2004.

[2] C. M. Simmonds and M. A. Beach, “Downlink calibration re-quirements for the TSUNAMI (II) adaptive antenna testbed,”in Proceedings of the 9th IEEE International Symposium on Per-sonal, Indoor and Mobile Radio Communications (PIMRC ’98),vol. 3, pp. 1260–1264, Boston, Mass, USA, September 1998.

[3] K. Sakaguchi, K. Kuroda, J.-I. Takada, and K. Araki, “Com-prehensive calibration for MIMO system,” in Proceedings ofthe 5th International Symposium on Wireless Personal Multime-dia Communications (WPMC 3’02), vol. 2, pp. 440–443, Hon-olulu, Hawaii, USA, October 2002.

[4] C. M. S. See, “Sensor array calibration in the presence of mu-tual coupling and unknown sensor gains and phases,” Elec-tronics Letters, vol. 30, no. 5, pp. 373–374, 1994.

[5] R. Sorace, “Phased array calibration,” IEEE Transactions onAntennas and Propagation, vol. 49, no. 4, pp. 517–525, 2001.

[6] K. R. Dandekar, L. Hao, and X. Guanghan, “Smart antenna ar-ray calibration procedure including amplitude and phase mis-match and mutual coupling effects,” in Proceedings of the IEEEInternational Conference on Personal Wireless Communications(ICPWC ’00), pp. 293–297, Hyderabad, India, December 2000.

[7] T. Kaiser, “When will smart antennas be ready for the market?Part I,” IEEE Signal Processing Magazine, vol. 22, no. 2, pp. 87–92, 2005.

[8] F. Rayal, “Why have smart antennas not yet gained tractionwith wireless network operators?” IEEE Antennas and Propa-gation Magazine, vol. 47, no. 6, pp. 124–126, 2005.

[9] G. Brown, “3G base station design and wireless network eco-nomics,” Unstrung Insider, vol. 5, no. 10, pp. 1–30, 2006.

[10] J. D. Fredrick, Y. Wang, and T. Itoh, “A smart antenna re-ceiver array using a single RF channel and digital beamform-ing,” IEEE Transactions on Microwave Theory and Techniques,vol. 50, no. 12, pp. 3052–3058, 2002.

[11] S. Ishii, A. Hoshikuki, and R. Kohno, “Space hopping schemeunder short range Rician multipath fading environment,”in Proceedings of the 52nd Vehicular Technology Conference(VTC ’00), vol. 1, pp. 99–104, Boston, Mass, USA, September2000.

[12] A. J. Cooper, “‘Fibre/radio’ for the provision of cord-less/mobile telephony services in the access network,” Electron-ics Letters, vol. 26, no. 24, pp. 2054–2056, 1990.

[13] G. Brown, “Open basestation bonanza,” Unstrung Insider,vol. 4, no. 7, pp. 1–20, 2005.

[14] T. Cooper and R. Farrell, “Value-chain engineering of a tower-top cellular base station system,” in Proceedings of the IEEE65th Vehicular Technology Conference (VTC ’07), pp. 3184–3188, Dublin, Ireland, April 2007.

[15] T. S. Cooper, R. Farrell, and G. Baldwin, “Array Calibration,”Patent Pending S2006/0482.

[16] T. Cooper, J. McCormack, R. Farrell, and G. Baldwin, “To-ward scalable, automated tower-top phased array calibration,”in Proceedings of the IEEE 65th Vehicular Technology Conference(VTC ’07), pp. 362–366, Dublin, Ireland, April 2007.

[17] Analog Devices Datasheet, “400 MSPS 14-Bit DAC 1.8VCMOS Direct Digital Synthesizer,” January 2003.

[18] T. S. Cooper, G. Baldwin, and R. Farrell, “Six-port precisiondirectional coupler,” Electronics Letters, vol. 42, no. 21, pp.1232–1234, 2006.

[19] Agilent EEsof, Palo Alto, Calif, USA. Advanced Design System,Momentum.

[20] CadSoft Computer, 801 South Federal Hwy., Suite 201, DelrayBeach, FL 33483-5185. Eagle.

[21] ECS Circuits, Unit 2, Western Business Park, Oak Close,Dublin 12, Ireland.

[22] Rhode & Schwartz Vertiriebs-GmbH, Muehldorfstrasse 15,81671 Muenchen, Germany.

[23] J. Hromkovic, Algorithmics for Hard Problems, Springer,Berlin, Germany, 2nd edition, 2004.


Research ArticleOptimal Design of Uniform Rectangular Antenna Arraysfor Strong Line-of-Sight MIMO Channels

Frode Bøhagen,1 Pal Orten,2 and Geir Øien3

1 Telenor Research and Innovation, Snarøyveien 30, 1331 Fornebu, Norway2 Department of Informatics, UniK, University of Oslo (UiO) and Thrane & Thrane, 0316 Oslo, Norway3 Department of Electronics and Telecommunications, Norwegian University of Science and Technology (NTNU),7491 Trandheim, Norway

Received 26 October 2006; Accepted 1 August 2007

Recommended by Robert W. Heath

We investigate the optimal design of uniform rectangular arrays (URAs) employed in multiple-input multiple-output communi-cations, where a strong line-of-sight (LOS) component is present. A general geometrical model is introduced to model the LOScomponent, which allows for any orientation of the transmit and receive arrays, and incorporates the uniform linear array as aspecial case of the URA. A spherical wave propagation model is used. Based on this model, we derive the optimal array designequations with respect to mutual information, resulting in orthogonal LOS subchannels. The equations reveal that it is the dis-tance between the antennas projected onto the plane perpendicular to the transmission direction that is of importance with respectto design. Further, we investigate the influence of nonoptimal design, and derive analytical expressions for the singular values ofthe LOS matrix as a function of the quality of the array design. To evaluate a more realistic channel, the LOS channel matrix isemployed in a Ricean channel model. Performance results show that even with some deviation from the optimal design, we getbetter performance than in the case of uncorrelated Rayleigh subchannels.

Copyright © 2007 Frode Bøhagen et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Multiple-input multiple-output (MIMO) technology is apromising tool for enabling spectrally efficient future wire-less applications. A lot of research effort has been put into theMIMO field since the pioneering work of Foschini and Gans[1] and Telatar [2], and the technology is already hitting themarket [3, 4]. Most of the work on wireless MIMO systemsseek to utilize the decorrelation between the subchannels in-troduced by the multipath propagation in the wireless envi-ronment [5]. Introducing a strong line-of-sight (LOS) com-ponent for such systems is positive in the sense that it booststhe signal-to-noise ratio (SNR). However, it will also have anegative impact on MIMO performance as it increases thecorrelation between the subchannels [6].

In [7], the possibility of enhancing performance byproper antenna array design for MIMO channels with astrong LOS component was investigated, and it was shownthat the performance can actually be made superior for pureLOS subchannels compared to fully decorrelated Rayleighsubchannels with equal SNR. The authors of the present

paper have previously studied the optimal design of uni-form linear arrays (ULAs) with respect to mutual informa-tion (MI) [8, 9], and have given a simple equation for theoptimal design. Furthermore, some work on the design ofuniform rectangular arrays (URAs) for MIMO systems is pre-sented in [10], where the optimal design for the special caseof two broadside URAs is found, and the optimal through-put performance was identified to be identical to the optimalHadamard bound. The design is based on taking the spheri-cal nature of the electromagnetic wave propagation into ac-count, which makes it possible to achieve a high rank LOSchannel matrix [11]. Examples of real world measurementsthat support this theoretical work can be found in [12, 13].

In this paper, we extend our work from [8], and use thesame general procedure to investigate URA design. We intro-duce a new general geometrical model that can describe anyorientation of the transmit (Tx), receive (Rx) URAs, and alsoincorporate ULAs as a special case. Again, it should be notedthat a spherical wave propagation model is employed, in con-trast to the more commonly applied approximate plane-wavemodel. This model is used to derive new equations for the


optimal design of the URAs with respect to MI. The resultsare more general than those presented in an earlier work, andthe cases of two ULAs [8] and two broadside URAs [10] canbe identified as two special cases. The proposed principle isbest suited for fixed systems, for example, fixed wireless ac-cess and radio relay systems, because the optimal design isdependent on the Tx-Rx distance and on the orientation ofthe two URAs. Furthermore, we include an analysis of the in-fluence of nonoptimal design, and analytical expressions forthe singular values of the LOS matrix are derived as a func-tion of the quality of the array design. The results are usefulfor system designers both when designing new systems, aswell as when evaluating the performance of existing systems.

The rest of the paper is organized as follows. Section 2describes the system model used. In Section 3, we presentthe geometrical model from which the general results are de-rived. The derivation of the optimal design equations is givenin Section 4, while the eigenvalues of the LOS channel matrixare discussed in Section 5. Performance results are shown inSection 6, while conclusions are drawn in Section 7.

2. SYSTEM MODEL

The wireless MIMO transmission system employs N Tx an-tennas and M Rx antennas when transmitting informationover the channel. Assuming slowly varying and frequency-flatfading channels, we model the MIMO transmission in com-plex baseband as [5]

r = √η ·Hs + n, (1)

where r ∈ CM×1 is the received signal vector, s ∈ CN×1

is the transmitted signal vector, H ∈ CM×N is the normal-ized channel matrix linking the Tx antennas with the Rx an-tennas, η is the common power attenuation over the chan-nel, and n ∈ CM×1 is the additive white Gaussian noise(AWGN) vector. n contains i.i.d. circularly symmetric com-plex Gaussian elements with zero mean and variance σ2

n , thatis, n ∼ CN (0M×1, σ2

n · IM),1 where IM is the M ×M identitymatrix.

As mentioned above, H is the normalized channel ma-trix, which implies that each element in H has unit averagepower; consequently, the average SNR is independent of H.Furthermore, it is assumed that the total transmit power isP, and all the subchannels experience the same path loss asaccounted for in η, resulting in the total average receivedSNR at one Rx antenna being γ = ηP/σ2

n . We apply s ∼

CN (0N×1, (P/N) · IN ), which means that the MI of a MIMOtransmission described by (1) becomes [2]2

I =U∑

p=1

log2

(1 +

γ

Nμp

)bps/Hz, (2)

1 CN (x, Y) denotes a complex symmetric Gaussian distributed randomvector, with mean vector x and covariance matrix Y.

2 Applying equal power Gaussian distributed inputs in the MIMO system iscapacity achieving in the case of a Rayleigh channel, but not necessarily inthe Ricean channel case studied here [14]; consequently, we use the termMI instead of capacity.

where U = min(M,N) and μp is the pth eigenvalue of Wdefined as

W =⎧⎨

⎩

HHH , M ≤ N ,

HHH, M > N ,(3)

where (·)H is the Hermitian transpose operator.3

One way to model the channel matrix is as a sum of twocomponents: a LOS component: and an non-LOS (NLOS)component. The ratio between the power of the two com-ponents gives the Ricean K-factor [15, page 52]. We expressthe normalized channel matrix in terms of K as

H =√

K

1 + K·HLOS +

√1

1 + K·HNLOS, (4)

where HLOS and HNLOS are the channel matrices containingthe LOS and NLOS channel responses, respectively. In thispaper, HNLOS is modeled as an uncorrelated Rayleigh matrix,that is, vec(HNLOS) ∼ CN (0MN×1, IMN ), where vec(·) is thematrix vectorization (stacking the columns on top of eachother). In the next section, the entries of HLOS will be de-scribed in detail, while in the consecutive sections, the con-nection between the URA design and the properties of HLOS

will be addressed. The influence of the stochastic channelcomponent HNLOS on performance is investigated in the re-sults section.

3. THE LOS CHANNEL: GEOMETRICAL MODEL

When investigating HLOS in this section, we only consider thedirect components between the Tx and Rx. The optimal de-sign, to be presented in Section 4, is based on the fact thatthe LOS components from each of the Tx antennas arrive atthe Rx array with a spherical wavefront. Consequently, thecommon approximate plane wave model, where the Tx andRx arrays are assumed to be points in space, is not applicable[11]; thus an important part of the contribution of this paperis to characterize the received LOS components.

The principle used to model HLOS is ray-tracing [7]. Ray-tracing is based on finding the path lengths from each of theTx antennas to each of the Rx antennas, and employing thesepath lengths to find the corresponding received phases. Wewill see later how these path lengths characterize HLOS, andthus its rank and the MI.

To make the derivation in Section 4 more general, we donot distinguish between the Tx and the Rx, but rather theside with the most antennas and the side with the fewest an-tennas (the detailed motivation behind this decision is givenin the first paragraph of Section 4). We introduce the nota-tion V = max(M,N), consequently we refer to the side withV antennas as the Vx, and the side with U antennas as theUx.

We restrict the antenna elements, both at the Ux and atthe Vx, to be placed in plane URAs. Thus the antennas are

3 μp also corresponds to the pth singular value of H squared.

Frode Bøhagen et al. 3

n1

d(1)U

(1, 0) (1, 1) (1, 2)

(0, 0) (0, 1) (0, 2)

d(2)U

n2

Figure 1: An example of a Ux URA with U = 6 antennas (U1 = 2and U2 = 3).

θ

φ

n1

x

y

z

Figure 2: Geometrical illustration of the first principal direction ofthe URA.

placed on lines going in two orthogonal principal directions,forming a lattice structure. The two principal directions arecharacterized with the vectors n1 and n2, while the uniformseparation in each direction is denoted by d(1) and d(2). Thenumbers of antennas at the Ux in the first and second princi-pal directions are denoted by U1 and U2, respectively, and wehave U = U1 · U2. The position of an antenna in the latticeis characterized by its index in the first and second principaldirection, that is, (u1,u2), where u1 ∈ {0, . . . ,U1 − 1} andu2 ∈ {0, . . . ,U2− 1}. As an example, we have illustrated a Uxarray with U1 = 2 and U2 = 3 in Figure 1. The same defini-tions are used at the Vx side for V1, V2, v1, and v2.

The path length between Ux antenna (u1,u2) and Vxantenna (v1, v2) is denoted by l(v1,v2)(u1,u2) (see Figure 4).Since the elements of HLOS are assumed normalized as men-tioned earlier, the only parameters of interest are the receivedphases. The elements of HLOS then become

(HLOS)m,n = e( j2π/λ)l(v1,v2)(u1,u2) , (5)

where (·)m,n denotes the element in row m and column n,and λ is the wavelength. The mapping between m, n, and(v1, v2), (u1,u2) depends on the dimension of the MIMO sys-tem, for example, in the case M > N , we get m = v1 · V2 +v2 + 1 and n = u1 · U2 + u2 + 1. The rest of this sectionis dedicated to finding an expression for the different pathlengths. The procedure employed is based on pure geometri-cal considerations.

α

n2

n1

x′

y′

Figure 3: Geometrical illustration of the second principal directionof the URA.

We start by describing the geometry of a single URA; af-terwards, two such URAs are utilized to describe the com-munication link. We define the local origo to be at the lowercorner of the URA, and the first principal direction as shownin Figure 2, where we have employed spherical coordinatesto describe the direction with the angles θ ∈ [0,π/2] andφ ∈ [0, 2π]. The unit vector for the first principal direc-tion n1, with respect to the Cartesian coordinate system inFigure 2, is given by [16, page 252]

n1 = sin θ cosφ nx + sin θ sinφ ny + cos θ nz, (6)

where nx, ny , and nz denote the unit vectors in their respec-tive directions.

The second principal direction has to be orthogonal tothe first; thus we know that n2 is in the plane, which is or-thogonal to n1. The two axes in this orthogonal plane are re-ferred to as x′ and y′. The plane is illustrated in Figure 3,where n1 is coming perpendicularly out of the plane, and wehave introduced the third angle α to describe the angle be-tween the x′-axis and the second principal direction. To fixthis plane described by the x′- and y′-axis to the Cartesiancoordinate system in Figure 2, we choose the x′-axis to beorthogonal to the z-axis, that is, placing the x′-axis in thexy-plane. The x′ unit vector then becomes

nx′ = 1∣∣n1 × nz

∣∣n1 × nz = sinφnx − cosφny. (7)

Since origo is defined to be at the lower corner of the URA,we require α ∈ [π, 2π]. Further, we get the y′ unit vector

ny′ = 1∣∣n1 × nx′

∣∣n1 × nx′

= cos θ cosφnx + cos θ sinφny − sin θnz.(8)

Note that when θ = 0 and φ = π/2, then nx′ = nx and ny′ =ny . Based on this description, we observe from Figure 3 thatthe second principal direction has the unit vector

n2 = cosαnx′ + sinαny′ . (9)

These unit vectors, n1 and n2, can now be employed todescribe the position of any antenna in the URA. The posi-tion difference, relative to the local origo in Figure 2, between


VxUx

x

z

y

l(v1,v2)(u1,u2)

R

(u1,u2) (v1, v2)

Figure 4: The transmission system investigated.

two neighboring antennas placed in the first principal direc-tion is

k(1) = d(1)n1

= d(1)( sin θ cosφnx + sin θ sinφny + cos θnz),

(10)

where d(1) is the distance between two neighboring antennasin the first principal direction. The corresponding positiondifference in the second principal direction is

k(2) = d(2)n2 = d(2)((cosα sinφ + sinα cos θ cosφ)nx

+ (sinα cos θ sinφ − cosα cosφ)ny

− sinα sin θnz),

(11)

where d(2) is the distance between the antennas in the secondprincipal direction. d(1) and d(2) can of course take differentvalues, both at the Ux and at the Vx; thus we get two pairs ofsuch distances.

We now employ two URAs as just described to model thecommunication link. When defining the reference coordi-nate system for the communication link, we choose the lowercorner of the Ux URA to be the global origo, and the y-axis istaken to be in the direction from the lower corner of the UxURA to the lower corner of the Vx URA. To determine thez- and x-axes, we choose the first principal direction of theUx URA to be in the yz-plane, that is, φU = π/2. The systemis illustrated in Figure 4, where R is the distance between thelower corner of the two URAs. To find the path lengths thatwe are searching for, we define a vector from the global origoto Ux antenna (u1,u2) as

a(u1,u2)U = u1 · k(1)

U + u2 · k(2)U , (12)

and a vector from the global origo to Vx antenna (v1, v2) as

a(v1,v2)V = R · ny + v1 · k(1)

V + v2 · k(2)V . (13)

All geometrical parameters in k(1) and k(2) (θ, φ, α, d(1), d(2))in these two expressions have a subscript U or V to distin-guish between the two sides in the communication link. Wecan now find the distance between Ux antenna (u1,u2) and

Vx antenna (v1, v2) by taking the Euclidean norm of the vec-tor difference:

l(v1,v2)(u1,u2) =∥∥∥a(v1,v2)

V − a(u1,u2)U

∥∥∥ (14)

= (l2x +(R + ly

)2+ l2z

)1/2(15)

≈ R + ly +l2x + l2z

2R. (16)

Here, lx, ly , and lz represent the distances between the twoantennas in these directions when disregarding the distancebetween the URAs R. In the transition from (15) to (16), weperform a Maclaurin series expansion to the first order of thesquare root expression, that is,

√1 + a ≈ 1 + a/2, which is

accurate when a� 1. We also removed the 2 · ly term in thedenominator. Both these approximations are good as long asR� lx, ly , lz.

It is important to note that the geometrical model justdescribed is general, and allows any orientation of the twoURAs used in the communication link. Another interestingobservation is that the geometrical model incorporates thecase of ULAs, for example, by employing U2 = 1, the Ux ar-ray becomes a ULA. This will be exploited in the analysis inthe next section. A last but very important observation is thatwe have taken the spherical nature of the electromagneticwave propagation into account, by applying the actual dis-tance between the Tx and Rx antennas when considering thereceived phase. Consequently, we have not put any restric-tions on the rank of HLOS, that is, rank(HLOS) ∈ {1, 2, . . . ,U}[11].

4. OPTIMAL URA/ULA DESIGN

In this section, we derive equations for the optimalURA/ULA design with respect to MI when transmitting overa pure LOS MIMO channel. From (2), we know that the im-portant channel parameter with respect to MI is the {μp}.Further, in [17, page 295], it is shown that the maximal MIis achieved when the {μp} are all equal. This situation oc-curs when all the vectors h(u1,u2) (i.e., columns (rows) of HLOS

when M > N (M ≤ N)), containing the channel response be-tween one Ux antenna (u1,u2) and all the Vx antennas, thatis,

h(u1,u2)

= [e( j2π/λ)l(0,0)(u1,u2) , e( j2π/λ)l(0,1)(u1,u2) , . . . , e( j2π/λ)l((V1−1),(V2−1))(u1,u2)]T

,(17)

are orthogonal to each other, resulting in μp = V , for p ∈{1, . . . ,U}. Here, (·)T is the vector transpose operator. Thisrequirement is actually the motivation behind the choice todistinguish between Ux and Vx instead of Tx and Rx. By bas-ing the analysis on Ux and Vx, we get one general solution,instead of getting one solution valid for M > N and anotherfor M ≤ N .

When the orthogonality requirement is fulfilled, all the Usubchannels are orthogonal to each other. When doing spa-tial multiplexing on these U orthogonal subchannels, the op-timal detection scheme actually becomes the matched filter,


that is, HHLOS. The matched filter results in no interference

between the subchannels due to the orthogonality, and at thesame time maximizes the SNR on each of the subchannels(maximum ratio combining).

A consequence of the orthogonality requirement is thatthe inner product between any combination of two differentsuch vectors should be equal to zero. This can be expressedas hH

(u1b ,u2b )h(u1a ,u2a ) = 0, where the subscripts a and b are em-

ployed to distinguish between the two different Ux antennas.The orthogonality requirement can then be written as

V1−1∑

v1=0

V2−1∑

v2=0

ej2π/λ(l(v1,v2)(u1a ,u2a )−l(v1,v2)(u1b

,u2b)) = 0. (18)

By factorizing the path length difference in the parenthesesin this expression with respect to v1 and v2, it can be writtenin the equivalent form

V1−1∑

v1=0

e j2π(β11+β12)v1 ·V2−1∑

v2=0

e j2π(β21+β22)v2 = 0, (19)

where βi j = βi j(ujb − uja), and the different βi js are definedas follows:4

β11 = d(1)V d(1)

U V1

λRcos θV cos θU , (20)

β12 = d(1)V d(2)

U V1

λR

[sin θV cosφV cosαU

− cos θV sinαU sin θU],

(21)

β21 = −d(2)V d(1)

U V2

λRsinαV sin θV cos θU , (22)

β22 = d(2)V d(2)

U V2

λR

[cosαU cosαV sinφV

+ cosαU sinαV cos θV cosφV

+ sinαV sinαU sin θV sin θU].

(23)

The orthogonality requirement in (19) can be simplified byemploying the expression for a geometric sum [16, page 192]and the relation sin x = (e jx − e− jx)/2 j [16, page 128] to

sin[π(β11 + β12

)]

sin[(π/V1)

(β11 + β12

)]

︸︷︷︸=ζ1

· sin[π(β21 + β22

)]

sin[(π/V2)

(β21 + β22

)]

︸︷︷︸=ζ2

= 0.

(24)

Orthogonal subchannels, and thus maximum MI, areachieved if (24) is fulfilled for all combinations of (u1a ,u2a)and (u1b ,u2b), except when (u1a ,u2a) = (u1b ,u2b).

4 This can be verified by employing the approximate path length from (16)in (18).

The results above clearly show how achieving orthogo-nal subchannels is dependent on the geometrical parame-ters, that is, the design of the antenna arrays. By investigat-ing (20)–(23) closer, we observe the following inner productrelation:

βi j = Vi

λRk

( j)TU k(i)

V ∀i, j ∈ {1, 2}, (25)

where k(i) = k(i)x nx +k(i)

z nz, that is, the vectors defined in (10)and (11) where the y-term is set equal to zero. Since solving(24) is dependent on applying correct values of βi j , we seefrom (25) that it is the extension of the arrays in the x- andz-direction that are crucial with respect to the design of or-thogonal subchannels. Moreover, the optimal design is inde-pendent of the array extension in the y-direction (directionof transmission). The relation in (25) will be exploited in theanalysis to follow to give an alternative projection view onthe results.

Both ζ1 and ζ2, which are defined in (24), aresin(x)/ sin(x/Vi) expressions. For these to be zero, the sin(x)in the nominator must be zero, while the sin(x/Vi) in the de-nominator is non-zero, which among other things leads torequirements on the dimensions of the URAs/ULAs, as willbe seen in the next subsections. Furthermore, ζ1 and ζ2 areperiodic functions, thus (24) has more than one solution. Wewill focus on the solution corresponding to the smallest ar-rays, both because we see this as the most interesting casefrom an implementation point of view, and because it wouldnot be feasible to investigate all possible solutions of (24).From (20)–(23), we see that the array size increases with in-creasing βi j , therefore, in this paper, we will restrict the anal-ysis to the case where the relevant |βi j| ≤ 1, which are found,by investigating (24), to be the smallest values that producesolutions. In the next four subsections, we will systematicallygo through the possible different combinations of URAs andULAs in the communications link, and give solutions of (24)if possible.

4.1. ULA at Ux and ULA at Vx

We start with the simplest case, that is, both Ux and Vx em-ploying ULAs. This is equivalent to the scenario we studied

in [8]. In this case, we have U2 = 1 giving β12 = β22 = 0, and

V2 = 1 giving ζ2 = 1, therefore, we only need to consider β11.Studying (24), we find that the only solution with our arraysize restriction is |β11| = 1, that is,

d(1)V d(1)

U = λR

V1 cos θV cos θU, (26)

which is identical to the result derived in [8]. The solution isgiven as a product dU dV , and in accordance with [8], we re-fer to this product as the antenna separation product (ASP).When the relation in (26) is achieved, we have the optimaldesign in terms of MI, corresponding to orthogonal LOS sub-channels.


Projection view

Motivated by the observation in (25), we reformulate (26) as

(d(1)V cos θV ) · (d(1)

U cos θU) = λR/V1. Consequently, we ob-serve that the the product of the antenna separations pro-jected along the local z-axis at both sides of the link shouldbe equal to λR/V1. The z-direction is the only direction ofrelevance due to the fact that it is only the array extensionin the xz-plane that is of interest (cf. (25)), and the fact thatthe first (and only) principal direction at the Ux is in the yz-plane (i.e., φU = π/2).

4.2. URA at Ux and ULA at Vx

Since Vx is a ULA, we have V2 = 1 giving ζ2 = 1, thus toget the optimal design, we need ζ1 = 0. It turns out that withthe aforementioned array size restriction (|βi j| ≤ 1), it is notpossible to find a solution to this problem, for example, byemploying |β11| = |β12| = 1, we observe that ζ1 = 0 formost combinations of Ux antennas, except when u1a + u2a =u1b + u2b , which gives ζ1 = V1. By examining this case a bitcloser, we find that the antenna elements in the URA that arecorrelated, that is, giving ζ1 = V1, are the diagonal elementsof the URA. Consequently, the optimal design is not possiblein this case.

Projection view

By employing the projection view, we can reveal the rea-son why the diagonal elements become correlated, and thuswhy a solution is not possible. Actually, it turns out that thediagonal of the URA projected on to the xz-plane is per-pendicular to the ULA projected on to the xz-plane when|β11| = |β12| = 1. This can be verified by applying (25) toshow the following relation:

(k(1)U − k(2)

U

)T︸︷︷︸

diagonal of URA

·k(1)V = 0. (27)

Moreover, the diagonal of the URA can be viewed as a ULA,and when two ULAs are perpendicular aligned in space, theASP goes towards infinity (this can be verified by employingθV → π/2 in (26)). This indicates that it is not possible to dothe optimal design when this perpendicularity is present.

4.3. ULA at Ux and URA at Vx

As mentioned earlier, a ULA at Ux gives U2 = 1, resulting in

(u2b − u2a) = 0, and thus β12 = β22 = 0. Investigating theremaining expression in (24), we see that the optimal designis achieved when |β11| = 1 if V1 ≥ U , giving ζ1 = 0, or|β21| = 1 if V2 ≥ U , giving ζ2 = 0, that is,

d(1)V d(1)

U = λR

V1 cos θV cos θUif V1 ≥ U , or (28)

d(2)V d(1)

U = λR

V2 sin θV∣∣ sinαV

∣∣ cos θU

if V2 ≥ U. (29)

Furthermore, the optimal design is also achieved if both theabove ASP equations are fulfilled simultaneously, and eitherq/V1 /∈ Z or q/V2 /∈ Z, for all q < U . This guarantees eitherζ1 = 0 or ζ2 = 0 for all combinations of u1a and u1b .

Projection view

A similar reformulation as performed in Section 4.1 can bedone for this scenario. We see that both ASP equations, (28)and (29), contain the term cos θU , which projects the antennadistance at the Ux side on the z-axis. The other trigonometricfunctions project the Vx antenna separation on to the z-axis,either based on the first principal direction (28) or based onthe second principal direction (29).

4.4. URA at Ux and URA at Vx

In this last case, when both Ux and Vx are URAs, we haveU1,U2,V1,V2 > 1. By investigating (20)–(24), we observethat in order to be able to solve (24), at least one βi j mustbe zero. This indicates that the optimal design in this caseis only possible for some array orientations, that is, valuesof θ, φ, and α, giving one βi j = 0. To solve (24) when oneβi j = 0, we observe the following requirement on the βi j s:|β11| = |β22| = 1 and V1 ≥ U1, V2 ≥ U2 or |β12| = |β21| = 1and V1 ≥ U2, V2 ≥ U1.

This is best illustrated through an example. For instance,we can look at the case where αV = 0, which results in β21 =0. From (24), we observe that when β21 = 0 and |β22| =1, we always have ζ2 = 0 if V2 ≥ U2, except when (u2b −u2a) = 0. Thus to get orthogonality in this case as well, weneed |β11| = 1 and V1 ≥ U1. Therefore, the optimal designfor this example becomes

d(1)V d(1)

U = λR

V1 cos θV cos θU, V1 ≥ U1, (30)

d(2)V d(2)

U = λR

V2∣∣ cosαU sinφV

∣∣ , V2 ≥ U2. (31)

The special case of two broadside URAs is revealed by fur-ther setting αU = 0, θU = 0, θV = 0, and φV = π/2 in (30)and (31). The optimal ASPs are then given by

d(1)V d(1)

U = λR

V1, V1 ≥ U1; d(2)

V d(2)U = λR

V2, V2 ≥ U2.

(32)

This corresponds exactly to the result given in [10], whichshows the generality of the equations derived in this workand how they contain previous work as special cases.

Projection view

We now look at the example where αV = 0 with a projec-tion view. We observe that in (30), both antenna separationsin the first principal directions are projected along the z-axisat Ux and Vx, and the product of these two distances shouldbe equal to λR/V1. In (31), the antenna separations along thesecond principal direction are projected on the x-axis at Ux


and Vx, and the product should be equal to λR/V2. Theseresults clearly show that it is the extension of the arrays inthe plane perpendicular to the transmission direction thatis crucial. Moreover, the correct extension in the xz-plane isdependent on the wavelength, transmission distance, and di-mension of the Vx.

4.5. Practical considerations

We observe that the optimal design equations from previ-ous subsections are all on the same form, that is, dV dU =λR/ViX , where X is given by the orientation of the arrays. Afirst comment is that utilizing the design equations to achievehigh performance MIMO links is best suited for fixed sys-tems (such as wireless LANs with LOS conditions,5 broad-band wireless access, radio relay systems, etc.) since the op-timal design is dependent on both the orientation and theTx-Rx distance. Another important aspect is the size of thearrays. To keep the array size reasonable,6 the product λRshould not be too large, that is, the scheme is best suited forhigh frequency and/or short range communications. Notethat these properties agree well with systems that have a fairlyhigh probability of having a strong LOS channel componentpresent. The orientation also affects the array size, for exam-ple, if X → 0, the optimal antenna separation goes towardsinfinity. As discussed in the previous sections, it is the arrayextension in the xz-plane that is important with respect toperformance, consequently, placing the arrays in this planeminimizes the size required.

Furthermore, we observe that in most cases, even if onearray is fully specified, the optimal design is still possible. Forinstance, from (30) and (31), we see that if d(1) and d(2) aregiven for one URA, we can still do the optimal design bychoosing appropriate values for d(1) and d(2) for the otherURA. This is an important property for centralized systemsutilizing base stations (BSs), which allows for the optimal de-sign for the different communication links by adapting thesubscriber units’ arrays to the BS array.

5. EIGENVALUES OF W

As in the previous two sections, we focus on the pure LOSchannel matrix in this analysis. From Section 4, we know thatin the case of optimal array design, we get μp = V for all p,that is, all the eigenvalues of W are equal to V . An interest-ing question now is: What happens to the μp s if the designdeviates from the optimal as given in Section 4? In our analy-sis of nonoptimal design, we make use of {βi j}. From above,we know that the optimal design, requiring the smallest an-tenna arrays, was found by setting the relevant |βi j| equal tozero or unity, depending on the transmission scenario. Since{βi j} are functions of the geometrical parameters, studying

5 This is of course not the case for all wireless LANs.6 What is considered as reasonable, of course, depends on the application,

and may, for example, vary for WLAN, broadband wireless access, andradio relay systems.

the deviation from the optimal design is equivalent to study-ing the behavior of {μp}Up=1, when the relevant βi js deviatefrom the optimal ones. First, we give a simplified expressionfor the eigenvalues of W as functions of βi j . Then, we lookat an interesting special case where we give explicit analyticalexpressions for {μp}Up=1 and describe a method for character-izing nonoptimal designs.

We employ the path length found in (16) in the HLOS

model. As in [8], we utilize the fact that the eigenvalues of thepreviously defined Hermitian matrix W are the same as for areal symmetric matrix W defined by W = BHWB, where Bis a unitary matrix.7 For the URA case studied in this paper,it is straightforward to show that the elements of W are (cf.(24))

(W)k,l = sin[π(β11 + β12

)]

sin[(π/V1)

(β11 + β12

)] · sin[π(β21 + β22

)]

sin[(π/V2)

(β21 + β22

)] ,

(33)

where k = u1aU2 +u2a +1 and l = u1bU2 +u2b +1. We can nowfind the eigenvalues {μp}Up=1 of W by solving det(W−IUμ) =0, where det(·) is the matrix determinant operator. Analyt-ical expressions for the eigenvalues can be calculated for allcombinations of ULA and URA communication links by us-ing a similar procedure to that in Section 4. The eigenvalueexpressions become, however, more and more involved forincreasing values of U .

5.1. Example: β11 = β22 = 0 or β12 = β21 = 0

As an example, we look at the special case that occurs whenβ11 = β22 = 0 or β12 = β21 = 0. This is true for some ge-ometrical parameter combinations, when employing URAsboth at Ux and Vx, for example, the case of two broadsideURAs. In this situation, we see that the matrix W from (33)can be written as a Kronecker product of two square matrices[10], that is,

W = W1 ⊗ W2, (34)

where

(Wi)k,l =

sin(πβi j(k − l)

)

sin(π(βi j /Vi)(k − l)

) . (35)

Here, k, l ∈ {1, 2, . . . ,Uj} and the subscript j ∈ {1, 2} is de-

pendent on βi j . If W1 has the eigenvalues {μ(1)p1 }Uj

p1=1, and W2

has the eigenvalues {μ(2)p2 }Uj

p2=1, we know from matrix theory

that the matrix W has the eigenvalues

μp = μ(1)p1 · μ(2)

p2 , ∀p1, p2. (36)

7 This implies that det(W−λI) = 0 ⇒ det(BHWB−λBH IB) = 0 ⇒ det(W−λI) = 0.


Expressions for μ(i)pi were given in [8] for Uj = 2 and Uj = 3.

For example, for Uj = 2 we get the eigenvalues

μ(i)1 = Vi +

sin(βi jπ

)

sin(βi j(π/Vi)

) , μ(i)2 = Vi −

sin(βi jπ

)

sin(βi j(π/Vi)

) .

(37)

In this case, we only have two nonzero βi js, which wefrom now on, denote β1 and β2, that fully characterize theURA design. The optimal design is obtained when both |βi|are equal to unity, while the actual antenna separation is toosmall (large) when |βi| > 1 (|βi| < 1). This will be applied inthe results section to analyze the design.

There can be several reasons for |βi| to deviate fromunity (0 dB). For example, the optimal ASP may be too largefor practical systems so that a compromise is needed, orthe geometrical parameters may be difficult to determinewith sufficient accuracy. A third reason for nonoptimal ar-ray design may be the wavelength dependence. A communi-cation system always occupies a nonzero bandwidth, whilethe antenna distance can only be optimal for one singlefrequency. As an example, consider the 10.5 GHz-licensedband (10.000–10.680 GHz [18]). If we design a system forthe center frequency, the deviation for the lower frequencyyields λlow/λdesign = fdesign/ flow = 10.340/10.000 = 1.034 =0.145 dB. Consequently, this bandwidth dependency onlycontribute to a 0.145 dB deviation in the |βi| in this case, andin Section 6, we will see that this has almost no impact on theperformance of the MIMO system.

6. RESULTS

In this section, we will consider the example of a 4×4 MIMOsystem with URAs both at Ux and Vx, that is, U1 = U2 =V1 = V2 = 2. Further, we set θV = θU = 0, which givesβ12 = β21 = 0; thus we can make use of the results from thelast subsection, where the nonoptimal design is characterizedby β1 and β2.

Analytical expressions for {μp}4p=1 in the pure LOS case

are found by employing (37) in (36). The square roots ofthe eigenvalues (i.e., singular values of HLOS) are plotted asa function of |β1| = |β2| in Figure 5. The lines representthe analytical expressions, while the circles are determinedby using a numerical procedure to find the singular valueswhen the exact path length from (15) is employed in HLOS.The parameters used in the exact path length case are as fol-

lows: φV = π/2, αU = π, αV = π, R = 500 m, d(1)U = 1 m,

d(2)U = 1 m, λ = 0.03 m, while d(1)

V and d(2)V are chosen to get

the correct values of |β1| and |β2|.The figure shows that there is a perfect agreement be-

tween the analytical singular values based on approximatepath lengths from (16), and the singular values found basedon exact path lengths from (15). We see how the singularvalues spread out as the design deviates further and furtherfrom the optimal (decreasing |βi|), and for small |βi|, we getrank(HLOS) = 1, which we refer to as a total design mis-match. In the figure, the solid line in the middle representstwo singular values, as they become identical in the presentcase (|β1| = |β2|). This is easily verified by observing the

−20 −15 −10 −5 0 5 10

|β1| = |β2| (dB)

0

0.5

1

1.5

2

2.5

3

3.5

4

Sin

gula

rva

lues

ofH

LOS,√

μi

u(1)1 u(2)

1

u(1)2 u(2)

2

u(1)1 u(2)

2 and u(1)2 u(2)

1

Numerical

Represents twosingular values

Figure 5: The singular values of HLOS for the 4 × 4 MIMO systemas a function of |β1| = |β2|, both exactly found by a numerical pro-cedure and the analytical from Section 5.

symmetry in the analytical expressions for the eigenvalues.For |βi| > 1, we experience some kind of periodic behavior;this is due to the fact that (24) has more than one solution.However, in this paper, we introduced a size requirementon the arrays, thus we concentrate on the solutions where|β| ≤ 1.

When K �= ∞ in (4), the MI from (2) becomes a ran-dom variable. We characterize the random MI by the MI cu-mulative distribution function (CDF), which is defined as theprobability that the MI falls below a given threshold, that is,F(Ith) = Pr[I < Ith] [5]. All CDF curves plotted in the nextfigures are based on 50 000 channel realizations.

We start by illustrating the combined influence of |βi|and the Ricean K-factor. In Figure 6, we show F(Ith) for theoptimal design case (|β1| = |β2| = 0 dB), and for the totaldesign mismatch (|β1| = |β2| = −30 dB).

The figure shows that the design of the URAs becomesmore and more important as the K-factor increases. This isbecause it increases the influence of HLOS on H (cf. (4)). Wealso observe that the MI increases for the optimal design casewhen the K-factor increases, while the MI decreases for in-creasing K-factors in the total design mismatch case. Thisillustrates the fact that the pure LOS case outperforms theuncorrelated Rayleigh case when we do optimal array design(i.e., orthogonal LOS subchannels).

In Figure 7, we illustrate how F(Ith) changes when wehave different combinations of the two |βi|. We see how theMI decreases when |βi| decreases. In this case, the Ricean K-factor is 5 dB, and from Figure 6, we know that the MI wouldbe even more sensitive to |βi| for larger K-factors. From thefigure, we observe that even with some deviation from the


4 6 8 10 12 14 16

Ith (bps/Hz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(I

th)

|β1| = |β2| = −30 dB|β1| = |β2| = 0 dB

K = 20 dB

K = 10 dB

K = −5 dB

K = −5 dB

K = 10 dB

K = 20 dB

Figure 6: The MI CDF for the 4×4 MIMO system when γ = 10 dB.

6 7 8 9 10 11 12 13 14 15

Ith (bps/Hz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(I

th)

|β1| = |β2| = 0 dB (optimal)|β1| = −3 dB and |β2| = 0 dB|β1| = |β2| = −3 dB

|β1| = |β2| = −20 dBRayleigh (K = −∞ dB)

Figure 7: The MI CDF for the 4×4 MIMO system when γ = 10 dBand K = 5 dB (except for the Rayleigh channel where K = −∞ dB).

optimal design, we get higher MI compared to the case ofuncorrelated Rayleigh subchannels.

7. CONCLUSIONS

Based on the new general geometrical model introduced forthe uniform rectangular array (URA), which also incorpo-rates the uniform linear array (ULA), we have investigatedthe optimal design for line-of-sight (LOS) channels with re-spect to mutual information for all possible combinations ofURA and ULA at transmitter and receiver. The optimal de-

sign based on correct separation between the antennas (dUand dV ) is possible in several interesting cases. Importantparameters with respect to the optimal design are the wave-length, the transmission distance, and the array dimensionsin the plane perpendicular to the transmission direction.

Furthermore, we have characterized and investigated theconsequence of nonoptimal design, and in the general case,we gave simplified expressions for the pure LOS eigenvaluesas a function of the design parameters. In addition, we de-rived explicit analytical expressions for the eigenvalues forsome interesting cases.

ACKNOWLEDGMENTS

This work was funded by Nera with support from the Re-search Council of Norway (NFR), and partly by the BEATSproject financed by the NFR, and the NEWCOM Networkof Excellence. Some of this material was presented at theIEEE Signal Processing Advances in Wireless Communica-tions (SPAWC), Cannes, France, July 2006.

REFERENCES

[1] G. J. Foschini and M. J. Gans, “On limits of wireless commu-nications in a fading environment when using multiple an-tennas,” Wireless Personal Communications, vol. 6, no. 3, pp.311–335, 1998.

[2] E. Telatar, “Capacity of multiantenna Gaussian channels,”Tech. Memo, AT&T Bell Laboratories, Murray Hill, NJ, USA,June 1995.

[3] T. Kaiser, “When will smart antennas be ready for the market?Part I,” IEEE Signal Processing Magazine, vol. 22, no. 2, pp. 87–92, 2005.

[4] T. Kaiser, “When will smart antennas be ready for the mar-ket? Part II—results,” IEEE Signal Processing Magazine, vol. 22,no. 6, pp. 174–176, 2005.

[5] D. Gesbert, M. Shafi, D.-S. Shiu, P. J. Smith, and A. Naguib,“From theory to practice: an overview of MIMO space-timecoded wireless systems,” IEEE Journal on Selected Areas inCommunications, vol. 21, no. 3, pp. 281–302, 2003.

[6] D. Gesbert, “Multipath: curse or blessing? A system perfor-mance analysis of MIMO wireless systems,” in Proceedingsof the International Zurich Seminar on Communications (IZS’04), pp. 14–17, Zurich, Switzerland, February 2004.

[7] P. F. Driessen and G. J. Foschini, “On the capacity formula formultiple input-multiple output wireless channels: a geometricinterpretation,” IEEE Transactions on Communications, vol. 47,no. 2, pp. 173–176, 1999.

[8] F. Bøhagen, P. Orten, and G. E. Øien, “Design of optimalhigh-rank line-of-sight MIMO channels,” IEEE Transactionson Wireless Communications, vol. 6, no. 4, pp. 1420–1425,2007.

[9] F. Bøhagen, P. Orten, and G. E. Øien, “Construction and ca-pacity analysis of high-rank line-of-sight MIMO channels,” inProceedings of the IEEE Wireless Communications and Network-ing Conference (WCNC ’05), vol. 1, pp. 432–437, New Orleans,La, USA, March 2005.

[10] P. Larsson, “Lattice array receiver and sender for spatially or-thonormal MIMO communication,” in Proceedings of the IEEE61st Vehicular Technology Conference (VTC ’05), vol. 1, pp.192–196, Stockholm, Sweden, May 2005.


[11] F. Bøhagen, P. Orten, and G. E. Øien, “On spherical vs. planewave modeling of line-of-sight MIMO channels,” to appear inIEEE Transactions on Communications.

[12] H. Xu, M. J. Gans, N. Amitay, and R. A. Valenzuela, “Exper-imental verification of MTMR system capacity in controlledpropagation environment,” Electronics Letters, vol. 37, no. 15,pp. 936–937, 2001.

[13] J.-S. Jiang and M. A. Ingram, “Spherical-wave model for short-range MIMO,” IEEE Transactions on Communications, vol. 53,no. 9, pp. 1534–1541, 2005.

[14] D. Hosli and A. Lapidoth, “How good is an isotropic Gaussianinput on a MIMO Ricean channel?” in Proceedings IEEE Inter-national Symposium on Information Theory (ISIT ’04), p. 291,Chicago, Ill, USA, June-July 2004.

[15] G. L. Stuber, Principles of Mobile Communication, Kluwer Aca-demic Publishers, Norwell, Mass, USA, 2nd edition, 2001.

[16] L. Rade and B. Westergren, Mathematics Handbook for Scienceand Engineering, Springer, Berlin, Germany, 5th edition, 2004.

[17] D. Tse and P. Viswanath, Fundamentals of Wireless Communi-cation, Cambridge University Press, Cambridge, UK, 1st edi-tion, 2005.

[18] IEEE 802.16-2004, “IEEE standard for local and metropolitanarea networks part 16: air interface for fixed broadband wire-less access systems,” October 2004.


Research ArticleJoint Estimation of Mutual Coupling, Element Factor,and Phase Center in Antenna Arrays

Marc Mowler,1 Bjorn Lindmark,1 Erik G. Larsson,1, 2 and Bjorn Ottersten1

1 ACCESS Linnaeus Center, School of Electrical Engineering, Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden2 Department of Electrical Engineering (ISY), Linkoping University, 58183 Linkoping, Sweden

Received 17 November 2006; Revised 20 June 2007; Accepted 1 August 2007

Recommended by Robert W. Heath Jr.

A novel method is proposed for estimation of the mutual coupling matrix of an antenna array. The method extends previous workby incorporating an unknown phase center and the element factor (antenna radiation pattern) in the model, and treating themas nuisance parameters during the estimation of coupling. To facilitate this, a parametrization of the element factor based on atruncated Fourier series is proposed. The performance of the proposed estimator is illustrated and compared to other methodsusing data from simulations and measurements, respectively. The Cramer-Rao bound (CRB) for the estimation problem is derivedand used to analyze how the required amount of measurement data increases when introducing additional degrees of freedom inthe element factor model. We find that the penalty in SNR is 2.5 dB when introducing a model with two degrees of freedom relativeto having zero degrees of freedom. Finally, the tradeoff between the number of degrees of freedom and the accuracy of the estimateis studied. A linear array is treated in more detail and the analysis provides a specific design tradeoff.

Copyright © 2007 Marc Mowler et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Adaptive antenna arrays in mobile communication systemspromise significantly improved performance [1–3]. How-ever, practical limitations in the antenna arrays, for instance,interelement coupling, are not always considered. The ar-ray is commonly assumed ideal which means that the radi-ation patterns for the individual array elements are modelledas isotropic or omnidirectional with a far-field phase corre-sponding to the geometrical location. Unfortunately, this isnot true in practice which leads to reduced performance asreported by [4–6]. One of the major contributors to the non-ideal behavior is the mutual coupling between the antennaelements of the array [7, 8] and the result is a reduced per-formance [9, 10].

To model the mutual coupling, a matrix representationhas been proposed whose inverse may be used to compen-sate the received data in order to extract the true signal [5].For basic antenna types and array configurations, the cou-pling matrix can be obtained from electromagnetic calcu-lations. Alternatively, calibration measurement data can becollected and the coupling matrix may be extracted from thedata [11–13]. In [14], compensation with a coupling matrixwas found superior to using dummy columns in the case of

a 4-column dual polarized array. One difficulty that ariseswhen estimating the coupling matrix from measurementsis that other parameters such as the element factor and thephase center of the antenna array need to be estimated. Thesehave been reported to influence the coupling matrix estimate[14].

Our work extends previous work [5, 11–14] by treatingthe radiation pattern and the array phase center as unknownsduring the coupling estimation. We obtain a robust and ver-satile joint method for estimation of the coupling matrix, theelement factor, and the phase center. The proposed methoddoes not require the user to provide any a priori knowledgeof the location of the array center or about the individual an-tenna elements.

The performance of the proposed joint estimator is com-pared via simulations of an 8-element antenna array to a pre-vious method developed by the authors [13]. In addition, weillustrate the estimation performance based on actual mea-surements on an 8-element antenna array. Furthermore, aCRB analysis is presented which can be used as a perfor-mance benchmark. Finally, a tradeoff between the complex-ity of the model and the performance of the estimator is ex-amined for an 8-element antenna array with element factorEtrue = cos(2θ).


θδx , δy

(a)

θ

(b)

Figure 1: Schematic drawing of the antenna array with (a) and without (b) the displacement of the phase center relative to the origin of thecoordinate system. Rotating the antenna array (solid) an angle θ gives a relative change (dashed) that depends on the distance to the origin(δx, δy).

2. DATA MODEL AND PROBLEM FORMULATION

Consider a uniform linear antenna array with M elementshaving an interelement spacing of d. A narrowband signal,s(t), is emitted by a point source with direction-of-arrival θrelative to the broadside of the receiving array. The data col-lected by the array with true array response a(θ) is [5, 14]

x = a(θ)s(t) + w, x ∈ CM×1. (1)

We model a(θ) as follows:

a(θ) � Ca(θ)e jk(δxcos θ+δysin θ) f (θ), a ∈ CM×1, (2)

where the following conditions hold.

(i) a(θ) is the true radiation pattern in the far-field includ-ing mutual coupling, edge effects, and mechanical er-rors. a(θ) is modelled as suggested by (2).

(ii) C is the M × M coupling matrix. This matrix iscomplex-valued and unstructured. The main focus ofthis paper is to estimate C.

(iii) {δx, δy} is the array phase center. Figure 1 illustrateshow the displacement of the antenna array is modelledby δxx + δy y from the origin. The proposed approachalso estimates {δx, δy}. Note that {δx, δy} is not in-cluded in the models used in [5, 14].

(iv) f (θ) ≥ 0 is the element factor (antenna radiationpattern describing the real-valued amplitude of theelectric field pattern) in direction θ without mutualcoupling and describes the amount of radiation fromthe individual antenna elements of the array in differ-ent directions θ. f (θ) is real-valued and includes nodirection-dependent phase shift [7]. The proposed es-timator also estimates f (θ). Note that f (θ) was not in-cluded in the models used in [5, 14]. All antenna ele-ments are implicitly assumed to have equal radiationpatterns when not in an array configuration, that is,f (θ) is the same for allM elements. When the elementsare placed in an array, the radiation patterns of eachelement are not the same due to the mutual coupling.However, f (θ) is allowed to be nonisotropic, which isalso the case in practice.

(v) a(θ) is a Vandermonde vector whose mth element is

am(θ) = e jkd sinθ(m−M/2−1/2). (3)

(vi) d is the distance between the antenna elements.(vii) k is the wave number.

(viii) w is i.i.d complex Gaussian noise with zero-mean andvariance σ2 per element.

Based on calibration data, when a signal, s(t) = 1, is trans-mitted from N known direction-of-arrivals1 {θ1, . . . , θN}, wecollect one data vector xn of measurements for each an-gle θn. The measurements are arranged in a matrix, X =[x1 · · · xN ], as follows:

X = CAD(

δx, δy)

E(

f (θ))

+ W,

A = [a(θ1) · · · a

(

θN)]

,

E(

f (θ)) = diag

{

f(

θ1)

, . . . , f(

θN)}

,

D(

δx, δy)=diag

{

e jk(δx cos θ1+δy sin θ1), . . . , e jk(δx cos θN+δy sin θN )}.(4)

The matrix W represents measurement noise, which is as-sumed to be i.i.d zero-mean complex Gaussian with varianceσ2 per element. In this paper, we propose a method to esti-mate C when δx, δy , and f (θ) are unknown.

One of the novel aspects of the proposed estimator is theinclusion of the element factor, f (θ), as a jointly unknownparameter during the estimation of the coupling matrix, C.This requires a parametrization that provides a flexible andmathematically appealing representation of an a priori un-known element factor. We have chosen to model f (θ) as alinear combination of sinusoidal basis functions accordingto

f (θ) =K∑

k=1

αkcos(k − 1)θ, |θ| < π

2, (5)

1 The orientation of the array is assumed to be perfectly known, while theexact position of the phase center is typically unknown during the an-tenna calibration.

Marc Mowler et al. 3

where K is a known (small) integer, and α are unknown andreal-valued constants. Equation (5) is effectively equivalentto a truncated Fourier series where the coefficients are to beestimated. Even though the chosen parametrization may as-sume negative values, it is introduced to allow the elementfactor, E, to assume arbitrary shapes that can match the truepattern of the measured antenna array. This can increase theaccuracy in the estimate of C compared to only assumingan omnidirectional element factor that would correspond toE = I .

Other alternative parameterizations of f (θ) exist as well.A piecewise constant function of θ is one example. The pro-posed parametrization, on the other hand, is particularly at-tractive since the basis functions are orthogonal and at thesame time smooth. Additionally, the unknown coefficients,αk, enter the model linearly. Based on (5), the element factorcan be expressed as

E =K∑

k=1

αkQk,

α = [α1 · · · αK]T

,

Qk = diag{

cos(k − 1)θ1, . . . , cos(k − 1)θN}

.

(6)

The coefficients, α, will be jointly estimated together with thecoupling matrix and phase center by the estimator proposedin this paper.

3. THE PROPOSED ESTIMATOR

We propose to estimate C, α, δx, and δy from X by using aleast-squares criterion2 on the data model expressed in (4)according to

minC,α,δx ,δy

∥

∥X− CAD(

δx, δy)

E(α)∥

∥

2F . (7)

Under the assumption of Gaussian noise, (7) is themaximum-likelihood estimator. The values of C,α, δx, δy

that minimize the Frobenius norm in (7) are found using aniterative approach. The coupling matrix, the matrix C is firstexpressed as if the other parameters were known followed bya minimization over the α parameters while keeping C and{δx, δy} fixed. A second minimization is made over {δx, δy}with C and α treated as constants after which the algorithmloops back to minimize over α again. The steps of the estima-tor are as follows:

(1) minimize (7) with respect to the coupling matrix, C,while the phase center, {δx, δy}, and the element fac-tor representation, α, are fixed. This is done using thepseudoinverse approach expressed by [4]

C = X(ADE)H[

ADE(ADE)H]−1

; (8)

2 The minimum of this cost function is not unique, see Section 4 for a dis-cussion of this.

(2) minimize (7) with respect to α while C, δx, and δy arefixed. In this step, the value of C found in the previ-ous step is used as the assumed constant value for thecoupling matrix; the minimization over α is then per-formed using the following manipulations: first, re-arrange the measurement matrix, X, and the expres-sion (4) in vectorized form as

x =[

vec{

Re{X}}vec{

Im{X}}]

,

Y =[

vec{

Re{

CADQ1

}} · · · vec{

Re{

CADQK

}}

vec{

Im{

CADQ1

}} · · · vec{

Im{

CADQK

}}

]

;

(9)

the least-squares criterion used is then expressed as

minα

∥

∥x − Yα∥

∥

2F (10)

with the solution

α = (YTY)−1

YTx, (11)

which gives the minimizing α parameters;(3) minimize (7) with respect to δx and δy while keeping

C and α fixed. Using the α parameters found in theprevious step, C, is expressed again according to (8).Assuming the other parameters to be constant, a two-dimensional gradient search is conducted to find theminimizing {δx, δy} of (7). Steps 1–3 are iterated until

‖X− CA DE‖2F is within a certain tolerance level.

To provide the algorithm with an initial estimate, in the firstiteration, we take D = I and α = [0, 1, 0, . . . , 0]T , which cor-responds to the element factor f (θ) = cosθ that was used in[13]. The initialization of the algorithm could of course bedone in many different ways and this may affect its perfor-mance. The choice considered here can be seen as a refine-ment of the algorithm previously proposed in [13].

Typically, about 5 iterations of the algorithm are requiredto reach a local optimum depending on the given tolerancelevel. Convergence to the global optimum can not be assured;however, the algorithm will converge since the cost function(7) is nonincreasing in each iteration. The topic has been ad-dressed in previous conference papers [13].

4. CRAMER-RAO BOUND ANALYSIS

The parameter K , corresponding to the number of termsused in the truncated Fourier series representation of the ele-ment factor, is key to the proposed estimator. This value willaffect the accuracy of the estimator. Increasing the value willgive a better match between the true element factor and theassumed model. At the same time, it will increase the com-plexity of the model and complicate the estimation. Anotheraspect is the fact that increasing the value K towards infin-ity may not be the best way of tuning the algorithm if thetrue value is much less. This would force the estimator to usea model far more complicated than needed, leading to esti-mation of additional parameters. Even though the parame-ters would be close to zero, the performance of the estima-tor is still affected as indicated by the CRB analysis in this


section. On the other hand, a smaller number of parametersmay give insufficient flexibility to the algorithm and force itinto a nonoptimal result.

To compensate for the affected performance associatedwith a large value of K , the number of measurements N of xncan be used as well as a higher signal-to-noise ratio (SNR).Let us now study the tradeoff between these three parame-ters by quantifying how the choice of K relates to N and theSNR (1/σ2). The Cramer-Rao bound (CRB) for the estima-tion problem in (7) is derived and will provide a lower boundon the variance of the unknown parameters [15–17]. All theelements of C are considered to be unknown complex-valuedparameters and therefore separated with respect to real andimaginary parts withCR

i j denoting the real part of the element

in the ith row and jth column, and CIi j denoting the corre-

sponding imaginary part. All the 2M2 + 2 + K real-valuedunknown parameters of (4) are collected into the vector

ξ =[

CR11 · · · CR

1M · · · CRMM · · · CI

i j δx δy α1 · · · αK]T

.

(12)

The CRB for the estimate of ξ is expressed by [15]

[CRB]= [IFisher]−1

, (13)

where IFisher is the Fisher information matrix given by

[

IFisher]

i j =2σ2

Re{

∂μH

∂ξi

∂μ

∂ξ j

}

, (14)

where μ is the expected value of x in (9). The problem (7)is unidentifiable as presented. The scaling ambiguity presentbetween C and α is solved by enforcing a constraint on theproblem. Here, we choose to constrain ‖C‖2

F = M as op-posed to other possibilities such as α1 = 1 or CR

11 = 1. Thelatter favors particular elements of C or α by forcing theseto be nonzero. This is undesirable and consequently ruledout. This constraint was also implemented as a normaliza-tion in the estimator presented in the previous section, whilenot mentioned there explicitly.

The CRB under parametric constraints is found by fol-lowing the results derived in [18]. The constraint is expressedas

g(ξ) = ‖C‖2F −M = Tr

{

CHC}−M = 0. (15)

Defining

[

G(ξ)]

l =∂g(ξ)∂ξl

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

2Re{

ci j}

if ξl = Re{

ci j}

,

2Im{

ci j}

if ξl = Im{

ci j}

,

0 otherwise,

(16)

the constrained CRB is given by [18]

[CRB] = U(UTIFisher U)−1 UT , (17)

where U is implicitly defined via G(ξ)U = 0. Note that theCRB is proportional to σ2, that is, the estimation accuracy isinversely proportional to the SNR.

θ

d hgp

Figure 2: Schematic overview of the antenna array consisting ofdipoles over an infinite ground plane. The distance from the groundplane is hgp and the spacing between the elements is d. The anglefrom the normal to the antenna axis is denoted by θ.

5. RESULTS

5.1. Dipole array example

First, let us consider the case of an 8-element dipole arrayover a ground plane, see Figure 2. In the absence of mutualcoupling, each element has the true radiation pattern

Etrue = sin(

khgpcos θ)

, (18)

where k is the wave number and hgp is the distance to theground plane. Using known expressions for mutual couplingbetween dipoles [7], it is straightforward to calculate the em-bedded element patterns. Figures 3(a) and 3(b) show the ra-diation pattern for a single element and an 8-element arraywith mutual coupling for hgp = 3λ/8. The interelement spac-ing is d = λ/2 and the true {δx, δy} are {0, 0}.

We now compare our proposed method using a trun-cated Fourier series expansion of the element factor, see (5),to results obtained by using a cosine-shaped (E = cosnθ)element pattern assumption as in [13]. In our proposedmethod, the algorithm described in Section 3 is used to es-timate C, {α1,α2,α3}, and {δx, δy}, which corresponds tochoosing K = 3 in the model for the element factor as ex-pressed in (5). Similarly, the parameter n in the expressionE = cosnθ is optimized as part of the procedure when us-ing the method of [13]. In both cases, we make a starting as-sumption that {δx, δy} = {0, 0}, which is also the true valuefor this case.

With no compensation, the uncompensated radiationpattern is represented in Figure 3(b). With compensation,where a cosine-shaped element pattern (E = cosnθ) isassumed, the resulting radiation pattern is displayed inFigure 3(c). The unknown values in addition to the mutual

coupling matrix were estimated to n = 0.5 and {δx, δy} ={0, 0}, and the performance is significantly improved overthe uncompensated case. By allowing the model for the el-ement factor to follow a truncated Fourier series (K = 3)according to our method, the performance of the mutualcoupling matrix estimate is improved even more resultingin the graph of Figure 3(d). For this case, our method esti-mated the values for the auxiliary unknown parameters to be

α = [−1.1 3.1 − 1.1]T and {δx, δy} = {0, 0}. The result alsoagrees well with the true (ideal) radiation pattern when themutual coupling is neglected as presented in Figure 3(a).

The phase error for the cases of uncompensated andcompensated phase diagrams are presented in Figure 4. Thetop graph represents the uncompensated case, while the


−20

−15

−10

−5

0

5

Rad

iati

onpa

tter

ns

(dB

)

−90 −60 −30 0 30 60 90

θ (degrees)

(a) True radiation patterns for a single element; f (θ)

−20

−15

−10

−5

0

5

Rad

iati

onpa

tter

ns

(dB

)

−90 −60 −30 0 30 60 90

θ (degrees)

(b) Uncompensated radiation patterns with mutual coupling;x1 · · · xM

−20

−15

−10

−5

0

5

Rad

iati

onpa

tter

ns

(dB

)

−90 −60 −30 0 30 60 90

θ (degrees)

(c) Radiation patterns obtained from compensated data using a Cmatrix estimated via E = cosnθ as the parametrization of the ele-ment factor [13]; C−1

cos x1 · · · C−1cosxM

−20

−15

−10

−5

0

5

Rad

iati

onpa

tter

ns

(dB

)

−90 −60 −30 0 30 60 90

θ (degrees)

(d) Compensated radiation patterns using our method with K = 3;C−1x1 · · · C−1xM

Figure 3: Compensation for mutual coupling for an array of 8 dipoles placed hgp = 3λ/8 over a ground plane.

middle and bottom graphs represent compensated cases witha cosine-shaped and the truncated Fourier series parameter-izations of the element factors, respectively. This also verifiesthat our method with K = 3 improves the result over con-straining the element factor to be cosine-shaped (E = cosnθ)as was proposed in previous work [13].

5.2. CRB for dipole array example

The CRB as a function of K (i.e., the order of theparametrization of f (θ)) for an 8-element array of dipolesover ground plane is presented in Figure 5. When generatingthis figure, we used the theoretical models of [7] for the cou-pling between antenna elements with inter-element spacingd = λ/2. The true data used are based on a scenario whenK = 1, D = I, and E = I. The SNR was 0 dB (σ2 = 1). Thefour curves show

(i) CRBC = Tr{{UC(UTCICUC)

−1UT

C}C}, the CRB of Cwhen both D and E are known;

(ii) CRBCD = Tr{{UCD(UTCDICDUCD)

−1UT

CD}C}, the CRBof C when D is unknown and E is known;

(iii) CRBCE = Tr{{UCE(UTCEICEUCE)

−1UT

CE}C}, the CRB ofC when D is known and E is unknown;

(iv) CRBCDE = Tr{{UCDE(UTCDEICDEUCDE)

−1UT

CDE}C}, theCRB of C when D and E are unknown.

The results in Figure 5 quantify the increase in achievable es-timation performance for the elements of C when increasingthe number of nuisance parameters in the model. In partic-ular, we see that the estimation problem becomes more dif-ficult when more α parameters are introduced in the model.For example, it is more difficult to estimate the coupling ma-trix C when the phase center δx, δy , is unknown. However,

6 EURASIP Journal on Wireless Communications and NetworkingX

−50 −25 0 25 50

θ (deg)

−10

0

10

(a)

C−1 co

snX

−50 −25 0 25 50

θ (deg)

−10

0

10

(b)

C−1

X

−50 −25 0 25 50

θ (deg)

−10

0

10

(c)

Figure 4: Uncompensated (a) and compensated (b, c) phase dia-grams with E = cosnθ and E = ∑

αkQk , respectively. The trueelement factor is Etrue = sin(2π(3/8)cosθ), which corresponds tohgp = 3λ/8.

for K ≥ 3, the difficulty of identifying α dominates overthe problem of estimating the phase center. Since the CRB isproportional to σ2, the increase in emitter power (i.e., SNR)required to maintain a given performance when the modelis increased with more unknowns can be seen in the fig-ure.

In Figure 6, the CRB as a function of K and N is stud-ied. From this figure, we can directly read out how muchhigher emitter power (or equivalently, lower σ2) is requiredto be able to maintain the same estimation performance forC when K or N vary. For example, if fixing N = 100, say,then going from K = 1 to K = 2 requires 1 dB additionalSNR. Going from K = 2 to K = 3 requires an increase ofthe SNR level by 1.5 dB. However, going from K = 3 to 4requires 10 dB extra SNR. Thus, K = 3 appears to be a rea-sonable choice. In practice, it is difficult to handle more thanK = 3 for the element factor in this case.

In Figure 6, we observe that in the limit when the numberof angles reaches N = 15, the problem becomes unidentifi-able. This is so because for N ≤ 15 the number of unknownparameters in the model exceeds the number of recordedsamples. From Figure 6 many other interesting observationscan be made. For instance, increasing N from 100 to 200, fora fixed K , is approximately equivalent to increasing the SNRwith 3 dB. This holds in general: for N � 1, doubling theSNR gives the same effect as doubling N .

−26

−24

−22

−20

−18

−16

−14

−12

1 2 3 4

K

CR

B(d

B)

CRBCDE

CRBCE

CRBCD

CRBC

Figure 5: The CRB for the elements of C under different assump-tions on whether D, E are known or not, and for different K . SNRis 0 dB.

−25

−20

−15

−10

15 50 100 150 200N

CR

B(d

B)

K = 1K = 2

K = 3K = 4

Figure 6: The CRB for the elements of C as a function of K and Nwhen SNR is 0 dB.

5.3. Measured results on a dual polarized array

Next, let us study the performance of the proposed estima-tor on an actual antenna array. Data from an 8-column an-tenna array (see Figure 7) were collected at 1900 MHz dur-ing calibration measurements with 180 measurement pointsdistributed evenly over the interval θ ∈ {−90◦ · · · 90◦}.Uncompensated radiation patterns with mutual coupling arepresented in Figure 8 for measured data of the array. The es-timator presented in this paper was used to estimate the cou-pling matrix. The estimated coupling matrix was then used


Figure 7: Eight-column dual polarized array developed by Pow-erwave Technologies Inc. Results are presented for this array inSection 5.3. Note that each vertical column of radiators forms anelement with pattern am(θ) in the horizontal plane.

to precompensate the data, after which radiation patterns canbe obtained.

To modify our estimator (Section 3) to the dual polarizedcase, we follow [14]. Considering a dual polarized array with±45◦ polarization and neglecting noise, (4) becomes

[

x−45◦co x−45◦

xp

x+45◦xp x+45◦

co

]

=[

C11 ADE1 C12ADE2

C21 ADE2 C22 ADE1

]

, (19)

where x−45◦co means measuring the incoming −45◦ polarized

signal with the antenna elements of the same polarization,while x−45◦

xp means the data measured at the −45◦ elementwhen the incoming signal is +45◦. To estimate the total cou-pling matrix,

Ctot =[

C11 C12

C21 C22

]

, (20)

the four blocks of (19) are treated independently accordingto

x−45◦co = C11ADE1, x−45◦

xp = C12ADE2,x+45◦xp = C21ADE2, x+45◦

co = C22ADE1.(21)

The D matrices of these four independent equations areequal. The E1 matrix represents the copolarization blocks of(19), namely, x−45◦

co and x+45◦co , simultaneously and is mod-

elled according to (5) with a set of α parameters estimated byour method. For the cross-polarization, we assume isotropicelement patterns, E2 = I. Once the equations in (21) aresolved, the total coupling matrix can be expressed using (20)and the radiation pattern of the measured data may be com-pensated by inverting the coupling matrix according to

⎡

⎣

x−45◦compensated

x+45◦compensated

⎤

⎦ = C−1tot

⎡

⎣

x−45◦measured

x+45◦measured

⎤

⎦ . (22)

Figure 8(a) shows the individual radiation patterns ofeach antenna element as measured during calibration (x).Figure 8(b) shows the radiation patterns after compensationby the coupling matrix (C−1

isoX) when isotropic conditions are

assumed by the estimator. This means assuming E = I, whichis equivalent to setting K = 1 in our algorithm. The radiationpatterns after compensation by the coupling matrix whenusing the proposed estimator with K = 3 are presented inFigure 8(c).

The results using an isotropic assumption on the elementfactor, Figure 8(b), shows an improvement over the uncom-pensated data of Figure 8(a). The graphs showing the copo-larization (solid) are smoother and more equal to each other,which is what is expected from an array with equal elementswhen no coupling is present. The cross-polarization (dashed)is suppressed significantly compared to the uncompensated

case. The estimated phase center is {δx, δy} = [0.2 0].Further improvement is achieved using our proposed

method, Figure 8(c). Using K = 3, our method estimates thecoefficients in the element factor representation (5) as α =[0.8 − 0.4 0.6] and the phase center as {δx, δy} = [0.3 0].The resulting compensated radiation pattern of Figure 8(c)is even closer to the ideal array response when no couplingis assumed. The copolarization graphs are almost overlap-ping in the ±60◦ interval showing the radiation patterns of8 equal elements with cosine-like element factors. The cross-polarization is also improved slightly over the isotropic case.

Phase diagrams representing the average phase error ofthe coupling matrix before and after compensation with thecoupling matrix are presented in Figure 8(d). The phase er-ror after compensation with K = 3 (Figure 8(d), bottom),modelling the phase shift and the element factor, is less thanwithout the compensation (Figure 8(d), top). Assuming anisotropic element factor (Figure 8(d), middle) gives a betterresult than without compensation but not as good as the re-sult of our method. This indicates that the validity of the es-timated coupling matrix based on phase considerations in-creases with the proposed estimator. Furthermore, the resultsof our method with K = 3 show a notable improvement overthe results presented in [14].

6. K-VALUE TRADEOFF BASED ONMONTE CARLO SIMULATIONS

We have seen in Section 5.2 that there is a tradeoff betweenK , N , and the SNR in terms of the CRB. In reality, the er-ror in the estimation is a combination of both model- andnoise-induced errors. Let us therefore study the overall per-formance of our proposed estimator using the root-mean-square error of the mutual coupling matrix:

RMS = 1M2

‖C− C‖F = 1M2

√

√

√

∑

i j

∣

∣Cij − Cij

∣

∣

2, (23)

where M2 is the number of elements in C. As an example,we use an 8-element linear array with spacing d = λ/2 anda true element factor given by αtrue = [0 0 1 0 . . . ]T . Simu-lations were conducted with our method for K = 1 . . . 5based on 1000 realizations and an SNR = 30 dB. The resultis shown in Figure 9. The CRB for the given SNR level is also


−25

−20

−15

−10

−5

0

Rad

iati

onpa

tter

ns

(dB

)

−50 0 50

θ (degrees)

(a) Measured uncompensated radiation patterns for the array of

Figure 7; {x−45◦measured, x+45◦

measured}T

−25

−20

−15

−10

−5

0

Rad

iati

onpa

tter

ns

(dB

)

−50 0 50

θ (degrees)

(b) Radiation patterns obtained from compensated data using a Cmatrix estimated via our algorithm setting K = 1 (i.e., forcing E ∝I); C−1

iso{x−45◦measured, x+45◦

measured}T

−25

−20

−15

−10

−5

0

Rad

iati

onpa

tter

ns

(dB

)

−50 0 50

θ (degrees)

(c) Compensated radiation patterns using our method with K = 3;

C−1{x−45◦measured, x+45◦

measured}T

−50 −25 0 25 50

θ (degrees)

−10

0

10

−10

0

10

−10

0

10

c)

b)

a)

(Deg

)

(d) Phase errors for the cases in (a), (b), and (c)

Figure 8: Radiation patterns obtained from compensated data with the coupling matrix C estimated in different ways. The measurementsare from the 8-element dual polarized array in Figure 7 with co- (solid) and cross- (dashed) polarization collected during calibration.

presented in the same graph and represents the impact of thenoise. This is seen as an increase of the CRB in the regionK > 3.

Because of the insufficient parametrization of E, the RMSis higher for smaller values of K than the true K . The RMSdecreases toward the point where K = 3, which is the truevalue of K . For higher values of K , there is no longer a modelerror and the noise is the sole contributor to the RMS. Thisis evident as an increase in RMS when K > 3. The sameestimation was also made with E = cosnθ [13]. A straightline represents this case in Figure 9 showing the difference inRMS, which is higher compared to using our method withK = 3. This shows that the optimum tradeoff for our pro-posed method, in this case, is K = 3. This gives the best

performance when comparing different values of K and theE = cosnθ assumption.

7. CONCLUSIONS

We have introduced a new method for the estimation of themutual coupling matrix of an antenna array. The main nov-elty over existing methods was that the array phase centerand the element factors were introduced as unknowns in thedata model, and treated as nuisance parameters in the esti-mation of the coupling as well, by being jointly estimated to-gether with the coupling matrix.

In a simulated test case, our method outperformedthe previously proposed estimator [13] for the case of an


0

0.01

0.02

0.03

0.04

0.05

0.06

1 2 3 4 5

K

RM

S

RMScos0.5

RMSK=3

CRB

Figure 9: CRB and RMS based on Monte Carlo simulations whenSNR is 30 dB. The true element factor is Etrue = cos2θ, which meansαtrue = [0 0 1 0 0] or Ktrue = 3.

8-element dipole array. The radiation pattern and phase er-ror were significantly improved leading to increased accuracyin any following postprocessing.

Based on a CRB analysis, the SNR penalty associated withintroducing a model for the element factor with two degreesof freedom (K = 3) was 2.5 dB relative to having zero de-grees of freedom. This means that an additional 2.5 dB morepower (or a doubling of the number of accumulated sam-ples) must be used to retain the estimation accuracy of thecoupling matrix compared to the case when the algorithmassumed omnidirectional elements. To add another degreeof freedom (set K = 4) costs another 10 dB.

Using measured calibration data from a dual polarizedarray, we found that the proposed method and the associatedestimator could significantly improve the quality of the esti-mated coupling matrix, and the result of subsequent com-pensation processing.

Finally, the tradeoff between the complexity of the pro-posed data model and the accuracy of the estimator was stud-ied via Monte Carlo simulations. For the case of an 8-elementlinear array, the optimum was found to be at K = 3 whichcoincides with the true value in that case.

ACKNOWLEDGMENTS

This material was presented in part at ICASSP 2007 [19]. ErikG. Larsson is a Royal Swedish Academy of Sciences ResearchFellow supported by a grant from the Knut and Alice Wallen-berg Foundation.

REFERENCES

[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communi-cation, Cambridge University Press, Cambridge, UK, 2005.

[2] A. Swindlehurst and T. Kailath, “A performance analysis ofsubspace-based methods in the presence of model errors—part I: the MUSIC algorithm,” IEEE Transactions on Signal Pro-cessing, vol. 40, no. 7, pp. 1758–1774, 1992.

[3] M. Jansson, A. Swindlehurst, and B. Ottersten, “Weighted sub-space fitting for general array error models,” IEEE Transactionson Signal Processing, vol. 46, no. 9, pp. 2484–2498, 1998.

[4] B. Friedlander and A. J. Weiss, “Effects of model errorson waveform estimation using the MUSIC algorithm,” IEEETransactions on Signal Processing, vol. 42, no. 1, pp. 147–155,1994.

[5] H. Steyskal and J. S. Herd, “Mutual coupling compensationin small array antennas,” IEEE Transactions on Antennas andPropagation, vol. 38, no. 12, pp. 1971–1975, 1990.

[6] J. Yang and A. L. Swindlehurst, “The effects of array calibrationerrors on DF-based signal copy performance,” IEEE Transac-tions on Signal Processing, vol. 43, no. 11, pp. 2724–2732, 1995.

[7] C. A. Balanis, Antenna Theory: Analysis and Design, John Wiley& Sons, New York, NY, USA, 1997.

[8] T. Svantesson, “The effects of mutual coupling using a lineararray of thin dipoles of finite length,” in Proceedings of the 9thIEEE SP Workshop on Statistical Signal and Array Processing(SSAP ’98), pp. 232–235, Portland, Ore, USA, September 1998.

[9] K. R. Dandekar, H. Ling, and G. Xu, “Effect of mutual cou-pling on direction finding in smart antenna applications,”Electronics Letters, vol. 36, no. 22, pp. 1889–1891, 2000.

[10] B. Friedlander and A. Weiss, “Direction finding in the pres-ence of mutual coupling,” IEEE Transactions on Antennas andPropagation, vol. 39, no. 3, pp. 273–284, 1991.

[11] T. Su, K. Dandekar, and H. Ling, “Simulation of mutual cou-pling effect in circular arrays for direction-finding applica-tions,” Microwave and Optical Technology Letters, vol. 26, no. 5,pp. 331–336, 2000.

[12] B. Lindmark, S. Lundgren, J. Sanford, and C. Beckman, “Dual-polarized array for signal-processing applications in wirelesscommunications,” IEEE Transactions on Antennas and Propa-gation, vol. 46, no. 6, pp. 758–763, 1998.

[13] M. Mowler and B. Lindmark, “Estimation of coupling, ele-ment factor, and phase center of antenna arrays,” in Proceed-ings of IEEE Antennas and Propagation Society InternationalSymposium, vol. 4B, pp. 6–9, Washington, DC, USA, July 2005.

[14] B. Lindmark, “Comparison of mutual coupling compensationto dummy columns in adaptive antenna systems,” IEEE Trans-actions on Antennas and Propagation, vol. 53, no. 4, pp. 1332–1336, 2005.

[15] S. M. Kay, Fundamentals of Statistical Signal Processing: Esti-mation Theory, Prentice-Hall, Upper Saddle River, NJ, USA,1993.

[16] H. L. Van Trees, Detection, Estimation, and Modulation Theory,Wiley-Interscience, New York, NY, USA, 2007.

[17] J. Gorman and A. Hero, “Lower bounds for parametric estima-tion with constraints,” IEEE Transactions on Information The-ory, vol. 36, no. 6, pp. 1285–1301, 1990.

[18] P. Stoica and B. C. Ng, “On the Crameer-Rao bound underparametric constraints,” IEEE Signal Processing Letters, vol. 5,no. 7, pp. 177–179, 1998.

[19] M. Mowler, E. G. Larsson, B. Lindmark, and B. Ottersten,“Methods and bounds for antenna array coupling matrix es-timation,” in Proceedings of IEEE International Conference onAcoustics, Speech, and Signal Processing (ICASSP ’07), vol. 2,pp. 881–884, Honolulu, Hawaii, USA, April 2007.


Research ArticleDiversity Characterization of Optimized Two-AntennaSystems for UMTS Handsets

A. Diallo,1 P. Le Thuc,1 C. Luxey,1 R. Staraj,1 G. Kossiavas,1 M. Franzen,2 and P.-S. Kildal3

1 Laboratoire d’Electronique, Antennes et Telecommunications (LEAT), Universite de Nice Sophia-Antipolis,CNRS UMR 6071, 250 rue Albert Einstein, Bat. 4, Les Lucioles 1, 06560 Valbonne, France

2 Bluetest AB, Gotaverksgatan 1, 41755 Gothenburg, Sweden3 Department of Signals and Systems, Chalmers University of Technology, 41296 Gothenburg, Sweden

Received 16 November 2006; Revised 20 June 2007; Accepted 22 November 2007


This paper presents the evaluation of the diversity performance of several two-antenna systems for UMTS terminals. All the mea-surements are done in a reverberation chamber and in a Wheeler cap setup. First, a two-antenna system having poor isolationbetween its radiators is measured. Then, the performance of this structure is compared with two optimized structures having highisolation and high total efficiency, thanks to the implementation of a neutralization technique between the radiating elements.The key diversity parameters of all these systems are discussed, that is, the total efficiency of the antenna, the envelope correlationcoefficient, the diversity gains, the mean effective gain (MEG), and the MEG ratio. The comparison of all these results is especiallyshowing the benefit brought back by the neutralization technique.

Copyright © 2007 A. Diallo et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Nowadays, wireless mobile communications are growing ex-ponentially in several fields of telecommunications. The newgeneration of mobile phones must be able to transfer largeamounts of data and consequently increasing the transferrate of these data is clearly needed. One solution is to imple-ment a diversity scheme at the terminal side of the commu-nication link. This can be done by multiplying the numberof the radiating elements of the handset. In addition, theseradiators must be highly isolated to achieve the best diver-sity performance. Also, the antenna engineers must take intoaccount the radiator’s environment of the handset to designsuitable multiantenna systems. In practice, the terminal canbe considered to operate in a so-called multipath propaga-tion environment: the electromagnetic field will take manysimultaneous paths between the transmitter and the receiver.In such a configuration, total efficiency, diversity gain, meaneffective gain (MEG), and MEG ratio are the most importantparameters for diversity purposes.

Only few papers are actually focusing on the design ofa specific technique to address the isolation problem of sev-eral planar inverted-F antennas (PIFAs) placed on the samefinite-sized ground plane and operating in the same fre-

quency bands. In [1, 2], the authors are evaluating the iso-lation between identical PIFAs when moving them all alonga mobile phone PCB for multiple-input multiple-output(MIMO) applications. The same kind of work is done in [3–6] for different antenna types. The best isolation values arealways found when the antennas are spaced by the largestavailable distance on the PCB, that is, one at the top edgeand the other at the bottom. Excellent studies can be foundin [7–16], but no specific technique to isolate the elements isdescribed in these papers. One solution is reported in [17],however, for two thin PIFAs for mobile phones operating indifferent frequency bands (GSM900 and DCS1800). It con-sists in inserting high-Q-value lumped LC components at thefeeding point of one antenna to achieve a blocking filter atthe resonant frequency of the other. This solution gives sig-nificant results in terms of decoupling but strongly reducesthe frequency bandwidth. Another very interesting solutionreported in [18, 19] consists in isolating the antennas by adecoupling network, at their feeding ports, this solution suf-fers from the fact that in small handsets available space is re-stricted. Finally, a promising solution is described in [20], butin this work the PIFAs are operating around 5 GHz.

Some authors of the current paper have already designedand fabricated several multiantenna structures for mobile


PCB 100× 40 mm2

UMTS PIFA

Feeding strip 1

Shorting strip 1

Shorting strip 2 Feeding strip 2

z y

x

Figure 1: 3D view of the initial two-antenna system.

phone applications. In [21], the isolation problem has beenaddressed for closely spaced PIFAs operating in very closefrequency bands with the help of a neutralization tech-nique. Recently, several two-antenna systems operating in theUMTS band (1920–2170 MHz) and especially including neu-tralization line to achieve high isolation between the feedingports of their radiating parts have been designed for diversityand MIMO applications [22]. Two prototypes have alreadybeen characterized in terms of scattering parameters, totalefficiency, and envelope correlation coefficient. The obtainedresults show that these structures have a strong potential foran efficient implementation of a diversity scheme at the mo-bile terminal side of a wireless link. However, to completelycharacterize these prototypes, some particular facilities andthe associated expertise are needed [23]. The antenna groupof Chalmers Institute of Technology possesses these capabil-ities through the Bluetest reverberation chamber [24].

This paper is the result of a short-term mission grantedby the COST 284. The antenna-design competencies of theLEAT have been combined with the reverberation chambermeasurement skills of the antenna group of Chalmers Insti-tute of Technology. Several prototypes have been measured atChalmers in terms of total efficiency, diversity gain, envelopecorrelation coefficient, and mean effective gain. Efficiency re-sults are compared with the same measurements obtainedthrough a homemade Wheeler Cap at the LEAT. The enve-lope correlation coefficient, the MEG, and the MEG ratio cal-culated from simulated values are also presented and com-pared [23, 25–27]. We focus on the comparison of the per-formance of an initial two-antenna system with two differ-ent neutralized structures and especially the benefit broughtback by the neutralization technique.

2. DESIGNED STRUCTURES ANDS-PARAMETER MEASUREMENTS

The multiantenna systems were designed using the electro-magnetic software tool IE3D [28]. The initial two-antennasystem is presented in Figure 1 (the design procedure was al-ready described in [22]). It consists of two PIFAs symmetri-cally placed on a 40× 100 mm2 PCB and separated by 0.12λ0

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Frequency (GHz)

−50

−40

−30

−20

−10

0

(dB

)

SimulatedMeasured

S21

S11/S22

Figure 2: Simulated and measured S-parameters of the initial two-antenna system.

PCB 100× 40 mm2

UMTS PIFA

Feeding strip 1

Shorting strip 1

Shorting strip 2 Feeding strip 2

z y

x

Neutralization line

Figure 3: 3D view of the two-antenna system with a suspended linebetween the PIFA shorting strips.

(18 mm at 2 GHz). They are fed by a metallic strip solderedto an SMA connector and shorted to the PCB by an iden-tical strip. Each PIFA is optimized to cover the UMTS band(1920–2170 MHz) with a return loss goal of−6 dB. The opti-mized dimensions are of 26.5 mm length and of 8 mm width.A prototype was fabricated using a 0.3-mm-thick nickel sil-ver material (conductivity σ = 4 × 106 S/m). In Figure 2,we present the simulated and the measured S-parameters ofthe structure. The absolute value S21 reaches a maximum of−5 dB in the middle of the UMTS band.

In the first attempt to improve the isolation between theradiating elements, a suspended line as a neutralization de-vice was inserted between the shorting strips of the two PI-FAs (see Figure 3). The optimization of this line was alreadyexplained in [21]. Figure 4 shows the S-parameters of thisnew structure. We can see a good matching and a strong im-provement of the isolation in the bandwidth of interest: themeasured S21 parameter always remains below−15 dB. How-ever, a different isolation can be obtained if we implement thesame neutralization technique between the two feeding strips

A. Diallo et al. 3

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Frequency (GHz)

−50

−40

−30

−20

−10

0

(dB

)

SimulatedMeasured

S21

S11/S22

Figure 4: Simulated and measured S-parameters of the two-antenna system with a line between the PIFA shorting strips.

PCB 100× 40 mm2

UMTS PIFA

Shorting strip 1

Feeding strip 1

Feeding strip 2Shorting strip 2

z y

x

Neutralization line

Figure 5: 3D view of the two-antenna system with a suspended linebetween the PIFA feeding strips.

of the PIFAs (see Figure 5). We can observe in Figure 6 that adeep null is now achieved in the middle of the UMTS band.Moreover, the measured S21 always remains below −18 dB inthe whole UMTS band. All these values seem to be very sat-isfactory for diversity purposes.

3. COMPARISON OF THE DIVERSITY PERFORMANCE

3.1. Total efficiency

Traditionally, the radiation performance of an antenna ismeasured outdoors or in an anechoic chamber. In order toobtain the total efficiency, we need to measure the radia-tion pattern in all directions in space and integrate the re-ceived power density to find the total radiated power. Thisgives the total efficiency when compared to the correspond-ing radiated power of a known reference antenna. This finalresult is obtained after a long measurement procedure. Thisparameter can be measured very much faster and easier ina reverberation chamber. However, it is necessary to mea-sure a reference case (a dipole antenna having an efficiency

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Frequency (GHz)

−50

−40

−30

−20

−10

0

(dB

)

SimulatedMeasured

S21

S11/S22

Figure 6: Simulated and measured S-parameters of the two-antenna system with the line between the PIFA feeding strips.

of 96% in our case) and then the antenna system under test(AUT). It is also important that the chamber is loaded in ex-actly the same way for these both measurements. For the ref-erence case, the transmission between the reference antennaand the excitation antennas is measured in the chamber withthe reference antenna in free space that means at least half awavelength away from any lossy objects and/or the metallicwalls of the chamber. As soon as the reference case is com-pleted, we can measure the AUT. From both measurements,we can then compute Pref (1) and PAUT (2):

Pref =∣∣S21, ref

∣∣

2

(

1− ∣∣S11∣∣

2)(

1− ∣∣S22, ref∣∣

2) , (1)

PAUT =∣∣S21, AUT

∣∣

2

(

1− ∣∣S11∣∣

2) (

1− ∣∣S22, AUT∣∣

2) , (2)

where S21 is the averaged transmission power level, S11 is thefree space reflection coefficient of the excitation antenna, andS22 is the free space reflection coefficient of the reference an-tenna (or the antenna under test). The − denotes averagingover n positions of the platform stirrer, polarization stirrer,and mechanical stirrers. The total efficiency can be then cal-culated from (3)

ηtot =(

1− ∣∣S22, AUT∣∣

2)PAUT

Pref. (3)

Figure 7 shows the total efficiency in dB of all the antennasystems (without the neutralization line (a), with the linebetween the feeding strips (b), and with the line betweenthe shorting strips (c)). The simulated curves have been ob-tained with the help of IE3D which uses the simulated scat-tering parameters. The experimental curves have been mea-sured in the reverberation chamber and with the help of ahomemade Wheeler-Cap setup [16]. With frequency averag-ing, the standard deviation of the efficiency measurements is


Table 1: Comparison of ηtot and the MEG of both antennas of the different structures at f = 2 GHz.

ηtot (dB) antenna1 MEG (dB) antenna1 ηtot (dB) antenna2 MEG (dB) antenna2

Sim. RC Sim. RC Sim. RC Sim. RC

Initial −0.816 −0.75 −3.826 −3.75 −0.81 −1.25 −3.826 −4.25

Line between thefeeding strips

−0.10 −0.2 −3.11 −3.2 −0.09 −0.5 −3.108 −3.5

Line between theshorting strips

−0.14 −0.35 −3.152 −3.35 −0.14 −0.65 −3.151 −3.65

Table 2: MEG ratio of the antennas for all the prototypes at f =2 GHz.

MEG1/MEG2

Initial 1,12

Line between the feeding strips 1,07

Line between the shorting strips 1,07

given as +/ − 0.5 dB in the reverberation chamber. The un-certainty of the homemade Wheeler Cap system is assumedto be quite the same. The total efficiency of both antennasfrom each prototype is presented. It can be seen that theyare slightly different in the two measurement cases (dottedlines and solid lines) due to the fact that the fabricated proto-types suffer from small inherent asymmetries. However, onlyone curve is presented for each simulation case due to perfectsymmetries and identical structure on the CAD software. Wecan observe that all these curves are in a good agreement es-pecially if we compare their maximums. The small frequencyshift observed in all the curves with the dotted lines is dueto the fact that the antenna was mechanically modified dur-ing transportation for measurement, and therefore frequencyis detuned. This effect impacts directly the S11 and then thefrequency location of the maximum of the total efficiency.The improvement brought by the neutralization techniqueis clearly shown: the maximum total efficiency of the neu-tralized antennas is around −0.25 dB, whereas the one of theinitial structure is less than −1 dB.

3.2. Mean effective gain and mean effective gain ratio

In order to characterize the performance of a multichannelantenna in a mobile environment, different parameters as theMEG and the MEG ratio are used. The total efficiency is theaverage antenna gain in the whole space. Equation (4) showsthat it can be calculated from the integration of the radiationpattern cuts

ηtot =∫ 2π

0

∫ π0

(

Gθ(θ,ϕ) +Gϕ(θ,ϕ))

sin θd θdϕ

4π, (4)

where Gθ and Gϕ are the antenna power gain patterns.The MEG is a statistical measure of the antenna gain in

a mobile environment. It is equal to the ratio of the mean

received power of the antenna and the total mean incident. Itcan be expressed by (5) as in [6]:

MEG =∫ 2π

0

∫ π

0

(XPR

1 + XPRGθ(θ,ϕ)Pθ(θ,ϕ)

+1

1 + XPRGϕ(θ,ϕ)Pϕ(θ,ϕ)

)

sin θd θdϕ,

(5)

where Pθ and Pϕ are the angular density functions of the inci-dent power, and XPR represents the cross-polarization powergain which is defined in (6):

XPR =∫ 2π

0

∫ π0 Pθ(θ,ϕ) sin θd θdϕ

∫ 2π0

∫ π0 Pϕ(θ,ϕ) sin θd θdϕ

. (6)

In the case where the antenna is located in a statistically uni-form Rayleigh environment (i.e., the case in the reverbera-tion chamber), we have XPR = 1 and Pθ = Pϕ = 1/4π. TheMEG is then equal to the total antenna efficiency divided bytwo or −3 dB [27]. Moreover, to achieve good diversity gain,the average received power from each antenna element mustbe nearly equal: this corresponds to getting the ratio of theMEG between the two antennas close to unity [29]. Table 1presents ηtotand the MEG of both antennas for the three pro-totypes at f = 2 GHz. The “Sim.” values have been computedusing the simulated radiation patterns while the reverbera-tion chamber results “RC” are taken from the previous mea-surements.

The neutralization line provides an enhancement of theηtot and the MEG as expected from the previous values. Theimprovement of the MEG is about 0.7 dB with regard to theinitial structure. Table 2 presents the MEG ratio between thetwo antennas of the different prototypes (computed fromthe RC MEG) at 2 GHz. It is seen that the antennas havecomparable-average-received power because these entire ra-tios are close to unity. Such a result was somewhat expecteddue to the symmetric antenna configuration of our proto-types. In fact, the MEG difference only shows here the proto-typing errors we made during the fabrication process. Nev-ertheless, all the results of this section confirm the benefit ofusing a neutralization technique between the radiators.

3.3. Correlation

For diversity and MIMO applications, the correlation be-tween the signals received by the antennas at the same side of

A. Diallo et al. 5

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency (GHz)

−10

−8

−6

−4

−2

0To

tale

ffici

ency

(dB

)

SimulationWheeler cap

Reverberation chamber

(a)

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency (GHz)

−10

−8

−6

−4

−2

0

Tota

leffi

cien

cy(d

B)



(b)

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency (GHz)

−10

−8

−6

−4

−2

0

Tota

leffi

cien

cy(d

B)



(c)

Figure 7: Total efficiency of the two-antenna structures: (a) with-out the neutralization line, (b) with the neutralization line betweenthe feeding strips, and (c) with the neutralization line between theshorting strips.

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency (GHz)

0

0.1

0.2

0.3

0.4

0.5

Env

elop

eco

rrel

atio

nco

effici

ent

S-parametersFar field


(a)

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency (GHz)

0

0.1

0.2

0.3

0.4

0.5

Env

elop

eco

rrel

atio

nco

effici

ent



(b)

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

Frequency (GHz)

0

0.1

0.2

0.3

0.4

0.5

Env

elop

eco

rrel

atio

nco

effici

ent



(c)

Figure 8: Envelope correlation coefficient versus frequency of thetwo-antenna systems: (a) without the neutralization line, (b) withthe line between the feeding strips, and (c) with the neutralizationline between the shorting strips.


−35 −30 −25 −20 −15 −10 −5 0 5 10

Relative received power (dB)

10−3

10−2

10−1

100

Cu

mu

lati

vepr

obab

ility

Diversity gainat 1%

(a)

−35 −30 −25 −20 −15 −10 −5 0 5 10


10−3

10−2

10−1

100

Cu

mu

lati

vepr

obab

ility

Diversity gainat 1%

(b)

−35 −30 −25 −20 −15 −10 −5 0 5 10


10−3

10−2

10−1

100

Cu

mu

lati

vepr

obab

ility

Diversity gainat 1%

(c)

Figure 9: Cumulative probability of the two-antenna systems overa 4 MHz bandwidth at 2 GHz: (a) without the neutralization line,(b) with the neutralization line between the feeding strips, and (c)with the neutralization line between the shorting strips.

−35 −30 −25 −20 −15 −10 −5 0 5 10


10−3

10−2

10−1

100

Cu

mu

lati

vepr

obab

ility

Diversity gainat 1%

(a)

−35 −30 −25 −20 −15 −10 −5 0 5 10


10−3

10−2

10−1

100

Cu

mu

lati

vepr

obab

ility

Diversity gainat 1%

(b)

−35 −30 −25 −20 −15 −10 −5 0 5 10


10−3

10−2

10−1

100

Cu

mu

lati

vepr

obab

ility

Diversity gainat 1%

(c)

Figure 10: Smoothed cumulative probability of the two-antennasystems over a 4 MHz bandwidth at 2 GHz: (a) without the neu-tralization line, (b) with the neutralization line between the feed-ing strips, and (c) with the neutralization line between the shortingstrips.

A. Diallo et al. 7

a wireless link is an important figure of merit. Usually, the en-velope correlation is presented to evaluate the diversity capa-bilities of a multiantenna system [30]. This parameter shouldbe preferably computed from 3D-radiation patterns [31, 32],but the process is tedious because sufficient pattern cuts mustbe taken into account. In the case of a two-antenna system,the envelope correlation ρe is given by (7) as in [31, 32]:

ρe =∣∣∫∫

4π

[�F1(θ,ϕ)•�F∗2 (θ,ϕ)dΩ]∣∣

2

∫∫

4π

∣∣�F1(θ,ϕ)

∣∣

2dΩ∫∫

4π

∣∣�F2(θ,ϕ)

∣∣

2dΩ

, (7)

where �Fi = (θ,ϕ) is the field radiation pattern of the antennasystem when the port i is excited, and • denotes the Hermi-tian product.

However, assuming that the structure will operate in auniform multipath environment, a convenient and quick al-ternative consists by using (8) (see [31–33]):

ρ12 =∣∣S∗11 S12 + S∗12 S22

∣∣

2

(

1− ∣∣S11∣∣

2 − ∣∣S21∣∣

2)(

1− ∣∣S22∣∣

2 − ∣∣S12∣∣

2) . (8)

It offers a simple procedure compared to the radiation pat-tern approach, but it should be emphasized that this equationis strictly valid when the three following assumptions are ful-filled:

(i) lossless antenna case that means having antennas withhigh efficiency and no mutual losses [29, 30];

(ii) antenna system is positioned in a uniform multipathenvironment which is not strictly the case in real envi-ronments, however, the evaluation of some prototypesin different real environments has already shown thatthere are no major differences in these cases [34];

(iii) load termination of the nonmeasured antenna is 50Ω.In reality, the radio front-end module does not alwaysachieve this situation, but the 50Ω evaluation proce-dure is commonly accepted [35, 36].

All these limitations are clearly showing that in real systemsthe envelope correlation calculated based on of the help ofthe Si j parameters is not the exact value, but nevertheless is agood approximation. In addition, it should be noted that an-tennas with an envelope correlation coefficient less than 0.5are recognized to provide significant diversity performance[30].

To measure the correlation between the antennas of oursystems in the reverberation chamber, each branch is con-nected to a separate receiver. The two different received sig-nals are recorded, and the envelope correlation can be di-rectly computed. Figure 8 presents the measured envelopecorrelation coefficients of all the antenna systems. They arecompared with those obtained using (7) (computation fromthe simulated IE3D complex 3D-radiation patterns) andwith those obtained using (8) (measured S-parameter val-ues). All these curves are in a moderate agreement, but it canbe seen that the envelope correlation coefficients of all theprototypes are always lower than 0.15 on the whole UMTSband: good performance in terms of diversity is thus ex-pected [1]. Here, it is however somewhat difficult to claim

that the neutralization technique provides an improvementof the correlation. It seems rather obvious that with suchspaced antennas operating in a uniform multipath environ-ment, low correlation is not very difficult to achieve.

3.4. Apparent diversity gain and actual diversity gain

The concept of diversity means that we make use of two ormore antennas to receive a signal and that we are able to com-bine the replicas of the received signal in a desirable way toimprove the communication link performance. One require-ment is high isolation between the antennas; otherwise thediversity gain will be low. The apparent diversity gain Gdiv app

relative to antenna1 and the actual diversity gain Gdiv act aredefined in (9)

Gdiv app = S/N

S1/N1,

Gdiv act = S/N

S1/N1ηtot1,

(9)

where ηtot1 is the total efficiency of antenna1.Note that these formulas are valid only if the noise signals

N1 (and N2 for the second antenna) are independent of thetotal efficiency. This is the case if the system noise is dom-inated by those of the receivers or if the antenna noise tem-perature is the same as the surrounding temperature. The lastcondition is often close to being satisfied in mobile systemsbecause the antenna is rather omnidirectional and picks upthermal noise mainly from the environment (ground, build-ings, trees, human) around the antenna, and less from thelow sky temperature.

We can see in Figure 9 the power samples of each two-antenna system (without the neutralization line (a), with theline between the feeding strips (b), and with the line betweenthe shorting strips (c)) averaged over a 20-MHz frequencyband at 2 GHz. We can observe that the combined signalcurves with the selection combining scheme (solid lines) aresteeper than the two curves of the antenna elements takenalone (dotted lines). This is the benefit of combining thetwo signals received by each antenna of the structure. By justlooking at the curves in Figure 9, the uncertainty is undoubt-edly very large. This is due to the obvious lack of samples atlow-probability levels coming from the measurement proce-dure.

The apparent diversity gain is determined by the power-level improvement at a certain probability level. In Figures9(a), 9(b), and 9(c), we have chosen 1% probability. It is thenthe difference between the strongest antenna element curveand the combined signal curve. The power improvement is7.6 dB for the system with low isolation, whereas it is 8.8 dBand 9.1 dB for the system with high isolation, respectively,for the line between the shorting strips and the line betweenthe feeding strips. As the total efficiency is not taken into ac-count in the apparent diversity gain, the improvement onlycomes from the fact that the radiation patterns are slightlydifferent in the case of the two neutralized structures. Es-pecially, an increase of the cross-polarization level occurs inthe radiation patterns of the neutralized structures due to the


Table 3: Summary of the measured and computed diversity gains of all the antenna systems.

PrototypesTotal efficiency

best branchApparent

diversity gain

Apparent diversitygain, smooth

curved

Actual diversitygain

Actual diversitygain, smooth

curved

Without any line −0.75 dB 7.6 dB 8.6 dB 6.3 dB 7.8 dB

Shorting strips link −0.35 dB 8.8 dB 9.2 dB 8 dB 8.8 dB

Feeding strips link −0.2 dB 9.1 dB 9.75 dB 8.6 dB 9.5 dB

fact that a strong current is flowing on the line. This increaseof the X-pol appears to be beneficial for the diversity gain.When taking into account the total efficiency of the anten-nas, we can compute the actual diversity gain as 6.3 dB forthe initial system, 8 dB and 8.6 dB for the neutralized short-ing strips and feeding strips systems, respectively. The datafrom Figure 9 were also processed with the smooth functionof MATLAB [37] in order to evaluate the validity of our mea-surements. Several “smooth steps” were tried out in this op-eration and the new curves are presented in Figure 10. It ap-pears that all the apparent diversity gains were formerly un-derestimated. The new actual diversity gains are now 7.8 dB,8.8 dB, and 9.5 dB for, respectively, the initial, the neutralizedshorting strips and feeding strips systems. A summary of allthese values can be found in Table 3.

It seems obvious that the neutralization technique en-hances the actual diversity gain. These results are consistentwith other publications [38] and even better due to the use ofhighly efficient antennas here. We should also point out thatthe apparent diversity gain and the actual diversity gain arenot so much different due to the same reason [39].

4. CONCLUSION

In this paper, we have presented different two-antenna sys-tems with poor and high isolations for diversity purposes.The reverberation chamber measurements at the antennagroup of Chalmers University of Technology have shown thateven if the envelope correlation coefficient of these systems isvery low, having antennas with high isolation will improvethe total efficiency and the effective diversity gain of the sys-tem. The same conclusions have been drawn regarding theMEG values. All these results point out the usefulness of oursimple solution to achieve efficient antenna systems at theterminal side of a wireless link for diversity or MIMO appli-cations. Next studies will focus on the effect of the users uponthe neutralization technique by positioning the antenna sys-tems next to a phantom head.

ACKNOWLEDGMENT

The authors express their gratitude to the COST284 projectfor providing the opportunity to make a short-term scientificmission from the LEAT to Chalmers Institute.

REFERENCES

[1] Z. Ying and D. Zhang, “Study of the mutual coupling, corre-lations and efficiency of two PIFA antennas on a small ground

plane,” in Proceedings of IEEE Antennas and Propagation Soci-ety International Symposium, vol. 3, pp. 305–308, Washington,DC, USA, July 2005.

[2] J. Thaysen and K. B. Jakobsen, “MIMO channel capacity ver-sus mutual coupling in multi antenna element system,” inProceedings of the 26th Annual Meeting & Symposium on An-tenna Measurement Techniques Association (AMTA ’04), StoneMountain, Ga, USA, October 2004.

[3] M. Karakoikis, C. Soras, G. Tsachtsiris, and V. Makios, “Com-pact dual-printed inverted-F antenna diversity systems forportable wireless devices,” IEEE Antennas and Wireless Prop-agation Letters, vol. 3, no. 1, pp. 9–14, 2004.

[4] K.-L. Wong, C.-H. Chang, B. Chen, and S. Yang, “Three-antenna MIMO system for WLAN operation in a PDA phone,”Microwave and Optical Technology Letters, vol. 48, no. 7, pp.1238–1242, 2006.

[5] S. H. Chae, S.-K. Oh, and S.-O. Park, “Analysis of mutualcoupling, correlations, and TARC in WiBro MIMO array an-tenna,” IEEE Antennas and Wireless Propagation Letters, vol. 6,pp. 122–125, 2007.

[6] H. T. Hui, “Practical dual-helical antenna array for diver-sity/MIMO receiving antennas on mobile handsets,” IEE Pro-ceedings: Microwaves, Antennas and Propagation, vol. 152,no. 5, pp. 367–372, 2005.

[7] J. Villanen, P. Suvikunnas, C. Icheln, J. Ollikainen, and P.Vainikainen, “Advances in diversity performance analysis ofmobile terminal antennas,” in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP ’04), Sendai,Japan, August 2004.

[8] M. Manteghi and Y. Rahmat-Samii, “Novel compact tri-bandtwo-element and four-element MIMO antenna designs,” inProceedings of the International Symposium on Antennas andPropagations (ISAP ’06), pp. 4443–4446, Albuquerque, NM,USA, July 2006.

[9] M. Manteghi and Y. Rahmat-Samii, “A novel miniaturized tri-band PIFA for MIMO applications,” Microwave and OpticalTechnology Letters, vol. 49, no. 3, pp. 724–731, 2007.

[10] D. Browne, M. Manteghi, M. P. Fits, and Y. Rahmat-Samii,“Experiments with compact antenna arrays for MIMO radiocommunications,” IEEE Transaction on Antennas and Propa-gation, vol. 54, no. 11, part 1, pp. 3239–3250, 2007.

[11] B. Lindmark and L. Garcia-Garcia, “Compact antenna arrayfor MIMO applications at 1800 and 2450 MHz,” Microwaveand Optical Technology Letters, vol. 48, no. 10, pp. 2034–2037,2006.

[12] R. G. Vaughan and J. B. Andersen, “Antenna diversity in mo-bile communications,” IEEE Transactions on Vehicular Tech-nology, vol. 36, no. 4, pp. 149–172, 1987.

[13] B. K. Lau, J. B. Andersen, G. Kristensson, and A. F. Molisch,“Impact of matching network on bandwidth of compact an-tenna arrays,” IEEE Transactions on Antennas and Propagation,vol. 54, no. 11, pp. 3225–3238, 2006.

A. Diallo et al. 9

[14] J. B. Andersen and B. K. Lau, “On closely coupled dipoles in arandom field,” IEEE Antennas and Wireless Propagation Letter,vol. 5, no. 1, pp. 73–75, 2006.

[15] J. W. Wallace and M. A. Jensen, “Termination-dependentdiversity performance of coupled antennas: network theoryanalysis,” IEEE Transactions on Antennas and Propagation,vol. 52, no. 1, pp. 98–105, 2004.

[16] M. A. J. Jensen and J. W. Wallace, “A review of antennasand propagation for MIMO wireless communications,” IEEETransactions on Antennas and Propagation, vol. 52, no. 11, pp.2810–2824, 2004.

[17] J. Thaysen and K. B. Jakobsen, “Mutual coupling reduction us-ing a lumped LC circuit,” in Proceedings of the 13th Interna-tional Symposium on Antennas (JINA ’04), pp. 492–494, Nice,France, November 2004.

[18] S. Dossche, S. Blanch, and J. Romeu, “Optimum antennamatching to minimise signal correlation on a two-port an-tenna diversity system,” IET Electronics Letters, vol. 40, no. 19,pp. 1164–1165, 2004.

[19] S. Dossche, J. Rodriguez, L. Jofre, S. Blanch, and J. Romeu,“Decoupling of a two-element switched dual band patchantenna for optimum MIMO capacity,” in Proceedings ofthe International Symposium on Antennas and Propagations(ISAP ’06), pp. 325–328, Albuquerque, NM, USA, July 2006.

[20] Y. Gao, X. Chen, C. Parini, and Z. Ying, “Study of a dual-element PIFA array for MIMO terminals,” in Proceedings ofthe International Symposium on Antennas and Propagations(ISAP ’06), pp. 309–312, Albuquerque, NM, USA, July 2006.

[21] A. Diallo, C. Luxey, P. Le Thuc, R. Staraj, and G. Kossi-avas, “Study and reduction of the mutual coupling betweentwo mobile phone PIFAs operating in the DCS1800 andUMTS bands,” IEEE Transactions on Antennas and Propaga-tion, vol. 54, no. 11, part 1, pp. 3063–3074, 2006.

[22] A. Diallo, C. Luxey, P. Le Thuc, R. Staraj, and G. Kossiavas,“Enhanced diversity antennas for UMTS handsets,” in Pro-ceedings of the European Conference on Antennas and Propa-gations (EuCAP ’06), Nice, France, November 2006.

[23] P.-S. Kildal and K. Rosengren, “Correlation and capacity ofMIMO systems and mutual coupling, radiation efficiency,and diversity gain of their antennas: simulations and mea-surements in a reverberation chamber,” IEEE CommunicationsMagazine, vol. 42, no. 12, pp. 104–112, 2004.

[24] http://www.bluetest.se/.[25] T. Bolin, A. Derneryd, G. Kristensson, V. Plicanic, and Z. Ying,

“Two-antenna receive diversity performance in indoor envi-ronment,” Electronics Letters, vol. 41, no. 22, pp. 1205–1206,2005.

[26] T. Taga, “Analysis for mean effective gain of mobile antennasin land mobile radio environments,” IEEE Transactions on Ve-hicular Technology, vol. 39, no. 2, pp. 117–131, 1990.

[27] K. Kalliola, K. Sulonen, H. Laitinen, O. Kivekas, J. Krogerus,and P. Vainikainen, “Angular power distribution and mean ef-fective gain of mobile antenna in different propagation envi-ronments,” IEEE Transactions on Vehicular Technology, vol. 51,no. 5, pp. 823–838, 2002.

[28] IE3D, Release 11.15, Zeland software, 2005.[29] C. C. Chlau, X. Chen, and C. Q. Parinl, “A compact four-

element diversity-antenna array for PDA terminals in a mimosystem,” Microwave and Optical Technology Letters, vol. 44,no. 5, pp. 408–412, 2005.

[30] S. C. K. Ko and R. D. Murch, “Compact integrated diversityantenna for wireless communications,” IEEE Transactions onAntennas and Propagation, vol. 49, no. 6, pp. 954–960, 2001.

[31] I. Salonen and P. Vainikainen, “Estimation of signal correla-tion in antenna arrays,” in Proceedings of the 12th InternationalSymposium Antennas (JINA ’02), vol. 2, pp. 383–386, Nice,France, November 2002.

[32] P. Brachat and C. Sabatier, “Reseau d’antennes a 6 Capteurs enDiversite de Polarisation,” in Proceedings of the 13th Interna-tional Symposium Antennas (JINA ’04), Nice, France, Novem-ber 2004.

[33] J. Thaysen and K. B. Jakobsen, “Envelope correlation in (N, N)MIMO antenna array from scattering parameters,” Microwaveand Optical Technology Letters, vol. 48, no. 5, pp. 832–834,2006.

[34] Z. Ying, V. Plicanic, T. Bolin, G. Kristensson, and A. Derneryd,“Characterization of Multi-Channel Antenna Performance forMobile Terminal by Using Near Field and Far Field Parame-ters,” COST 273 TD (04)(095), Goteborg, Sweden, June 2004.

[35] A. Derneryd and G. Kristensson, “Signal correlation includingantenna coupling,” Electronics Letters, vol. 40, no. 3, pp. 157–159, 2004.

[36] A. Derneryd and G. Kristensson, “Antenna signal correlationand its relation to the impedance matrix,” Electronics Letters,vol. 40, no. 7, pp. 401–402, 2004.

[37] http://www.mathworks.fr/.[38] K. Rosengren and P.-S. Kildal, “Diversity performance of a

small terminal antenna for UMTS,” in Proceedings of NordicAntenna Symposium (Antenn ’03), Kalmar, Sweden, May 2003.

[39] P.-S. Kildal, K. Rosengren, J. Byun, and J. Lee, “Definitionof effective diversity gain and how to measure it in a rever-beration chamber,” Microwave and Optical Technology Letters,vol. 34, no. 1, pp. 56–59, 2002.


Research ArticleOptimal Design of Nonuniform Linear Arrays in CellularSystems by Out-of-Cell Interference Minimization

S. Savazzi,1 O. Simeone,2 and U. Spagnolini1

1 Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milano, Italy2 Center for Wireless Communications and Signal Processing Research (CCSPR), New Jersey Institute of Technology,University Heights, Newark, NJ 07102-1982, USA

Received 13 October 2006; Accepted 11 July 2007

Recommended by Monica Navarro

Optimal design of a linear antenna array with nonuniform interelement spacings is investigated for the uplink of a cellular system.The optimization criterion considered is based on the minimization of the average interference power at the output of a con-ventional beamformer (matched filter) and it is compared to the maximization of the ergodic capacity (throughput). Out-of-cellinterference is modelled as spatially correlated Gaussian noise. The more analytically tractable problem of minimizing the inter-ference power is considered first, and a closed-form expression for this criterion is derived as a function of the antenna spacings.This analysis allows to get insight into the structure of the optimal array for different propagation conditions and cellular layouts.The optimal array deployments obtained according to this criterion are then shown, via numerical optimization, to maximize theergodic capacity for the scenarios considered here. More importantly, it is verified that substantial performance gain with respectto conventionally designed linear antenna arrays (i.e., uniform λ/2 interelement spacing) can be harnessed by a nonuniform opti-mized linear array.

Copyright © 2007 S. Savazzi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Antenna arrays have emerged in the last decade as a power-ful technology in order to increase the link or system capac-ity in wireless systems. Basically, the deployment of multipleantennas at either the transmitter or the receiver side of awireless link allows the exploitation of two contrasting ben-efits: diversity and beamforming. Diversity relies on fadinguncorrelation among different antenna elements and pro-vides a powerful means to combat the impairments causedby channel fluctuations. In [1] it has been shown that a sig-nificant increase in system capacity can be achieved by theuse of antenna diversity combined with optimum combin-ing schemes. Independence of fading gains associated to theantennas array can be guaranteed if the scattering environ-ment is rich enough and the antenna elements are sufficientlyspaced apart (at least 5–10 λ, where λ denotes the carrierwavelength) [2]. On the other hand, when fading is highlycorrelated, as for sufficiently small antenna spacings, beam-forming techniques can be employed in order to mitigatethe spatially correlated noise. Interference rejection throughbeamforming is conventionally performed by designing auniform linear array with half wavelength interelement spac-

ings, so as to guarantee that the angle of arrivals can bepotentially estimated free of aliasing. Moreover, beamform-ing is effective in propagation environments where there is astrong line-of-sight component and the system performanceis interference-limited [3].

In this paper, we consider the optimization of a linearnonuniform antenna array for the uplink of a cellular sys-tem. The study of nonuniform linear arrays dates back tothe seventies with the work of Saholos [4] on radiation pat-tern and directivity. In [5] performance of linear and cir-cular arrays with different topologies, number of elements,and propagation models is studied for the uplink of an inter-ference free system so as to optimize the network coverage.The idea of optimizing nonuniform-spaced antenna arraysto enhance the overall throughput of an interference-limitedsystem was firstly proposed in [6]. Therein, for flat fadingchannels, it is shown that unequally spaced arrays outper-form equally spaced array by 1.5–2 dB. Here, different from[6], a more realistic approach that explicitly takes into ac-count the cellular layout (depending on the reuse factor)and the propagation model (that ranges from line-of-sightto richer scattering according to the ring model [7]) is ac-counted for.


θ1θ1

θ3

θ2

θ3

θ0Δ12

Δ23

Δ12

(1)

(2)

(3)� = 2 kmθ3 = θ1

� = 2 km(3)

(2)

(1)

Setting A: reuse 3 Setting B: reuse 7

Interferer

User

Figure 1: Two cellular systems with hexagonal cells and trisectorialantennas at the base stations (reuse factor F = 3, setting A, on theright and F = 7, setting B, on the left). The array is equipped withN = 4 antennas. Shaded sectors denote the allowed areas for userand the three interferers belonging to the first ring of interference(dashed lines identify the cell clusters of frequency reuse).

Δ12 Δ23 ΔN

2−1,

N

2

· · · · · ·

Figure 2: Nonuniform symmetric array structure for N even.

For illustration purposes, consider the interference sce-narios sketched in Figure 1. Therein, we have two differentsettings characterized by hexagonal cells and different reusefactors (F = 3 for setting A and F = 7 for setting B, frequencyreuse clusters are denoted by dashed lines). The base stationis equipped with a symmetric antenna array1 containing aneven number N of directional antennas (N = 4 in the exam-ple) to cover an angular sector of 120 deg, other BS antennaarray design options are discussed in [8]. Each terminal isprovided with one omnidirectional antenna. On the consid-ered radio resource (e.g., time-slot, frequency band, or or-thogonal code), it is assumed there is only one active user inthe cell, as for TDMA, FDMA, or orthogonal CDMA. Theuser of interest is located in the respective sector accordingto the reuse scheme. The contribution of out-of-cell inter-ferers is modelled as spatially correlated Gaussian noise. InFigure 1, the first ring of interference is denoted by shadedcells. The problem we tackle is that of finding the antennaspacings in vector Δ = [Δ12 Δ23]T (as shown in the example)

1 The symmetric array assumption (as in the array structure of Figure 2)has been made mainly for analytical convenience in order to simplify theoptimization problem. However, it is expected that for a scenario with asymmetric layout of interference (such as setting A), the assumption of asymmetric array does not imply any loss of optimality, while, on the otherhand, for an asymmetric layout (such as setting B), capacity gains couldbe in principle obtained by deploying an asymmetric array.

so as to optimize given performance metrics, as detailed be-low.

Two criteria are considered, namely, the minimization ofthe average interference power at the output of a conven-tional beamformer (matched filter) and the maximizationof the ergodic capacity (throughput). Since in many appli-cations the position of users and interferers is not knowna priori at the time of the antenna deployment or the in-cell/out-cell terminals are mobile, it is of interest to evaluatethe optimal spacings not only for a fixed position of usersand interferers but also by averaging the performance met-ric over the positions of user and interferers within their cells(see Section 2).

Even if the ergodic capacity criterion has to be consideredto be the most appropriate for array design in interference-limited scenario, the interference power minimization is ana-lytically tractable and highlights the justification for unequalspacings. Therefore, the problem of minimizing the inter-ference power is considered first and a closed-form expres-sion for this criterion is derived as a function of the antennaspacings (Section 4). This analysis allows to get insight intothe structure of the optimal array for different propagationconditions and cellular layouts avoiding an extensive numer-ical maximization of the ergodic capacity. The optimal ar-ray deployments obtained according to the two criteria areshown via numerical optimization to coincide for the con-sidered scenarios (Section 5). More importantly, it is veri-fied that substantial performance gain with respect to con-ventionally designed linear antenna arrays (i.e., uniform λ/2interelement spacing) can be harnessed by an optimized ar-ray (up to 2.5 bit/s/Hz for the scenarios in Figure 1).

2. PROBLEM FORMULATION

The signal received by theN antenna array at the base stationserving the user of interest can be written as

y = h0x0 +M∑

i=1

hixi + w, (1)

where h0 is the N × 1 vector describing the channel gainsbetween the user and the N antennas of the base sta-tion, x0 is the signal transmitted by the user, hi and xi arethe corresponding quantities referred to the ith interferer(i = 1, . . . ,M), w is the additive white Gaussian noise withE[wwH] = σ2I. The channel vectors h0 and {hi}Mi=1 are un-correlated among each other and assumed to be zero-meancomplex Gaussian (Rayleigh fading) with spatial correlationR0 = E[h0hH0 ] and {Ri = E[hihHi ]}Mi=1, respectively. Thecorrelation matrices are obtained according to a widely em-ployed geometrical model that assumes the scatterers as dis-tributed along a ring around the terminal, see Figure 3. Thismodel was thoroughly studied in [2, 7] and a brief reviewcan be found in Section 3. According to this model, the spa-tial correlation matrices of the fading channel depend on

(1) the set of N/2 antenna spacings (N is even) Δ =[Δ12 Δ23 · · · ΔN/2,N/2+1]T , where Δi j is the distancebetween the ith and the jth element of the array (the

S. Savazzi et al. 3

Φ0

φ

θ0

θi

Φidi

p

q

Δpq d0

r0

ri

InterfererUser

Figure 3: Propagation model for user and interferers: the scatterersare distributed on a rings of radii ri around the terminals.

array is assumed to be symmetric as shown in Figure 2,extension to an odd number of antennas N is straight-forward);

(2) the relative positions of user and interferers with re-spect to the base station of interest (these latter pa-rameters can be conveniently collected into the vectorη = [ηT0 ηT1 · · · ηTM]T , where, as detailed in Figure 3,vector η0 = [d0 θ0]T parametrizes the geometrical lo-cation of the in-cell user and vectors ηi = [di θi]T (i =1, . . . ,M) describe the location of the interferers (i =1, . . . ,M));

(3) the propagation environment is described by the angu-lar spread of the scattered signal received by the basestation (φ0 for the user and φi (i = 1, . . . ,M) for theinterferers); notice that for ideally φi → 0 all scatter-ers come from a unique direction so that line-of-sight(LOS) channel can be considered. Shadowing can bepossibly modelled as well, see Section 3 for further dis-cussion.

2.1. Interference power minimization

From (1), the instantaneous total interference power at theoutput of a conventional beamforming (matched filter) is [9]

P(

h0,Δ, η) = hH0 Qh0, (2)

where

Q = Q(Δ, η1, . . . ,ηM

) =M∑

i=1

Ri(Δ, ηi

)+ σ2IN (3)

accounts for the spatial correlation matrix of the interferersand for thermal noise with power σ2. Notice that, for clar-ity of notation, we explicitly highlighted that the interfer-ence correlation matrices depend on the terminals’ locations

η and the antenna spacings Δ through nonlinear relation-ships. The first problem we tackle is that of finding the set ofoptimal spacings Δ that minimizes the average (with respectto fading) interference power, P (Δ, η) = Eh0 [P (h0,Δ, η)],that is,

(Problem-1) : Δ = arg minΔ

P (Δ, η) (4)

for a fixed given position η of user and interferers. Problem1 is relevant for fixed system with a known layout at the timeof antenna deployment. Moreover, its solution will bring in-sight into the structure of the optimal array, which can be tosome extent generalized to a mobile scenario. In fact, in mo-bile systems or in case the position of users and interferers isnot known a priori at the time of the antenna deployment,it is more meaningful to minimize the average interferencepower for any arbitrary position of in-cell user (η0) and out-of-cells interferers (η1,η2, . . . ,ηM). Denoting the averagingoperation with respect to users and interferers positions byEη[P (Δ, η)], the second problem (9) can be can be stated as

(Problem-2) : Δ = arg minΔ

Eη[P (Δ, η)

]. (5)

2.2. Ergodic capacity maximization

The instantaneous capacity for the link between the user andthe BS reads [2]

C(

h0,Δ, η) = log2

(1 + hH0 Q−1h0

)[ bit/s/Hz], (6)

and depends on both the antenna spacings Δ and the termi-nals’ locations η. For fast-varying fading channels (comparedto the length of the coded packet) or for delay-insensitiveapplications, the performance of the system from an infor-mation theoretic standpoint is ruled by the ergodic capacityC(Δ,η). The latter is defined as the ensemble average of theinstantaneous capacity over the fading distribution,

C(Δ,η

) = Eh0

[C(

h0,Δ, η)]. (7)

According to the alternative performance criterion hereinproposed, the first problem (4) is recasted as

(Problem 1) : Δ = arg maxΔ

C(Δ,η), (8)

and therefore requires the maximization of the ergodic ca-pacity for a fixed given position η of user and interferers.As before, denoting the averaging operation with respect tousers and interferers positions by Eη[C(Δ,η)], the secondproblem (5) can be modified accordingly:

(Problem 2) : Δ = arg maxΔ

Eη[C(Δ,η)

]. (9)

Different from the interference power minimization ap-proach, in this case, functional dependence of the perfor-mance criterion (7) on the antenna spacings Δ is highly non-linear (see Section 3 for further details) and complicated bythe presence of the inverse matrix Q−1 that relies upon Δ andη. This implies both a large-computational complexity for


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

40.5

45

36

31.5

27

22.5

Δ12 λ

SIRr2

(dB

)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

16

17

15

13

12

11

14

Δ12 λ

Erg

odic

capa

city

(bps

/Hz)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b)

Figure 4: Setting-A: rank-2 approximation of the signal-to-interference ratio SIRr2(Δ,η) (23) versus Δ12/λ and Δ23/λ (a) compared withergodic capacity C(Δ,η) (b) (r = 50 m). Circles denote optimal solutions.

the numerical optimization of (8) and (9), and the impossi-bility to get analytical insight into the properties of the op-timal solution. When the number of antenna array is suffi-ciently small (as in Section 5), optimization can be reason-ably dealt with through an extensive search over the opti-mization domain and without the aid of any sophisticatednumerical algorithm. On the contrary, in case of an arraywith a larger number of antenna elements, more efficient op-timization techniques (e.g., simulated annealing) may be em-ployed to reduce the number of spacings to be explored andthus simplify the optimization process. Below we will prove(by numerical simulations) that the limitations of the aboveoptimization (8)-(9) are mitigated by the criteria (4)-(5) stillpreserving the final result.

3. SPATIAL CORRELATION MODEL

We consider a propagation scenario where each terminal, beit the user or an interferer, is locally surrounded by a largenumber of scatterers. The signals radiated by different scat-terers add independently at the receiving antennas. The scat-terers are distributed on a ring of radius r0 around the ter-minal (ri, i = 1, . . . ,M for the interferers) and the resultingangular spread of the received signal at the base station is de-noted by φ0 � r0/d0 (or φi � ri/di), as in Figure 3. Becauseof the finite angular spreads {φi}Mi=0, the propagation modelappears to be well suited for outdoor channels.

In [7], the spatial correlation matrix of the resultingRayleigh distributed fading process at the base station is com-puted by assuming a parametric distribution of the scatterersalong the ring, namely, the von Mises distribution (variable0 ≤ ϑ < 2π runs over the ring, see Figure 3):

f (ϑ) = 12πI0(κ)

exp[κ cos(ϑ)

]. (10)

By varying parameter κ, the distribution of the scatterersranges from uniform ( f (ϑ) = 1/(2π) for κ = 0) to a Diracdelta around the main direction of the cluster ϑ = 0 (forκ → ∞). Therefore, by appropriately adjusting parameter κand the angular spreads for each user and interferers φi, apropagation environment with a strong line-of-sight com-ponent (φi � 0 and/or κ → ∞) or richer scattering (largerφi with κ small enough) can be modelled. The (normal-ized) spatial correlation matrix has the general expressionfor both user and interferers (for the (p, q)th element withp, q = 1, . . . ,N and i = 0, 1, . . . ,M) [7]:

[Ri(θi,Δ

)]pq = exp

[j2πλΔpq sin

(θi)]

·I0(√κ2 − ((2π/λ)Δpqφi cos

(θi))2)

I0(κ).

(11)

It is worth mentioning that spatial channel models based ondifferent geometries such as elliptical or disk models [10, 11]may be considered as well by appropriately modifying thespatial correlation (11). Effects of mutual coupling (not ad-dressed in this paper) between the array elements may be in-cluded in our framework too, see [12, 13].

From (11), the spatial correlation matrices Ri of the userand interferers can be written as

Ri(ηi,Δ

) = ρiRi(θi,Δ

)with ρi = K

dαi, (12)

where K is an appropriate constant that accounts for receiv-ing and transmitting antenna gain and the carrier frequency,and α is the path loss exponent. The contribution of shad-owing in (12) will be considered in Section 5.3 as part of anadditional log-normal random scaling term.

S. Savazzi et al. 5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

26.5

27

26

25.5

25

24.5

24

23.5

23

22.5

Δ12 λ

SIRr1

(dB

)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

12

12.5

11.5

11

10.5

Δ12 λ

Erg

odic

capa

city

(bps

/Hz)

Δ23

λ

0 0.5 1.5 2 2.5 3 3.5 4 4.5 51

(b)

Figure 5: Setting-A: rank-1 approximation (a) of the signal-to-interference ratio, SIRr1(Δ,η), versus Δ12/λ and Δ23/λ. Dashed lines denotethe optimality conditions (24) obtained by the rank-1 approximation. As a reference, ergodic capacity is shown (b), for an angular spreadapproaching zero.

4. REDUCED-RANK APPROXIMATION FORTHE INTERFERENCE POWER

According to a reduced-rank approximation for the spa-tial correlation matrices of user and interferers Ri for i =0, 1, . . . ,M, in this section, we derive an analytical closedform expression for the interference power (2) to ease theoptimization of the antenna spacings Δ. In Section 4.1, weconsider the case where the angular spread for users and in-terferers φi is small so that a rank-1 approximation of the spa-tial correlation matrices can be used. This first case describesline-of-sight channels. Generalization to channel with richerscattering is given in Section 4.2.

4.1. Rank-1 approximation (line-of-sight channels)

If the angular spread is small for both user and inter-ferers2 (i.e., φi � 1 for i = 0, 1, . . . ,M), the asso-ciated spatial correlation matrices {Ri}Mi=0, can be con-veniently approximated by enforcing a rank-1 constraint.For φi � 1, the following simplification holds in (11):

I0(√κ2 − ((2π/λ)Δpqφi cos(θi))2)/I0(κ) � 1. Therefore, the

spatial correlation matrices (12) can be approximated as (wedrop the functional dependency for simplicity of notation)

Ri � ρ · vivHi , (13)

2 Rank-1 approximation for the out-of-cell interferers is quite accuratewhen considering large reuse factors as the angular spread experiencedby the array is reduced by the increased distance of the out-of-cell inter-ferers.

where

vi(Δ, j) =[

1 exp[− jω

(θi)Δ12] · · · exp

[− jω(θi)Δ1N

]]T

(14)

and ωi(θ) = 2π/λ sin(θi).From (13), the channel vectors for user and interfer-

ers can be written as hi = γ√ρivi, where γ ∼ CN (0, 1).

Therefore, within the rank-1 approximation, the interferencepower reads (the additive noise contribution σ2IN has beendropped since it is immaterial for the optimization problem)

P1(Δ,η) = vH0

( M∑

i=1

ρivivHi

)v0, (15)

therefore, optimal spacings with respect to Problem 1 (4) canbe written as

Δ = arg minΔ

P1(Δ,η), (16)

where the subscript is a reminder of the rank-1 approxima-tion. The advantage of the rank-1 performance criterion (15)is that it allows to derive an explicit expression as a functionof the parameters of interest. In particular, after tedious butstraightforward algebra, we get

P1(Δ,η)

=M∑

i=1

[ρiN + ρi

L∑

j=1

4S(l j , θ0, θi

)+ ρi

C∑

k=1

2S(ck, θ0, θi

)]

,

(17)


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

51

48

45

42

39

36

Δ12 λ

SIRr2

(dB

)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 19

18

16

15

14

13

17

Δ12 λ

Erg

odic

capa

city

(bps

/Hz)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b)

Figure 6: Setting B: rank-2 approximation of the signal-to-interference ratio, SIRr2(Δ,η), (23) versus Δ12/λ and Δ23/λ (a) compared withergodic capacity C(Δ,η) (b) (r = 50 m). Circles denote optimal solutions.

where S(x, θn, θm) = cos[2πx(sin(θn)−sin(θm))], L =((

N2

)−

N/2)/2 is the number of “lateral spacings” li = Δi, j for

i = 1, . . . ,N/2−1, j = i+1, . . . ,N− i, C = N/2 is the numberof “central spacings” ci = Δi,N−i for i = 1, . . . ,N/2.

As a remark, notice that if there exists a set of antennaspacing Δ such that the user vector v0 is orthogonal to the Minterference vectors {vi}Mi=1, then this nulls the interferencepower, P1(Δ,η) = 0, and thus implies that Δ is a solution to(16) (and therefore to (4)).

4.2. Rank-a (a > 1) approximation

In a richer scattering environment, the conditions on the an-gular spread φi � 1 that justify the use of rank-1 approx-imation can not be considered to hold. Therefore, a rank-aapproximation with a > 1 should be employed (in general)for the spatial correlation matrix of both user and interferers:

Ri �a∑

k=1

ρ(k)i · v(k)

i v(k)Hi (18)

for i = 0, 1, . . . ,M. The set of vectors {v(k)i }ak=1 in (18) is re-

quired to be linearly independent. In this paper, we limit theanalysis to the case a = 2, which will be shown in Section 5to account for a wide range of practical environments. The

expression of vectors v(k)i from (11) with respect to the an-

tenna spacings is not trivial as for the rank-1 case. However,in analogy with (14), we could set

v(k)i =

[1 exp

[− jω(k)

i Δ12

]· · · exp

[− jω(k)

i Δ1N

]]T,

(19)

where the wavenumbers ωi = [ω(1)i ,ω(2)

i ] for user and inter-ferers have to be determined according to different criteria.In order to be consistent with the rank-1 case considered in

the previous section, here we minimize the Frobenius norm

of approximation error matrix ‖Ri −∑a

k=1 ρ(k)i · v(k)

i v(k)Hi ‖2

with respect to ω = [ω(1)i ,ω(2)

i ] vector and ρ = [ρ(1)i , ρ(2)

i ]vectors. For instance, for a uniform distribution of the scat-terers along the ring (i.e., κ = 0), it can be easily proved thatthe optimal rank-2 approximation (for i = 0, . . . ,M) resultsin

ω(1)i = ωi

(θi)

+ ϕi, ω(2)i = ωi

(θi)− ϕi, (20)

where ϕi = 2π/λ · φi cos(θi) and ρ(1)i = ρ(2)

i = ρi/2.As for the rank-1 case in (17), after some alge-

braic manipulations, the performance criterion P2(Δ,η) =Eh0 [hH0 Qh0] admits an explicit expression in terms of the pa-rameters of interest:

P2(Δ,η) =M∑

i=1

[ρiN + ρi

L∑

j=1

4S(l j , θ0, θi

) · Ti(l j)

+ ρi

C∑

k=1

2S(ck, θ0, θi

)Ti(ck)]

,

(21)

where Ti(x) = cos(ϕ0x) cos(ϕix); notice that in practicalenvironments, the angular spread for the in-cell user, ϕ0,is larger than the out-of-cell interferers angular spreads,ϕ1, . . . ,ϕM (see Section 5). Therefore, the optimization prob-lem (4) can be stated as

Δ = arg minΔ

P2(Δ,η). (22)

5. NUMERICAL RESULTS

In this section, numerical results related to the layouts inFigure 1 (N = 4, M = 3, F = 3 for setting A and F = 7for setting B with a cell diameter � = 2 km) are presented.Both the interference power minimization problems (4), (5)

S. Savazzi et al. 7

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 50

47

44

41

38

35

32

29

26

Δ12 λ

SIRr1

(dB

)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

18

17

16

15

14

13

12

Δ12 λ

Erg

odic

capa

city

(bps

/Hz)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b)

Figure 7: Setting-B: rank-1 approximation (a) of the signal-to-interference ratio SIRr1(Δ,η) versus Δ12/λ and Δ23/λ. Circular marker de-notes the optimal solution (24) obtained by the rank-1 approximation. As a reference, ergodic capacity is shown (b), for an angular spreadapproaching zero.

and the ergodic capacity optimization problems (8), (9) forProblems 1 and 2, respectively, are considered and com-pared for various propagation environments. For Problem1, user and interferers are located at the center of their re-spective allowed sectors (η, as in Figure 1), instead, for Prob-lem 2 average system performances are computed over the al-lowed positions (herein uniformly distributed) of users andinterferers.

Exploiting the rank-a-based approximation (rank-1 andrank-2 approximations in (17) and (21), resp.), the inter-ference power (for fixed user and interferers position η asfor Problem 1, or averaged over terminal positions as forProblem 2) is minimized with respect to the array spac-ings and the resulting optimal solutions are compared tothose obtained through maximization of ergodic capacity.Herein, we show that the proposed approach based on in-terference power minimization is reliable in evaluating theoptimal spacings that also maximize the ergodic capacityof the system. Since the number of antenna array is lim-ited to N = 4, ergodic capacity optimization can be car-ried out through an extensive search over the optimizationdomain.

The channels of user and interferers are assumed to becharacterized by the same scatterer radius ri = r (for therank-2 case) and r → 0 (for the rank-1 case) with κ = 0.Furthermore, the path loss exponent is α = 3.5. The signal-to-background noise ratio (for the ergodic capacity simu-lations) is set to Nρ0/σ2 = 20 dB. For the sake of visual-ization, the rank-a approximation of the interference poweris visualized (in dB scale) as the signal-to-interference ratio(SIR):

SIRra(Δ,η) =[

ρ0

Pa(Δ,η)

]

dB. (23)

5.1. Setting A (F = 3)

Assuming at first fixed position η for user and interfer-ers (Problem 1), Figure 4(b) shows the exact ergodic capac-ity C(Δ,η) for r = 50 m (and thus the angular spread isφ0 = 5.75 deg, φ1 = φ3 = 0.87 deg, φ2 = 0.82 deg) andFigure 4(a) shows the rank-2 SIR approximation SIRr2(Δ,η)(23) versus Δ12 and Δ23 for setting A. According to both op-timization criteria, the optimal array has external spacingΔ12 � 1.26 λ and internal spacing Δ23 � 3.6 λ. It is interestingto compare this result with the case of a line-of-sight channelthat is shown in Figure 5. In this latter scenario, the optimalspacings are easily found by solving the rank-1 approximateproblem (16) as (k = 0, 1, . . .)

Δ12 = (2k + 1)Ψ(θ1)

with any Δ23 ≥ 0 (24a)

or Δ12 + Δ23 = (2k + 1)Ψ(θ1), (24b)

where Ψ(θ1) = λ/(2 sin(θ1)) � 0.6 λ as θ1 = θ2 = 52 deg.Conditions (24) guarantee that the channel vector of the useris orthogonal to the channel vectors of the first and third in-terferers (the second is aligned so that mitigation of its inter-ference is not feasible). Moreover, the optimal spacings forthe line-of-sight scenario (24) form a grid (see Figure 5(a))that contains the optimal spacings for the previous case inFigure 4 with larger angular spread. Notice that, for everypractical purpose, the solutions to the ergodic capacity maxi-mization (Figure 5(b)) are well approximated by SIRr1(Δ,η)maximization in (23). As a remark, we might observe thatwith line-of-sight channels, there is no advantage of deploy-ing more than two antennas (Δ12 = 0 or Δ23 = 0 satisfythe optimality conditions (24)) to exploit the interferencereduction capability of the array. Instead, for larger angu-lar spread than the line-of-sight case, we can conclude that


14.5

16

17.5

19

20.5

22

23.5

25Eη[S

IRr 2

](d

B)

Δ12

λ= Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(a)

4

5.5

7

8.5

10

11.5

13

Erg

odic

capa

city

(bps

/Hz)

Δ12

λ= Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b)

Figure 8: Setting B: rank 2 approximation of the signal-to-interference ratio Eη[SIRr2(Δ,η)] (a) and ergodic capacity Eη[C(Δ,η)] (b) aver-aged with respect to the position of user and interferers within the corresponding sectors for Δ12 = Δ23.

(i) large enough spacings have to be preferred to accommo-date diversity; (ii) contrary to the line-of-sight case, there isgreat advantage of deploying more than two antennas (ap-proximately 5-6 bit/s/Hz) whereas the benefits of deployingmore than three antennas are not as relevant (0.6 bit/s/Hzfor an optimally designed three-element array with uniformspacing 3.6 λ); (iii) compared to the λ/2-uniformly spaced ar-ray, optimizing the antenna spacings leads to a performancegain of approximately 2.5 bit/s/Hz.

Let us now turn to the solution of Problem 2 (9). In thiscase, the optimal set of spacings Δ should guarantee the bestperformance on average with respect to the positions of userand interferers within the corresponding sectors. It turns outthat the optimal spacings are Δ12 = Δ23 � 1.9 λ for both op-timization criteria (not shown here), and the (average) per-formance gain with respect to the conventional adaptive ar-rays with Δ12 = Δ23 = λ/2 has decreased to approximately0.5 bit/s/Hz. This conclusion is substantially different for sce-nario B as discussed below.

5.2. Setting B (F = 7)

For Problem 1, the exact ergodic capacity C(Δ,η) for r =50 m (and angular spread φ0 = 5.75 deg, φ1 = 0.34 deg,φ2 = 0.56 deg, and φ3 = 0.58 deg) and the rank-2 approx-imation SIRr2(Δ,η) (23) are shown versus Δ12 and Δ23, inFigure 6, for setting B. In this case, the optimal linear min-imum length array consists, as obtained by both optimiza-tion criteria, by uniform 2.2 λ spaced antennas. Optimal de-sign of linear minimal length array leads to a 2.5 bit/s/Hzcapacity gain with respect to the capacity achieved throughan array provided with four uniformly λ/2 spaced antennas.Similarly as before, we compare this result with the case of aline-of-sight channel (Figure 7(a)), where the optimal spac-ings, solution to the rank-1 approximate problem (16), are

Δ12 = Ψ(θ3) � 0.7 λ (external spacing) and Δ23 = 3Ψ(θ3) �2.2 λ (internal spacing) θ3 = 43.5 deg. In this case, the solu-tions (confirmed by the ergodic capacity maximization, seeFigure 7(b)) guarantee that the channel vector of the user isorthogonal to the channel vector of the third (predominant)interferer (the second is almost aligned so that mitigation ofits interference is not feasible, the first one has a minor im-pact on the overall performances). As pointed out before,a larger angular spread than the line-of-sight case require-larger spacings to exploit diversity.

As for Problem 2 (9), in Figure 8, we compare the ana-lytical rank-2 approximation Eη[SIRr2(Δ,η)] averaged overthe position of users and interferers with the exact aver-aged ergodic capacity for a uniform-spaced antenna array.The minimal length optimal solutions turn out again to beΔ12 = Δ23 � 2.2 λ. Moreover, we can conclude that in thisinterference layout the capacity gain with respect to the ca-pacity achieved through an array provided with four uni-formly λ/2 spaced antennas is 2.5 bit/s/Hz.

5.3. Impact of nonequal power interferingdue to shadowing effects

In this section, we investigate the impact of nonequal in-terfering powers caused by shadowing on the optimal an-tenna spacings. This amounts to include in the spatial cor-relation model (12) a log-normal variable for both user andinterferers as ρi = (K/di

α) · 10Gi/10 and Gi ∼ N (0, σ2Gi

) fori = 0, 1, . . . ,M. All shadowing variables {Gi}Mi=0 affect receiv-ing power levels and are assumed to be independent. Figure 9shows the ergodic capacity averaged over the shadowing pro-cesses for setting B and r = 50 m (as in Figure 6), when thestandard deviation of the fading processes are σG0 = 3 dB forthe user (e.g., as for imperfect power control) and σGi = 8 dB(i = 1, . . . ,M) for the interferers. By comparing Figure 9 with

S. Savazzi et al. 9

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

18

17

16

15

14

13

Δ12 λ

Erg

odic

capa

city

(bps

/Hz)

Δ23

λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 9: Setting B: ergodic capacity C(Δ,η) averaged with respectto the distribution of shadowing (r = 50 m). Circle denotes the op-timal solution.

Figure 6, we see that the overall effect of shadowing is that ofreducing the ergodic capacity but not to modify the optimalantenna spacings; similar results can be attained by analyzingthe interference power (not shown here).

6. CONCLUSION

In this paper, we tackled the problem of optimal design oflinear arrays in a cellular systems under the assumption ofGaussian interference. Two design problems are considered:maximization of the ergodic capacity (through numericalsimulations) and minimization of the interference power atthe output of the matched filter (by developing a closed formapproximation of the performance criterion), for fixed andvariable positions of user and interferers. The optimal ar-ray deployments obtained according to the two criteria areshown via numerical optimization to coincide for the con-sidered scenarios. The analysis has been validated by studyingtwo scenarios modelling cellular systems with different reusefactors. It is concluded that the advantages of an optimizedantenna array as compared to a standard design depend onboth the interference layout (i.e., reuse factor) and the prop-agation environment. For instance, for an hexagonal cellularsystem with reuse factor 7, the gain can be on average as highas 2.5 bit/s/Hz. As a final remark, it should be highlightedthat optimizing the antenna array spacings in such a way toimprove the quality of communication (by minimizing theinterference power) may render the antenna array unsuitablefor other applications where some features of the propaga-tion are of interest, such as localization of transmitters basedon the estimation of direction of arrivals.

REFERENCES

[1] J. H. Winters, J. Salz, and R. D. Gitlin, “The impact of antennadiversity on the capacity of wireless communication systems,”

IEEE Transactions on Communications, vol. 42, no. 234, pp.1740–1751, 1994.


[3] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmitbeamforming and power control for cellular wireless systems,”IEEE Journal on Selected Areas in Communications, vol. 16,no. 8, pp. 1437–1449, 1998.

[4] J. Saholos, “A solution of the general nonuniformly spaced an-tenna array,” Proceedings of the IEEE, vol. 62, no. 9, pp. 1292–1294, 1974.

[5] J.-W. Liang and A. J. Paulraj, “On optimizing base station an-tenna array topology for coverage extension in cellular radionetworks,” in Proceedings of the IEEE 45th Vehicular TechnologyConference (VTC ’95), vol. 2, pp. 866–870, Chicago, Ill, USA,July 1995.

[6] R. Jana and S. Dey, “3G wireless capacity optimization forwidely spaced antenna arrays,” IEEE Personal Communica-tions, vol. 7, no. 6, pp. 32–35, 2000.

[7] A. Abdi and M. Kaveh, “A space-time correlation model formultielement antenna systems in mobile fading channels,”IEEE Journal on Selected Areas in Communications, vol. 20,no. 3, pp. 550–560, 2002.

[8] P. Zetterberg, “On Base Station antenna array structures fordownlink capacity enhancement in cellular mobile radio,”Tech. Rep. IR-S3-SB-9622, Department of Signals, Sensors& Systems Signal Processing, Royal Institute of Technology,Stockholm, Sweden, August 1996.

[9] H. L. Van Trees, Optimum Array Processing, Wiley-Intersci-ence, New York, NY, USA, 2002.

[10] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J.H. Reed, “Overview of spatial channel models for antenna ar-ray communication systems,” IEEE Personal Communications,vol. 5, no. 1, pp. 10–22, 1998.

[11] T. Fulghum and K. Molnar, “The Jakes fading model incorpo-rating angular spread for a disk of scatterers,” in Proceedingsof the 48th IEEE Vehicular Technology Conference (VTC ’98),vol. 1, pp. 489–493, Ottawa, Ont, Canada, May 1998.

[12] I. Gupta and A. Ksienski, “Effect of mutual coupling on theperformance of adaptive arrays,” IEEE Transactions on Anten-nas and Propagation, vol. 31, no. 5, pp. 785–791, 1983.

[13] N. Maleki, E. Karami, and M. Shiva, “Optimization of antennaarray structures in mobile handsets,” IEEE Transactions on Ve-hicular Technology, vol. 54, no. 4, pp. 1346–1351, 2005.


Research ArticleCapacity Performance of Adaptive Receive Antenna SubarrayFormation for MIMO Systems

Panagiotis Theofilakos and Athanasios G. Kanatas

Wireless Communications Laboratory, Department of Technology Education and Digital Systems, University of Piraeus,80 Karaoli & Dimitriou Street, 18534 Piraeus, Greece

Received 15 November 2006; Accepted 1 August 2007

Recommended by R. W. Heath Jr.

Antenna subarray formation is a novel RF preprocessing technique that reduces the hardware complexity of MIMO systems whilealleviating the performance degradations of conventional antenna selection schemes. With this method, each RF chain is not allo-cated to a single antenna element, but instead to the complex-weighted and combined response of a subarray of elements. In thispaper, we derive tight upper bounds on the ergodic capacity of the proposed technique for Rayleigh i.i.d. channels. Furthermore,we study the capacity performance of an analytical algorithm based on a Frobenius norm criterion when applied to both Rayleighi.i.d. and measured MIMO channels.

Copyright © 2007 P. Theofilakos and A. G. Kanatas. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. INTRODUCTION

The interest in multiple-input multiple-output (MIMO) an-tenna systems has exploded over the last years because oftheir potential of achieving remarkably high spectral effi-ciency. However, their practical application has been limitedby the increased manufacture cost and energy consumptionof the RF chains (performing the frequency transition be-tween microwave and baseband) and analog-to-digital con-verters, the number of which is proportional to the numberof antenna elements.

This high degree of hardware complexity has motivatedthe introduction of antenna selection schemes, which judi-ciously choose a subset from all the available antenna ele-ments for processing and thus decrease the number of nec-essary RF chains. Both analytical [1–11] and stochastic [12]algorithms for antenna selection have been proposed. How-ever, when a limited number of frequency converters areavailable, antenna selection schemes suffer from severe per-formance degradations in most fading channels.

In order to alleviate the performance degradations ofconventional antenna selection, antenna subarray formation(ASF) has been recently introduced [13]. With this method,each RF chain is not allocated to a single antenna element,but instead to a combined and complex-weighted response ofa subarray of antenna elements. Even though additional RF

switches (for selecting the antenna elements that participatein each subarray), variable RF phase shifters, or/and variablegain-linear amplifiers (performing the complex-weighting)are required with respect to antenna selection schemes,the proposed method achieves decreased receiver hard-ware complexity, since less frequency converters and analog-to-digital converters are required with respect to the fullsystem.

Antenna subarray formation actually performs a lineartransformation in the RF domain in order to reduce thenumber of necessary RF chains while taking advantage ofthe responses of all antenna elements. Since it is a linear pre-processing technique that can be generally applied jointly toboth receiver and transmitter, antenna subarray formationcan be viewed as a special case of linear precoder-decoderjoint designs [14–19]. Indeed, the fundamental mathemat-ical models for both techniques are exactly the same; how-ever, in conventional linear precoding-decoding schemes,preprocessing is performed in the baseband by digital sig-nal processors that are not subject to the practical con-straints and hardware nonidealities imposed by the RF com-ponents (namely the number of available RF chains, variablephase shifters, or/and variable gain-linear amplifiers) andthus no restrictions on the structure of the preprocessing ma-trices are required. Instead of decoupling the MIMO chan-nel into independent subchannels (eigenmodes), ASF aims


at constructing subchannels (namely, subarrays) that are asmutually independent as possible and deliver the largestreceive power gain, under the aforementioned constraints.Note that an RF preprocessing technique for reducing hard-ware costs has also been introduced in [20], but withoutgrouping antenna elements into subarrays.

Initially, antenna subarray formation was introducedwith the restriction that each antenna element participatesin one subarray only. For this special case of ASF, the prob-lem of selecting the elements and the weights for the subar-ray formation has been addressed in [13], where an evolu-tionary optimization technique is used. In [21], we have in-troduced an analytical algorithm based on a Frobenius normcriterion. Recognizing that cost-effective analog amplifiers inRF with satisfactory noise figure are practically unavailable,we have also suggested a phase-shift-only design of the tech-nique [22]. Taking into consideration that the performanceof ASF may be adversely affected by hardware nonidealities,such as insertion loss, calibration, and phase-shifting errors(which are not an issue in conventional precoder-decoderschemes), we have presented simulation results in [23] thatindicate the robustness of ASF to such nonidealities.

In this paper, we elaborate on the capacity performanceof ASF and the Frobenius-norm-based algorithm. In partic-ular, we derive a theoretical upper bound on the ergodic ca-pacity of the technique for Rayleigh i.i.d. channels. Moreover,we demonstrate the performance of the technique and the al-gorithm through extensive computer simulations and appli-cation to measured channels.

The rest of the paper is organized as follows: Section 2 ex-plains the proposed technique and its mathematical formu-lation in more detail, provides capacity calculations for theresulted system and introduces some special ASF schemes. InSection 3, tight theoretical upper bounds on the ergodic ca-pacity of the technique are derived. Section 4 presents an an-alytical algorithm for ASF and its extensions for several ASFschemes. The capacity performance of the technique and theproposed algorithm is demonstrated in Section 5 through ex-tensive computer simulations. Finally, the paper is concludedwith a summary of results.

2. THE ANTENNA SUBARRAYFORMATION TECHNIQUE

In this section, we first present the antenna subarray for-mation technique and its mathematical formulation. After-wards, we provide capacity calculations for the resulted sys-tem. Finally, some special schemes of ASF are introduced,which are dependent on the number of phase shifters or/andvariable gain-linear amplifiers available at the receiver.

2.1. MIMO system model

Consider a flat fading, spatial multiplexing MIMO systemwith MT elements at the transmitter and MR > MT elementsat the receiver. Unless otherwise stated, the MR×MT channeltransfer matrix H is assumed to be perfectly known to thereceiver, but unknown to the transmitter.

In spatial multiplexing systems, independent data streamsare transmitted simultaneously by each antenna. Thereceived vector for MR receive elements is given by

y = Hs + n, (1)

where n is the zero-mean circularly symmetric complexGaussian noise vector with covariance matrix Rn = N0IMR

and s is the transmitted vector. Assuming that the total trans-mitter power is P, the covariance matrix for the transmittedvector is constrained as

tr{E[

ssH]} = P, (2)

and the intended average signal-to-noise ratio per antenna atthe receiver is

ρ = P

N0. (3)

2.2. General mathematical formulation ofantenna subarray formation

Antenna Subarray Formation can be applied with any num-ber of RF chains available at the receiver. However, withoutloss of generality, we assume that the receiver is equippedwith exactly MT RF chains. This assumption is frequentlymade in antenna selection literature and is justified by thewell-known fact that, when the number of receiving RFchains becomes larger than the number of transmit anten-nas, the number of parallel spatial data pipes that can beopened is constrained by the number of transmit antennas.Thus, the receiver RF chains in excess cannot be exploited toincrease the throughput, but can only offer increased diver-sity order [24]. This assumption is meaningful when the fullsystem channel matrix is of full column rank.

The process of subarray formation, complex weightingand combining at the receiver is linear and thus can be ade-quately described by the transformation matrix A. In partic-ular, the received vector after antenna subarray formation y isfound by left multiplying the received vector for MR antennaelements with AH, that is,

y = AHy. (4)

Thus, the response of the jth subarray y j (i.e., the jthentry of y) is

y j = αHj y =

MR∑

i=1

a∗i j yi, (5)

where α j denotes the jth column of A. Clearly, the responseof the jth subarray y j is a linear combination of the responsesof theMR receiving antenna elements and the conjugated en-tries of α j are the corresponding complex weights. Thus, (4)is an adequate mathematical formulation of the subarray for-mation process, provided that we furthermore enforce thefollowing restriction on the entries of A:

ai j = 0, if i�∈ S j , (6)

P. Theofilakos and A. G. Kanatas 3

Tx ...

MTantennaelements

Mobile radiochannel

H

MRantennaelements ...

AH

y

...N

RF chains

ρN

ρ2

ρ1

y = AHy

Figure 1: System model of receive antenna subarray formation.

with S j denoting the set of receive antenna element indicesthat participate in the jth subarray.

Throughout this paper we assume that the transforma-tion matrix A is adapted to the instantaneous channel state.Thus, we should have written A(H), denoting the depen-dence on the full system channel matrix H. However, to fa-cilitate notation, we just write A which henceforth impliesA(H).

By substituting (1) into (4), the received vector after sub-array formation becomes

y = AHHs + AHn. (7)

Apparently, the combined effect of the propagation chan-nel and the receive antenna subarrays on the transmitted sig-nal is described by the effective channel matrix

H = AHH. (8)

The effective noise component in (7) is

n = AHn, (9)

which is zero-mean circularly symmetric complex Gaussianvector (ZMCSCGV) [25] with covariance matrix:

Rnn = E[

nnH] = N0AHA. (10)

The block model of the resulted system is displayed inFigure 1.

2.3. Capacity of receive antenna subarray formation

Depending on the time-variation of the channel, there aredifferent quantities that characterize the capacity of theresulted system. In this paragraph we apply well-knowninformation-theoretic results for MIMO systems to RASFsystems and elaborate the capacity of the proposed techniquewhen different assumptions for channel-time variation aremade.

2.3.1. Deterministic capacity

Deterministic capacity is a meaningful quantity when thestatic channel model is adopted, which implies that the chan-nel matrix, despite being random, once chosen it is held fixed

for the whole transmission. In this case, the Shannon capac-ity of RASF is given in terms of mutual information betweenthe transmitter vector s and the received vector after subarrayformation y as

CRASF = maxp(s)

tr(Rs)=P

I(

s; y) = max

p(s)

[H(

y |H)−H(y | s, H

)],

(11)

where H(x) is the entropy of x, p(s) denotes the distributionof s and tr(Rs) = P is the power constraint on the transmit-ter. Recognizing that the transmitted symbols are indepen-dent from noise, assuming that s is ZMCSCGV [25, 26] andtaking into account that n∼NC(0,N0AHA), we find that

CRASF = maxp(s)

tr(Rs)=P

I(

s; y)

= log2 det(πeRy)− log2det

(πeN0AHA

),

(12)

where Ry = E[yyH] = AHHRsHHA + N0AHA is the covari-ance matrix of y. After some mathematical manipulations,(12) becomes

CRASF= maxRs

tr(Rs)=Plog2 det

[IMT +

1N0

RsHHA(

AHA)−1

AHH]. (13)

Since the transmitter does not know the channel and tak-ing into account the power constraint, it is reasonable to as-sume that

Rs = P

MTIMT . (14)

Thus, the Shannon capacity of receive antenna subarrayformation with equal power allocation at the transmitter is

CRASF = log2 det[

IMT +ρ

MTHHA(

AHA)−1

AHH]. (15)

The capacity of the resulted system is upper bounded bythe capacity of the full system, that is

CRASF ≤ CFS = log2 det(

IMR +ρ

MTHHH). (16)

Proof of this result is given in Appendix A.

2.3.2. Ergodic capacity

In time-varying channels with no delay constraints, ergodiccapacity is a meaningful quantity, defined as the probabilisticaverage of the static channel capacity over the distribution ofthe channel matrix H. The ergodic capacity for RASF is givenby

CRASF = EH

[log2 det

(IMT +

ρ

MTHHA(

AHA)−1

AHH)].

(17)


ρ

ρ

... ......

ρ

MRantennaelements

AH

Mu

ltip

lexe

r

arg(αMR )|αMR |

arg(α2)|α2|

|α1| arg(α1)

Linear combining

MR vgLNAs andphase shifters

ρN

ρ2

ρ1

... N

RF chains

(a)

ρ

......

...

...

...ρ

Mu

ltip

lexe

rMRantennaelements

AH

K < MRMT vgLNAsand phase shifters

ρN

ρ2

ρ1

NRF chains

|αMR ,N |arg(αMR ,N )

|αMR ,2|arg(αMR ,2)

|αMR ,1|arg(αMR ,1)

|α1N |arg(α1N )

|α12|arg(α12)

|α11|arg(α11)

Linear combining

(b)

ρ

ρ

......

...

...

Mu

ltip

lexe

r

MRantennaelements

AH

− arg (αMR ,N )

− arg (αMR ,2)

− arg (αMR ,1)

− arg (α1N )

− arg (α12)

− arg (α11)

Linear combining

ρN

ρ2

ρ1

... NRF chains

K < MRMT

phase shifters

(c)

Figure 2: Receiver structures for several receive antenna subarray formation (ASF) schemes: (a) strictly-structured ASF (SS-ASF), (b)relaxed-structured ASF (RS-ASF) and (c) reduced hardware complexity ASF (RHC-ASF).

2.3.3. Outage capacity

Outage capacity is a meaningful quantity in slowly varyingchannels. Assuming a fixed transmission rate R, there is anassociated probability Pout (bounded away from zero) thatthe received data will not be received correctly, or equiva-lently that mutual information will be less than transmissionrate R. Outage capacity for RASF is therefore defined as

CRASF = R : Pr{

log2 det(

IMT +ρ

MTHHA(

AHA)−1

AHH)<R}

= Pout.(18)

2.4. Receive antenna subarray formation schemes

In general, no more constraints on the transformation ma-trix A are required. However, depending on the number ofavailable phase shifters or/and variable gain-linear amplifiers(which determine the number of its nonzero entries), fur-ther restrictions on matrix A may be necessary. Motivatedby these practical considerations, we have introduced severalvariations of antenna subarray formation [22], namely, thefollowing.

(1) Strictly-Structured ASF (SS-ASF), in which each an-tenna element is allowed to participate in one subar-ray only. Thus, each row of the transformation matrixA may contain only one nonzero element, whereas norestriction is enforced on the columns of A. With thisscheme, exactly MR phase shifters and variable gain-linear amplifiers are required at the receiver.

(2) Relaxed-Structured ASF (RS-ASF), where no restric-tions on matrix A are imposed, except for the num-ber of its nonzero entries, which is a fixed system de-sign parameter that determines the number of phaseshifters and variable gain-linear amplifiers available tothe receiver.

(3) Reduced Hardware Complexity ASF (RHC-ASF), whichis a phase-shift-only design of the technique. Whilecost-effective variable gain-linear amplifiers with sat-isfactory noise figure are not practically available, theeconomic design and manufacture of variable phase-shifters for the microwave frequency is feasible due tothe rapid advances in MMIC technology. Therefore,this scheme reduces even further the hardware com-plexity of the receiver with negligible capacity loss, asit will be demonstrated in Section 5.

An efficient algorithm for determining the transforma-tion matrix A for all the aforementioned schemes will be pre-sented in detail in Section 4. Figure 2 presents the receiver ar-chitecture for each of the ASF schemes.

3. AN UPPER BOUND ON THE ERGODICCAPACITY OF ANTENNA SUBARRAY FORMATIONFOR I.I.D. RAYLEIGH CHANNELS

In this section, we derive an upper bound on the ergodic ca-pacity of the technique for i.i.d. Rayleigh fading channels, thetightness of which will be verified by extensive computer sim-ulations in Section 5.


A well-known upper bound on the (deterministic) capac-ity of the full system is given by

CFS ≤MT∑

i=1

log2

(1 +

ρ

MTγi

), (19)

where γi are independent chi-squared variates with 2MR de-grees of freedom. The equality holds in the “very artificialcase” when the transmitted signal vector components “areconveyed over MT “channels” that are uncoupled and eachchannel has a separate set of MR receive antennas” [27].In other words, when the full MIMO system is consistedof MT separable and independent parallel SIMO systems,each performing maximum ratio combining (MRC) at thereceiver.

In our case, we consider as well that the resulted systemis consisted of MT separable and independent parallel SIMOsystems. We suppose that the jth SIMO system is formed bythe jth transmit antenna element and the jth receive subar-ray; thus, for each subarray, only one signal component is re-ceived and processed without any interference from the oth-ers. Of course, this scheme is practically infeasible; however,it must lead to an upper bound of the resulted system capac-ity.

A subarray corresponds to an independent SIMO systemand is actually formed by choosing a subset of antenna el-ements, the responses of which are linearly combined andfed to an RF chain. Thus, generalized selection combining(i.e., combining the responses of a subset of antenna ele-ments) is performed in each SIMO system. The maximumSNR (which also achieves maximum capacity) in this caseis obtained with the hybrid selection maximum ratio com-bining scheme (HS/MRC). Furthermore, in this section, weassume that each subarray is formed using a predefined andfixed number of antenna elements (let it be k j antenna ele-ments for the jth subarray). Therefore, a capacity bound forantenna subarray formation can be obtained by

Cbound =MT∑

j=1

log2

(1 + ξ j

). (20)

Assuming that there are no delay constraints, the channelis ergodic and therefore it is meaningful to derive an upperbound on ergodic capacity as

Cbound =MT∑

j=1

E[

log2

(1 + ξ j

)]. (21)

The expectation in (21) can be found [28] by

c j∧= E[

log2

(1 + ξ j

)]=∫ ∞

0log2(1 + ξ)·pξ j (ξ)dξ. (22)

Since ξ j is actually the postprocessing SNR of HS/MRCwhen k j out of MR elements are chosen, its probability den-sity function is [29]

pξ j (ξ) =(MR

kj

)[(MT

ρ

)kj ξkj−1e−(MT/ρ)ξ(kj − 1

)!

+MT

ρ

MR−kj∑

l=1

(−1)kj+l−1

(MR − kj

l

)

×(kjl

)kj−1

e−(MT/ρ)ξ

×(

e−(MTl/ρkj )ξ−kj−2∑

m=0

1m!

(− l·MT

ρ·kjξ)m)]

.

(23)

Substituting (23) into (22) and defining the integral

In(x)∧=∫ ∞

0tn−1 ln(1 + t)e−xtdt x > 0; n = 1, 2, . . . ,

(24)

we get

c j= 1ln 2

(MR

kj

)[(MT

ρ

)kj Ikj(MT/ρ

)

(kj − 1

)!

+MT

ρ

MR−kj∑

l=1

(−1)kj+l−1

(MR−kj

l

)(kjl

)kj−1

×[

I1

(MT

ρ

{1 +

l

k j

})−

kj−2∑

m=0

1m!

×(− l·MT

ρ·kj

)mIm+1(MT/ρ

)]]

,

(25)

which, in fact, is the average channel capacity achieved whenemploying HS/MRC in a SIMO system with MR receiving an-tenna elements and k j branches.

The integral In(x) can be evaluated by [30]

In(x) = (n− 1)!·ex·n∑

q=1

Γ(−n + q, x)xq

, (26)

which for n = 1 reduces to

I1(x) = exE1(x)x

. (27)

Note that E1(x) is the exponential integral of first-orderfunction defined by

E1(x) =∫ ∞

x

e−t

tdt (28)

and Γ(α, x) is the complementary incomplete gamma func-tion (or Prym’s function) defined as

Γ(α, x) =∫ ∞

xtα−1e−tdt. (29)


For q positive integer, Γ(−q, x) can be calculated by

Γ(−q, x) = (−1)n

n!

[

E1(x)− e−xq−1∑

m=0

(−1)mm!xm+1

]

. (30)

Thus, the ergodic capacity bound for receive antennasubarray formation can be analytically obtained by

Cbound = 1ln 2

MT∑

j=1

(MR

kj

)

×[(

MT

ρ

)kj Ikj(MT/ρ

)

(kj − 1

)!

+MT

ρ

MR−kj∑

l=1

(−1)kj+l−1

×(MR − kj

l

)(kjl

)kj−1

×[

I1

(MT

ρ

{1 +

l

k j

})−

kj−2∑

m=0

1m!

×(− l·MT

ρ·kj

)mIm+1(MT/ρ

)]]

.

(31)

A simpler expression than (25) can be derived by rec-ognizing that log2(·) is a concave function and applyingJensen’s inequality to (21),

c j=E[

log2

(1 + ξ j

)]≤ log2

(1 + E[ξ j]). (32)

It is known for HS/MRC [29] that

E[ξ j] = ρ

MTkj

(

1 +MR∑

l=kj+1

1l

)

. (33)

Thus, (21) becomes

Cbound ≤MT∑

j=1

log 2

[

1 +ρ

MTkj

(

1 +MR∑

l=kj+1

1l

)]

, (34)

which has a much simpler form than (31) while being almostas tight as computer simulations have demonstrated.

Before concluding this section, we note that analyzing theresulted system into parallel SIMO systems each perform-ing HS/MRC results into capacity bounds of RS-ASF, sinceHS/MRC requires both phase shifters and variable gain am-plifiers. Capacity bounds for RHC-ASF could be derived ina similar manner by considering MT parallel SIMO systemseach performing HS/EGC. Since HS/MRC delivers the bestperformance amongst all hybrid selection schemes, the up-per bound on the ergodic capacity of RS-ASF is also an upperbound on the ergodic capacity of any ASF scheme, includingRHC-ASF.

4. ALGORITHM FOR ANTENNASUBARRAY FORMATION

In this section, we present a novel, analytical algorithm forreceive antenna subarray formation, based on a Frobenius

norm criterion. We first develop the algorithm for SS-ASFand then provide extensions for RS-ASF and RHC-ASF. Thecapacity performance of the algorithms will be demonstratedin Section 5.

4.1. Starting point for the algorithm

The starting point for determining the transformation ma-trix A will be an optimal solution to the unconstrained prob-lem of maximizing the deterministic capacity in (15). Asshown in Appendix A, (15) can be maximized when Ao= U,where the columns of U are the MT dominant left singularvectors of the full channel matrix H. Therefore, the entries ofthe transformation matrix A will be

ai j ={ui j if i ∈ S j

0 otherwise,(35)

with ui j being the (i, j) entry of matrix U. Alternatively,

A = SU, (36)

where denotes the Hadamard (elementwise) matrix prod-uct and the entries of S are

si j ={

1 i ∈ S j

0 otherwise.(37)

4.2. Frobenius norm based algorithm for SS-ASF

We first develop an algorithm for SS-ASF and afterwards ex-tend it for other receive ASF schemes. Due to the additionalconstraints of SS-ASF, the capacity of the resulted system isgiven by

CRASF = log2 det(

IMT +ρ

MTHHAAHH

)

= log2 det(

IMT +ρ

MTHHH).

(38)

In order to retain the capacity calculations to the in-tended system SNR measured at the output of every receiverantenna element, A is now subject to the following normal-ization:

AHA = IMT . (39)

Intuitively, the desired transformation matrix A shouldbe such that the distance between the two subspaces definedby Hopt = UHH (i.e., the effective channel matrix obtainedfrom the optimal solution to the unconstrained problem)and H = AHH is minimized. As a result, we employ the fol-lowing minimum distance distortion metric:

ε(A) =∥∥∥Hopt − H

∥∥∥

2

F=∥∥∥(U− A)HH

∥∥∥

2

F. (40)

Defining E∧= U−A and F

∧= EHH, (40) can be written as

ε(A) = ‖F‖2F =

N∑

j=1

(MT∑

i=1

∣∣ f ji∣∣2)

=MT∑

j=1

∥∥f j∥∥2

, (41)


Table 1: Frobenius-norm-based algorithm for RASF.

Algorithm stepsComplexity(K , MR, MT , and H are given)

(In case of SS-ASF, K :=MR)

Obtain the SVD of full system channel matrix H. H = UΣVH O(12MTM

2R + 9M3

R

)

Compute the decision metrics gi j that willdetermine if the ith antenna element willparticipate in the jth subarray.

For i: = 1 to MR

O(M2

TMR

)For j: = 1 to MT

gi j := U(i, j)·‖H(i, :)‖2

end

end

Initialize with every ai j = 0 and all S j empty.S j := ∅ (∀ j =1, . . . ,MT)

S j : set of indices of antenna elements that partic-ipate in the jth subarray.

A := 0MR×MT ; n: = 0

Repeat the following until matrix A is filled withK nonzero elements:

While n < K

O(KMRMT

)(i) let(i0,j0)

be the indices of the largest gi jelement over 1 ≤ i ≤ MR and 1 ≤ j ≤ MT ,provided that ai j = 0;

(i0, j0) = arg max

(i, j)ai j=0

(gi j)

Sj0 := Sj0 ∪ {i0}for SS-ASF only, i�∈

⋃

j

S j ; A(i0, j0)

:= U(i0, j0)

(ii) set ai0 j0 = ui0 j0 , that is, the i0th antennaelement participates in the j0th subarray;

n: = n + 1

end

for SS-ASF only, normalize A so thatFor SS-ASF only:

AHA = IMT .

For j = 1:MT

A(:, j) := A(:, j)/‖A(:, j)‖end

where f j denotes the jth row of F, being equal to f j = eHj H,

and e j is the jth column of matrix E.Recognizing that the ith row of matrix F can be written as

a linear combination of the rows hi of the full system channelmatrix H and taking into account that

ei j∧= ui j − ai j =

{ui j i�∈ S j

0 i ∈ S j ,(42)

the distortion metric becomes

ε(A)=MT∑

j=1

∥∥∥∥∥

∑

i∈Sj

e∗i jhi

∥∥∥∥∥

2

=MT∑

j=1

∥∥∥∥∥

∑

i�∈Sj

u∗i jhi

∥∥∥∥∥

2

≤MT∑

j=1

∑

i�∈Sj

∣∣ui j∣∣2∥∥hi∥∥2

,

(43)

where the upper bound on the right-hand side follows fromthe triangular inequality. As a result, the objective is to mini-mize the upper bound on the distortion metric in (43).

Since the selection of the elements of the transformationmatrix A is based on matrix U, it is trivial to conclude thatminimizing the upper bound in (43) is equivalent to maxi-mizing

p =MT∑

j=1

∑

i∈Sj

∣∣ui j∣∣2∥∥hi∥∥2

, (44)

which upper-bounds the power of the effective channel ma-

trix ‖H‖2F. Indeed, after mathematical manipulations similar

to those in (41)–(43), it follows that

∥∥H∥∥2

F =MT∑

j=1

∥∥∥∥∥

∑

i∈Sj

u∗i jhi

∥∥∥∥∥

2

≤MT∑

j=1

∑

i∈Sj

∣∣ui j∣∣2∥∥hi∥∥2 = p, (45)

where h j denotes the jth row of H and α j is the jth column ofmatrix A. Consequently, minimizing an upper bound on theminimum distance distortion metric is equivalent to maxi-mizing an upper bound on the power of the effective channelmatrix. The latter may not be the optimal way to maximizecapacity in spatial multiplexing systems, but it should resultinto an increased capacity performance, since it is knownthat [24]

CSS-ASF ≥ log2 det(

1 +ρ

MT

∥∥H∥∥2

F

). (46)

The proposed algorithm appoints the receiver antenna el-ements to the appropriate subarray, so that the metric (44)is maximized. Finally, A is normalized as in (39). Table 1presents the algorithm steps in more detail.


4.3. Extension of the algorithm for RS-ASF

The capacity of RS-ASF given by (15) is lower bounded bythe capacity formula (38) for SS-ASF, that is,

CRS-ASF ≥ log2 det(

IMT +ρ

MTHHAAHH

). (47)

Proof of this result and indications for the tightness ofthe bound are provided in Appendix B.

Thus, in the case of RS-ASF we also use the Frobeniusnorm based algorithm initially developed for SS-ASF. The al-gorithm terminates when the transformation matrix A con-tains exactly K nonzero elements, where K < MRMT is a sys-tem design parameter that determines the number of vari-able gain-linear amplifiers and phase shifters available to thereceiver.

The computational complexity of the proposed algo-rithm (see Table 1) is dominated by the initial cost of the sin-gular value decomposition, that is, O(M3

R) when MR � MT ,whereas the complexity of Gorokhov et al. algorithm [4] andof the alternative implementation proposed in [5] for an-tenna selection is O(M2

TM2R) and O(M2

TMR), respectively.

4.4. Extention of the algorithm for RHC-ASF

The transformation matrix�A for RHC-ASF (a phase-shift-

only design of antenna subarray formation) can be obtainedfrom the transformation matrix A for RS-ASF by applyingthe following formula to its entries:

�ai j=⎧⎨

⎩exp(− j | ai j

)if i ∈ S j

0 otherwise.(48)

Intuitively, RHC-ASF follows the notion of equal gaincombining. A similar procedure for obtaining a phase-shift-only RF preprocessing technique has been followed in [20].

5. SIMULATION RESULTS

In this section, we present extensive computer simulation re-sults that demonstrate the capacity performance of receiveASF technique, the tightness of the ergodic capacity boundsderived in Section 3, and the performance of the proposedalgorithm.

5.1. Upper bound on ergodic capacity for ASF

We first deal with the ergodic capacity bounds of ASF forRayleigh i.i.d. channels derived in Section 3, namely, (31)and (34). Henceforth, we refer to (34) as “simpler theoreticalcapacity bound,” in order to distinguish it from (31). We con-sider a flat-fading Rayleigh i.i.d. MIMO channel withMR = 8receiving and MT = 2 transmitting antenna elements and as-sume that the receiver is equipped with N = MT = 2 RFchains.

Figure 3 presents the ergodic capacity bounds of RS-ASFover a wide range of SNRs when K = 8 variable gain-linearamplifiers and phase shifters are available at the receiver and

4

6

8

10

12

14

16

18

20

22

Erg

odic

capa

city

(bps

/Hz)

5 10 15 20 25

Average SNR (dB)

Exhaustive search ASFFull system (exact capacity)Antenna selection (exact capacity)Theoretical capacity bound of ASF (34)Theoretical capacity bound of full system (34)Simpler theoretical capacity bound for ASF (37)

Full system (8× 2)

Antenna selection

ASF

Figure 3: Ergodic capacity bounds for ASF and capacity of exhaus-tive search ASF when MR = 8, MT = 2, and K = 8 variable gain-linear amplifiers and phase shifters are available at the receiver (4antenna elements in each subarray). Results are compared to an er-godic capacity bound and exact ergodic capacity of the full system.

exactly k∧= K/N = 4 receiving antenna elements partici-

pate in each subarray. For purposes of reference, the ergodiccapacity of the exhaustive search solution of RS-ASF is alsoshown. The exhaustive search solution is obtained by consid-

ering all the(MR

k

)Npossible combinations of subarray for-

mation, that is, all possible combinations for the structure ofmatrix S as defined in (37), assuming that A is obtained as in(36). Apparently, both capacity bounds are very tight to theexhaustive search solution.

When each subarray contains MR antenna elements, thecapacity bound of the MIMO system is found by analyzing itinto MT parallel SIMO systems. Each of these parallel systemsreduces to a MRC diversity system and therefore the ergodiccapacity bound of the full system will be obtained by (31).This observation is verified in Figure 3.

5.2. Frobenius-norm-based algorithm

In this paragraph we demonstrate the capacity performanceof the Frobenius-norm-based algorithm for various schemesof receive ASF in terms of outage capacity (when the slowly-varying block fading channel model is adopted) and ergodiccapacity (when the channel is assumed ergodic). The pro-posed algorithm is applied to both Rayleigh i.i.d. and mea-sured MIMO channels.

5.2.1. Rayleigh i.i.d. channels

We consider Rayleigh i.i.d. MIMO channels with MT = 2elements at the transmitter and assume that the receiver is


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b.(c

apac

ity>

absc

issa

)

9 10 11 12 13 14 15

Capacity (bps/Hz)

Antenna selectionFrobenius norm based algorithm for RASF (K = 8)Exhaustive search RASF (K = 8)Full system (8× 2)

Figure 4: Empirical complementary cdf of the capacity of theresulted system when the Frobenius-norm-based algorithm forstrictly structured receive antenna subarray formation (SS-ASF) isapplied to a 8×2 Rayleigh i.i.d. channel with SNR = 15 dB. The per-formance of the algorithm is compared with the exhaustive searchsolution for SS-ASF, the full system (8 × 2), and Gorokhov et al.decremental algorithm for antenna selection.

equipped with MT = 8 elements, N = MT = 2 RF chains,and K = 8 phase shifters or/and variable gain-linear ampli-fiers.

Figure 4 presents the complementary cdf of the capacityof the resulted system for SS-ASF when the SNR is at 15 dB.Clearly, SS-ASF outperforms Gorokhov et al. algorithm forantenna selection [4], which is quasi optimal in terms of ca-pacity performance. Moreover, the performance of the pro-posed algorithm is very close to the exhaustive search solu-tion. Thus, the SS-ASF technique delivers a significant capac-ity increase with respect to conventional antenna selectionschemes. The same results are verified in Figure 5, where theergodic capacity of the resulted system over a wide range ofSNRs is plotted.

5.2.2. Measured channel

In order to examine the performance in realistic conditions,we have applied the proposed algorithm to measured MIMOchannel transfer matrices. Measurements were conducted us-ing a vector channel sounder operating at the center fre-quency of 5.2 GHz with 120 MHz measurement bandwidthin short-range outdoor environments with LOS propagationconditions. A more detailed description of the measurementsetup can be found in [31]. The transmitter has MT = 4equally spaced antenna elements and the receiver is equippedwith MR = 16 receiving elements and N = MT = 4 RFchains. The interelement distance for both the transmittingand receiving antenna arrays is d = 0, 4λ.

4

6

8

10

12

14

16

18

20

22

Erg

odic

capa

city

(bps

/Hz)

5 10 15 20 25

Average SNR (dB)

Exhaustive search RASFFrobenius norm based algorithm for SS-ASFFull system (8× 2)Antenna selection

Figure 5: Performance evaluation of strictly structured ASF (SS-ASF) applied to an 8× 2 MIMO Rayleigh i.i.d. channel, in terms ofergodic capacity. The performance of the algorithm is compared tothe exhaustive search solution for receive ASF, the full system (8×2),and Gorokhov et al. decremental algorithm for antenna selection.

Figure 6 displays the complementary cdf of the capacityof the resulted system when the Frobenius-norm-based al-gorithm is applied to several schemes of receive ASF and forvarious values of K (i.e., the number of phase shifters or/andvariable gain-linear amplifiers). Clearly, all ASF schemes out-perform conventional antenna selection.

Solid black lines correspond to RS-ASF (or SS-ASF forK =MR = 16) and dashed black lines to RHC-ASF. Compar-ing the solid with the dashed lines for the same value of K , itis evident that RHC-ASF delivers capacity performance veryclose to RS-ASF. Therefore, the expensive variable gain-linearamplifiers can be abolished from the design of ASF with neg-ligible capacity loss.

For K = 48, the capacity performance of RS-ASF andRHC-ASF is very close to the full system, despite the fact thatin ASF the receiver is equipped with only N = MT = 4 RFchains (whereas the full system hasMR = 16 RF chains). Evenwhen K = 32, the capacity loss with respect to the full sys-tem is still quite low (10% outage capacity loss of RHC-ASFis less than 1.5 bps/Hz at 15 dB). Similar results are observedfor a wide range of signal-to-noise ratios (Figure 7). Conse-quently, the proposed algorithm can deliver near-optimal ca-pacity performance with respect to the full system while re-ducing drastically the number of necessary RF chains.

6. CONCLUSIONS

In this paper, we have developed a tight theoretical up-per bound on the ergodic capacity of antenna subarrayformation and have presented an analytical algorithm for


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b.(c

apac

ity>

absc

issa

)

11 12 13 14 15 16 17 18 19 20 21

Capacity (bps/Hz)

SS-ASF(K = 16)

RS-ASF(K = 16)

Antenna selection

Full system(16× 4)

RS-ASF(K = 48)

RHC-ASF(K = 48)

RS-ASF(K = 32)

RHC-ASF(K = 32)

Figure 6: Empirical complementary cdf of the capacity of the re-sulted system when the Frobenius-norm-based algorithm for sev-eral schemes of receive antenna subarray formation (ASF) is ap-plied to a 16 × 4 measured channel with SNR = 15 dB. In particu-lar, the RASF schemes studied are strictly structured ASF (SS-ASF),relaxed-structured ASF (RS-ASF), and reduced hardware complex-ity ASF (RHC-ASF). K denotes the number of phase shifters or/andvariable gain-linear amplifiers available to the receiver. The perfor-mance of the algorithm is compared to the full system (16× 4) andGorokhov et al. decremental algorithm for antenna selection.

5

10

15

20

25

30

35

Erg

odic

capa

city

(bps

/Hz)

5 10 15 20 25

Average SNR (dB)

Antenna selection

ASF (K = 16)

ASF (K = 32)

Full system (16× 4)

Figure 7: Performance evaluation of Frobenius-norm-based algo-rithm for several schemes of receive antenna subarray formation(RASF) applied to a 16×4 MIMO measured channel, in terms of er-godic capacity. In particular, the RASF schemes studied are strictlystructured ASF (SS-ASF), relaxed-structured ASF (RS-ASF) (solidlines), and reduced hardware complexity ASF (RHC-ASF) (dottedlines). K denotes the number of phase shifters or/and variable gain-linear amplifiers available to the receiver. The performance of thealgorithm is compared to the full system (16× 4) and Gorokhov etal. decremental algorithm for antenna selection.

adaptively grouping receive array elements to subarrays. Ap-plication in Rayleigh i.i.d. and measured channels demon-strates significant capacity performance, which can becomenear optimal with respect to the full system, depending on

the number of available phase shifters or/and variable gain-linear amplifiers. Furthermore, it has been shown that aphase-shift-only design of the technique is feasible with neg-ligible performance penalty. Thus, it has been establishedthat antenna subarray formation is a promising RF prepro-cessing technique that reduces hardware costs while achiev-ing incredible performance enhancement with respect toconventional antenna selection schemes.

APPENDICES

A.

Let A = UAΣAVHA be a singular value decomposition [32] of

matrix A. We get

A(

AHA)−1

AH = UAΣAVHA

(VAΣ

2AVH

A

)−1VAΣAUH

A

= UAΣAVHA VAΣ

-2A VH

A VAΣAUHA

= UAUHA .

(A.1)

Thus, the capacity formula in (15) becomes

CRASF = log2 det(

IMT +ρ

MTHHUAUH

A H). (A.2)

Applying the known formula for determinants [32]

det (I + AB) = det (I + BA) (A.3)

to (A.2), we get

CRASF = log2 det(

IMT +ρ

MTUH

A HHHUA

)(A.4)

which can be written as

CRASF=MT∑

m=1

log2

(1 +

ρ

MTλm(

UHA HHHUA

)), (A.5)

where λm(X) denotes the mth eigenvalue of square matrix Xin descending order. Poincare separation theorem [32] statesthat

λm(

UHA HHHUA

) ≤ λm(

HHH) (A.6)

with equality occurring when the columns of UA are the MT

dominant left singular vectors of H. Thus,

CRASF ≤MT∑

k=1

log2

(1 +

ρ

MTλk(

HHH))

= log2 det(

IMR +ρ

MTHHH)= CFS,

(A.7)


where equality occurs when

UA =[

u1 u2 · · · uMT

](A.8)

and uk is the kth dominant singular vector of H. Therefore,an optimal solution to the unconstrained (i.e., without thesubarray formation constraints in (6) capacity maximizationproblem is

Ao =[

u1 u2 · · · uMT

]Q, (A.9)

where Q = ΣAVHA is a matrix with orthogonal rows and

columns.

B.

Let A = UAΣAVHA be a singular value decomposition of the

transformation matrix A. Exploiting Hadarmard’s inequal-ity for determinants [32] and after some trivial mathematicalmanipulations, it follows that

det(Σ2

A

) = det(

VAΣ2AVH

A

) = det(

AHA) ≤

MT∏

k=1

[AHA]kk

=MT∏

k=1

aHk ak =

MT∏

k=1

∥∥ak∥∥2 ≤ 1,

(B.1)

where ak denotes the kth column of the transforma-tion matrix A. The last inequality in (B.1) follows from‖ak‖ ≤ ‖uk‖ = 1, with uk being the kth left singular vectorof the full system channel matrix, and it is justified by thefact that the entries of matrix A are obtained as in (35).

In the high SNR regime, after substituting forA = UA

∑AVH

A and taking into account (B.1), it is validto write

det(

IMT +ρ

MTHHAAHH

)≈det(ρ

MTHHUAΣ

2AUH

A H)

=det(Σ2

A

)det(ρ

MTHHUAUH

A H)

≤det(ρ

MTHHUAUH

A H).

(B.2)

Recognizing that the right-hand side of (B.2) is an ap-proximation of (A.2), that is, the capacity of the RASF sys-tem, in the high SNR regime, the validity of the bound in(47) is proven.

Note that the same approximation for the capacity ofMIMO systems at high SNR has been widely used (see, e.g.,[24]). Simulation results in Figure 8 demonstrate that thebound is quite tight.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

b.(c

apac

ity>

absc

issa

)

16 18 20 22 24 26 28 30

Capacity (bps/Hz)

Capacity bound for RS-ASF with K = 32RS-ASF using K = 32 phase shifters and VGAsCapacity bound for RS-ASF with K = 48RS-ASF using K = 48 phase shifters and VGAs

5.6%1.2%

Figure 8: Comparison between capacity bound (47) for relaxedstructured ASF and true capacity (15) of the resulted system interms of empirical complementary cdf, when applied to a 16 × 4MIMO Rayleigh i.i.d. channel with SNR = 15 dB. Proof of thisbound can be found in Appendix B.

ACKNOWLEDGMENT

This work has been partially funded by Antenna Centre ofExcellence (ACE2) research programme, under the EU 6thFramework Programme.

REFERENCES

[1] D. A. Gore, R. U. Nabar, and A. J. Paulraj, “Selecting an opti-mal set of transmit antennas for a low rank matrix channel,”in Proceedings of IEEE Interntional Conference on Acoustics,Speech, and Signal Processing (ICASSP ’00), vol. 5, pp. 2785–2788, Istanbul, Turkey, June 2000.

[2] R. S. Blum and J. H. Winters, “On optimum MIMO with an-tenna selection,” IEEE Communications Letters, vol. 6, no. 8,pp. 322–324, 2002.

[3] A. F. Molisch, M. Z. Win, Y.-S. Choi, and J. H. Winters, “Ca-pacity of MIMO systems with antenna selection,” IEEE Trans-actions on Wireless Communications, vol. 4, no. 4, pp. 1759–1772, 2005.

[4] A. Gorokhov, D. A. Gore, and A. J. Paulraj, “Receive antennaselection for MIMO spatial multiplexing: theory and algo-rithms,” IEEE Transactions on Signal Processing, vol. 51, no. 11,pp. 2796–2807, 2003.

[5] M. Gharavi-Alkhansari and A. B. Gershman, “Fast antennasubset selection in MIMO systems,” IEEE Transactions on Sig-nal Processing, vol. 52, no. 2, pp. 339–347, 2004.

[6] D. A. Gore and A. J. Paulraj, “MIMO antenna subset selectionwith space-time coding,” IEEE Transactions on Signal Process-ing, vol. 50, no. 10, pp. 2580–2588, 2002.

[7] R. W. Heath Jr., S. Sandhu, and A. J. Paulraj, “Antenna se-lection for spatial multiplexing systems with linear receivers,”IEEE Communications Letters, vol. 5, no. 4, pp. 142–144, 2001.


[8] D. A. Gore, R. W. Heath Jr., and A. J. Paulraj, “Transmit se-lection in spatial multiplexing systems,” IEEE CommunicationsLetters, vol. 6, no. 11, pp. 491–493, 2002.

[9] M. A. Jensen and M. L. Morris, “Efficient capacity-based an-tenna selection for MIMO Systems,” IEEE Transactions on Ve-hicular Technology, vol. 54, no. 1, pp. 110–116, 2005.

[10] A. F. Molisch, M. Z. Win, and J. H. Winter, “Reduced-complexity transmit/receive-diversity systems,” IEEE Transac-tions on Signal Processing, vol. 51, no. 11, pp. 2729–2738, 2003.

[11] L. Dai, S. Sfar, and K. B. Letaief, “Receive antenna selection forMIMO systems in correlated channels,” in Proceedings of theIEEE International Conference on Communications (ICC ’04),vol. 5, pp. 2944–2948, Paris, France, June 2004.

[12] P. D. Karamalis, N. D. Skentos, and A. G. Kanatas, “Selectingarray configurations for MIMO systems: an evolutionary com-putation approach,” IEEE Transactions on Wireless Communi-cations, vol. 3, no. 6, pp. 1994–1998, 2004.

[13] P. D. Karamalis, N. D. Skentos, and A. G. Kanatas, “Adaptiveantenna subarray formation for MIMO systems,” IEEE Trans-actions on Wireless Communications, vol. 5, no. 11, pp. 2977–2982, 2006.

[14] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding forwireless communication,” IEEE Transactions on Communica-tions, vol. 46, no. 3, pp. 357–366, 1998.

[15] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundantfilterbank precoders and equalizers—I: unification and opti-mal designs,” IEEE Transactions on Signal Processing, vol. 47,no. 7, pp. 1988–2006, 1999.

[16] H. Sampath, P. Stoica, and A. J. Paulraj, “Generalized linearprecoder and decoder design for MIMO channels using theweighted MMSE criterion,” IEEE Transactions on Communi-cations, vol. 49, no. 12, pp. 2198–2206, 2001.

[17] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, andH. Sampath, “Optimal designs for space-time linear precodersand decoders,” IEEE Transactions on Signal Processing, vol. 50,no. 5, pp. 1051–1064, 2002.

[18] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rxbeamforming design for multicarrier MIMO channels: a uni-fied framework for convex optimization,” IEEE Transactions onSignal Processing, vol. 51, no. 9, pp. 2381–2401, 2003.

[19] C. Mun, J.-K. Han, and D.-H. Kim, “Quantized principal com-ponent selection precoding for limited feedback spatial multi-plexing,” in Proceedings of the IEEE International Conference onCommunications (ICC ’06), pp. 4149–4154, Istanbul, Turkey,June 2006.

[20] X. Zhang, A. F. Molisch, and S.-Y. Kung, “Variable-phase-shift-based RF-baseband codesign for MIMO antenna selection,”IEEE Transactions on Signal Processing, vol. 53, no. 11, pp.4091–4103, 2005.

[21] P. Theofilakos and A. G. Kanatas, “Frobenius norm based re-ceive antenna subarray formation for MIMO systems,” in Pro-ceedings of the1st European Conference on Antennas and Propa-gation (EuCAP ’06), vol. 626, Nice, France, November 2006.

[22] P. Theofilakos and A. G. Kanatas, “Reduced hardware com-plexity receive antenna subarray formation for MIMO systemsbased on frobenius norm criterion,” in Proceedings of the 3rdInternational Symposium on Wireless Communication Systems(ISWCS ’06), Valencia, Spain, September 2006.

[23] P. Theofilakos and A. G. Kanatas, “Robustness of receiveantenna subarray formation to hardware and signal non-idealities,” in Proceedings of the 65th IEEE Vehicular Technol-ogy Conference (VTC ’07), pp. 324–328, Dublin, Ireland, April2007.

[24] O. Oyman, R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Char-acterizing the statistical properties of mutual informationin MIMO channels,” IEEE Transactions on Signal Processing,vol. 51, no. 11, pp. 2784–2795, 2003.

[25] F. D. Neeser and J. L. Massey, “Proper complex random pro-cesses with applications to information theory,” IEEE Trans-actions on Information Theory, vol. 39, no. 4, pp. 1293–1302,1993.

[26] T. M. Cover and J. A. Thomas, Elements of Information Theory,John Wiley & Sons, New York, NY, USA, 1991.


[28] A. Papoulis and S. U. Pillai, Probability, Random Variablesand Stochastic Processes, McGraw-Hill, New York, NY, USA,4th edition, 2002.

[29] M. K. Simon and M.-S. Alouini, Digital Communication overFading Channels, John Wiley & Sons, New York, NY, USA,1st edition, 2000.

[30] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleighfading channels under different adaptive transmission anddiversity-combining techniques,” IEEE Transactions on Vehic-ular Technology, vol. 48, no. 4, pp. 1165–1181, 1999.

[31] N. D. Skentos, A. G. Kanatas, P. I. Dallas, and P. Constantinou,“MIMO channel characterization for short range fixed wire-less propagation environments,” Wireless Personal Communi-cations, vol. 36, no. 4, pp. 339–361, 2006.

[32] R. A. Horn and C. R. Johnson, Matrix Analysis, CambridgeUniversity Press, Cambridge, UK, 1985.


Research ArticleCapacity of MIMO-OFDM with Pilot-Aided Channel Estimation

Ivan Cosovic and Gunther Auer

DoCoMo Euro-Labs, Landsberger Straße 312, 80687 Munchen, Germany

Received 31 October 2006; Revised 9 July 2007; Accepted 4 October 2007


An analytical framework is established to dimension the pilot grid for MIMO-OFDM operating in time-variant frequency selec-tive channels. The optimum placement of pilot symbols in terms of overhead and power allocation is identified that maximizesthe training-based capacity for MIMO-OFDM schemes without channel knowledge at the transmitter. For pilot-aided channelestimation (PACE) with perfect interpolation, we show that the maximum capacity is achieved by placing pilots with maximumequidistant spacing given by the sampling theorem, if pilots are appropriately boosted. Allowing for realizable and possibly sub-optimum estimators where interpolation is not perfect, we present a semianalytical method which finds the best pilot allocationstrategy for the particular estimator.

Copyright © 2007 I. Cosovic and G. Auer. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Systems employing multiple transmit and receive antennas,known as multiple-input multiple-output (MIMO) systems,promise significant gains in channel capacity [1–3]. Togetherwith orthogonal frequency division multiplexing (OFDM),MIMO-OFDM is selected for the wireless local area network(WLAN) standard IEEE 802.11n [4], and for beyond 3rdgeneration (B3G) mobile communication systems [5].

As multiple signals are transmitted from different trans-mit antennas simultaneously, coherent detection requires ac-curate channel estimates of all transmit antennas’ signals atthe receiver. The most common technique to obtain chan-nel state information is via pilot-aided channel estimation(PACE) where known training symbols termed pilots aremultiplexed with data. For PACE, channel estimates are ex-clusively generated by means of pilot symbols, and these es-timates are then processed for the detection of data sym-bols as if they were the true channel response. A sophisti-cated pilot design should strike a balance between the at-tainable accuracy of the channel estimate and the resourcesconsumed by pilot symbols. An appropriate means to op-timize this trade-off is to maximize the channel capacity ofpilot-aided schemes. As the training overhead grows propor-tionally to the number of transmitted spatial streams [6], theattainable capacity gains for MIMO-OFDM are traded withthe bandwidth and energy consumed by a growing numberof pilot symbols to estimate the MIMO channels.

A lower bound for the attainable capacity for multipleantenna systems with pilot-aided channel estimation for theblock fading channel was derived in [7]. The capacity lowerbound was then used to optimize the energy allocation andthe fraction of resources consumed by pilots. The capac-ity lower bound of [7] was extended to single carrier sys-tems operating in frequency-selective channels [8–10], andto spatially correlated MIMO channels [11]. Furthermore,in [8, 12] the capacity achieving pilot design for MIMO-OFDM over frequency-selective channels was studied. ForOFDM, pilot symbols inserted in the frequency domain (be-fore OFDM modulation) sample the channel, allowing to re-cover the channel response for data bearing subcarriers bymeans of interpolation. This implies that, besides pilot over-head and power allocation, also the placement of pilots isto be optimized. It was found that equidistant placement ofpilot symbols not only minimizes the mean squared error(MSE) of the channel estimates [13], but also maximizes thecapacity [8].

All previous work deriving the capacity for training-based schemes for single-carrier [7–11] as well as for multi-carrier systems [8, 12] considered time-invariant channels,where the channel was assumed static for the block of trans-mitted symbols. Furthermore, work on OFDM was limitedto perfect interpolation [8, 12]. That is, additive white Gaus-sian noise (AWGN) is the only source of channel estimationerrors. This implies that, in the absence of noise channel es-timates perfectly match the true channel response. However,


Tx Rx

Pilot

Data

Pilot

Data

Pilot

Data

OFDMmod

OFDMmod

OFDMmod

1

2

NT

...

...... ...

......

OFDMdemod

OFDMdemod

OFDMdemod

1

2

NR

Ch

ann

eles

tim

atio

n&

dete

ctio

n

Figure 1: MIMO-OFDM system employing NT transmit and NR receive antennas.

from the sampling theorem it is well known that an infinitenumber of pilots is necessary for perfect interpolation [14].In the context of OFDM, perfect interpolation is observedonly if the channel model is sample spaced [15], that is, allchannel taps are integer multiples of the sampling duration.In practice, however, pilot sequences are finite while chan-nel taps are nonsample spaced, ultimately leading to reducedcapacity bounds. Whereas in related work for OFDM onlytime-invariant channels and perfect interpolation are consid-ered, in this work real world problems are taken into account,and a more realistic capacity bound is achieved.

Furthermore, previous work on maximizing the trainingbased capacity was exclusively dedicated to minimum meansquared error (MMSE) channel estimation [7–12]. Appliedto PACE the MMSE criterion finds the best tradeoff betweenthe attainable interpolation accuracy and the mitigation ofnoise [16]. While MMSE estimates are favorable in termsof performance, knowledge of the 2nd order statistics is re-quired for implementation, together with a computationallyexpensive matrix inversion [16]. Unlike previous work, ourobjective is not to find the optimum estimator that achievescapacity, rather we aim to identify the optimum pilot designthat maximizes capacity for a given and possibly subopti-mum estimator.

The present paper addresses the above mentioned limi-tations of previous work; its main contributions are summa-rized as follows.

(i) The work from [7], conducted for block fading chan-nels, is extended to time-variant frequency-selectivechannels. Assuming perfect interpolation and placingpilot symbols with maximum possible distance in timeand frequency still satisfying the sampling theorem,the capacity is shown to approach that for the blockfading channel, provided that the size of the block ischosen according to the maximum pilot spacing im-posed by the sampling theorem.

(ii) Previous work in [7, 8, 12] is extended to arbitrarylinear estimators. Capacity-achieving pilot design forrealizable, and possibly suboptimum, channel estima-tion schemes is therefore possible, as for example,interpolation by finite impulse response (FIR) filter-ing [16–18], discrete Fourier transform (DFT) based

interpolation [19, 20], or linear interpolation [21].We derive a closed form expression of the trainingbased-capacity for perfect interpolation, and propose asemianalytical procedure for practical estimation tech-niques.

(iii) For a particular class of estimators, namely FIR inter-polation filters, we demonstrate that the pilot grid thatmaximizes capacity is mostly independent of the cho-sen channel model, as long as the maximum channeldelay and the maximum Doppler frequency are withina certain range. This is an appealing property, as a so-phisticated pilot design should be valid for an as wideas possible range of channel conditions.

The remainder of this paper is structured as follows. InSection 2, the system model is introduced. In Section 3, theestimation error model is established and analyzed, whereasbounds on the achievable capacity of the optimized pi-lot design are derived in Section 4. Numerical examples inSection 5 verify the developed framework in terms of pilotboost and overhead, as well as number of transmit antennas.

2. SYSTEM MODEL

Consider a MIMO-OFDM system with NT transmit and NR

receive antennas as illustrated in Figure 1. We assume that NT

spatial streams are transmitted and that channel knowledge isnot available at the transmitter. Denote with Nc the numberof used subcarriers, and with L the number of OFDM sym-bols per frame. OFDM modulation is performed by NDFT-point (NDFT ≥ Nc) inverse DFT (IDFT), followed by inser-tion of a cyclic prefix (CP) of NCP samples. Assuming perfectorthogonality in time and frequency, the received signal ofsubcarrier n of the �th OFDM symbol block and νth receiveantenna is given by

Y (ν)n,� =

NT∑

μ=1

√Ed

NTX

(μ)n,� H

(μ,ν)n,� + Z(ν)

n,� ,

0 ≤ n < Nc, 0 ≤ � < L, 0 ≤ ν < NR.

(1)

In (1), X(μ)n,� , H

(μ,ν)n,� , and Z(ν)

n,� , denote the normalized transmit-

ted symbol over transmit antenna μ with E{|X (μ)n,� |2} = 1, the

I. Cosovic and G. Auer 3

channel transfer function (CTF) between transmit antenna μand receive antenna ν, and AWGN at the νth receive antennawith zero mean and variance N0, respectively. An energy pertransmitted data symbol of Ed/NT and a normalized average

channel gain, E{|H(μ,ν)n,� |2} = σ

2H = 1 is assumed.

The discrete CTF H(μ,ν)n,� , is obtained by sampling H(μ,ν)(f ,

t) at frequency f = n/T and time t = �Tsym, where Tsym

= (Nc + NCP)Tspl and T = NcTspl represent the OFDMsymbol duration with and without the cyclic prefix, andTspl is the sample duration. Considering a frequency selec-tive time-variant channel, modeled by a tapped delay linewith Q0 nonzero taps with channel impulse response (CIR),

h(μ,ν)(τ, t)=∑ Q0q=1h

(μ,ν)q (t)δ(t−τ(μ,ν)

q ), the CTF is described by

H(μ,ν)n,� = H(μ,ν)

(n

T, �Tsym

)=

Q0∑

q=1

h(μ,ν)q,� exp

(− j2πτ

(μ,ν)q

n

T

),

(2)

where h(μ,ν)q,� = h

(μ,ν)q (�Tsym) denotes the complex valued

channel tap q, assumed to be constant over one OFDM sym-

bol block, with associated tap delay τ(μ,ν)q . Intersymbol inter-

ference is avoided by ensuring that NCPTspl ≥ τmax, whereτmax denotes the maximum delay of the CIR. Then an arbi-

trary CIR is supported for which all channel taps h(μ,ν)q,� are

contained within the range 0≤τ(μ,ν)q ≤τmax, and the received

signal is given by (1).The 2nd-order statistics are determined by the two-

dimensional (2D) correlation function R(μ,ν)[Δn,Δ�] =E{H(μ,ν)

n,� (H(μ,ν)n+Δn,�+Δ�

)∗}, composed of two independent cor-relation functions in frequency and time, R(μ,ν)[Δn,Δ�] =R

(μ,ν)f [Δn]R

(μ,ν)t [Δ�]. Both R

(μ,ν)f [Δn] = E{H(μ,ν)

n,� (H(μ,ν)n+Δn,�)

∗}and R

(μ,ν)t [Δ�] = E{H(μ,ν)

n,� (H(μ,ν)n,�+Δ�

)∗} are strictly band-limi-

ted [18]. That is, the inverse Fourier transform of R(μ,ν)f [Δn]

described by the power delay profile is essentially nonzeroin the range [0, τmax], where τmax is the maximum channel

delay. Likewise, the Fourier transform of R(μ,ν)t [Δ�] describ-

ing time variations due to mobile velocities is given by theDoppler power spectrum, nonzero within [− fD,max, fD,max],where fD,max is the maximum Doppler frequency. No fur-ther assumptions regarding the distribution of h(μ,ν)(τ, t) areimposed. To this end, the CIR may possibly be nonsample

spaced, that is, tap delays τ(μ,ν)q in (2) may not be placed at

integer multiples of the sampling duration.In order to recover the transmitted information, pilot

symbols are commonly used for channel estimation. Chan-nel estimation schemes for MIMO-OFDM based on the leastsquares (LS) and MMSE criterion are studied in [22, 23]and [6, 24, 25], respectively. We assume that channel stateinformation about all NT × NR channels is required at thereceiver. To enable this, pilots belonging to different trans-mit antennas are orthogonally separated in time and/or fre-quency. Thus, the problem of MIMO-OFDM channel esti-mation breaks down to estimating the channel of a singleantenna OFDM system. Note, there are other possibilitiesto orthogonally separate the pilots, but they lead to higher

complexity and/or at least the same pilot overhead [6]. Fur-thermore, pilots belonging to the same transmit antennaare equidistantly spaced in time and frequency within theOFDM frame [17]. This is motivated by the findings in [8],where it is shown that equidistant placement of pilots min-imizes the harmonic mean of the MSE of channel estimatesover all subcarriers and thus maximizes the capacity. Figure 2illustrates the resulting placement of pilots for four spatialstreams, arranged in a rectangular shaped pattern. Extensionto other regular pilot patters, such as a diamond shaped grid[26], is straightforward.

Resources are constraint to bandwidth and energy. Un-like the orthogonally separated pilots, data symbols are spa-

tially multiplexed. One frame is assigned N(μ)p pilot and Nd

data symbols per spatial dimension which amounts to (cf.,Figure 2)

NcL = Nd +NT∑

μ=1

N(μ)p . (3)

The resulting pilot overhead of the μth antenna is defined by

Ω(μ)p = N

(μ)p

NcL. (4)

With an energy per transmitted data symbol of Ed/NT, thetotal transmit energy over all NT antennas equals

Etot = EdNd +NT∑

μ=1

E(μ)p N

(μ)p , (5)

where E(μ)p is the energy per pilot symbol of the μth transmit

antenna.

The accuracy of the channel estimates may be improved

by a pilot boost S(μ)p . With an energy per transmitted pilot set

to E(μ)p = S

(μ)p Ed, the signal-to-noise ratio (SNR) at the input

of the channel estimation unit is improved by a factor of S(μ)p .

On the other hand, the useful transmit energy of the payloadinformation is reduced, if the overall transmit energy in (5) iskept constant. The energy dedicated to pilot symbols is deter-

mined by the pilot overhead per antenna Ω(μ)p , and the pilot

boost per antenna S(μ)p . Including the pilot overhead, the ra-

tio of the energy per symbol Ed of a system with pilots, to theenergy per symbol E0 of an equivalent system with the sameframe size Nc × L, same transmit energy Etot, but without pi-lot symbols is

Ed

E0= 1

1 +∑ NT

μ=1 Ω(μ)p

(S

(μ)p − 1

) . (6)

The ratio Ed/E0 is a measure for the pilot insertion loss rela-tive to a reference system assuming no overhead due to pilots.Note, (6) is obtained exploiting (4), (5), and the constraintEtot = NcLE0.

In the following, we assume that for each transmit an-tenna the same number of pilot symbols and the same boost-

ing level are used, that is, Np = N(μ)p , Ωp=Ω(μ)

p , and Sp=S(μ)p ,


Antenna 1

6sy

mbo

ls

10 subcarriers

(a)

Antenna 2

(b)

Antenna 3

DataNullPilot

(c)

Antenna 4

(d)

Figure 2: Example for placing orthogonal pilots over antenna-specific OFDM subframes, NT = 4, N (1)p = N (2)

p = N (3)p = N (4)

p = 6, andNd = 36.

μ={1, . . . ,NT}. Now, the total pilot overhead in (4) amountsto NTΩp and (6) simplifies to

Ed

E0= 1

1 + NTΩp(Sp − 1

) . (7)

3. ESTIMATION ERROR MODELING

The channel estimation unit outputs an estimate of the CTF,

H(μ,ν)n,� , denoted by H

(μ,ν)n,� = wH y

(μ,ν). Let Mf and Mt denotethe number of pilot symbols in frequency and time used to

generate H(μ,ν)n,� . The MtMf × 1 column vector y

(μ,ν) containsthe received pilots from transmit antenna μ to receive an-tenna ν. The MtMf × 1 column vector w represents an ar-bitrary linear estimator.

3.1. Parametrization of the MSE

Channel estimation impairments are quantified by the MSE

of the estimation error ε(μ,ν)n,� = H

(μ,ν)n,� − H

(μ,ν)n,� . With the as-

sumption that all transmit and receive antennas are mutuallyuncorrelated, the MSE is independent of μ and ν, denoted

by σ2ε [n, �] = E[|ε(μ,ν)

n,� |2]. The MSE of an arbitrary 2D pilot-aided scheme is given by

σ2ε [n, �] = E

[∣∣ε(μ,ν)n,�

∣∣2]= E

[∣∣H(μ,ν)n,� − H

(μ,ν)n,�

∣∣2]

= E[∣∣H(μ,ν)

n,�

∣∣2]− 2R

{wHr

(μ,ν)yH [n, �]

}+ wHR

(μ,ν)y y w.

(8)

The 2D correlation functions r(μ,ν)yH [n, �] = E{ y (μ,ν)(H

(μ,ν)n,� )∗}

and R(μ,ν)y y =E{y (μ,ν)( y (μ,ν))H} represent the cross-correlation

between y(μ,ν) and the desired response H

(μ,ν)n,� , and the auto-

correlation matrix of the received pilots, y (μ,ν), respectively,

[16]. The autocorrelation matrix is composed of R(μ,ν)y y =

R(μ,ν)

hh+ I/γp, where Rhh = E{h(μ,ν)

(h(μ,ν)

)H} is the autocor-relation matrix of the CTF at pilot positions excluding theAWGN term, and I denotes the identity matrix, all of di-mension MfMt ×MfMt. With the pilot insertion loss of (7),the SNR at pilot positions amounts to γp = SpEd/N0 =γ0Sp/(1 +NTΩp(Sp− 1)), where γ0 = E0/N0 denotes the SNRof a reference system assuming perfect channel knowledgeand no overhead due to pilots.

The MSE in (8) is dependent on n and �. In order to allowfor a tractable model, we choose to average the MSE over theentire sequence, so σ2

ε[n, �]→σ2ε.

The channel estimates generated by a linear estimator wcan be decomposed into a signal and noise part, denoted by

H(μ,ν)n,� = wH h

(μ,ν)+ wH z

(ν), where h(μ,ν)

and z(ν) account for

CTF and AWGN vectors at pilot positions. Likewise, the es-

timation error ε(μ,ν)n,� can be separated into an interpolation

error H(μ,ν)n,� − wH h

(μ,ν)and a noise error wH z

(ν). Assuming

that CTF H(μ,ν)n,� and AWGN Z(ν)

n,� are uncorrelated, the MSEalso separates into a noise and interpolation error:

σ2ε = E

[∣∣H(μ,ν)n,� −wH h

(μ,ν) −wH z(ν)∣∣2

]

= E[|H(μ,ν)

n,� −wH h(μ,ν)∣∣2

]+ E[∣∣wH z

(ν)∣∣2].

(9)


We note that this separation of the MSE is possible for any

linear estimator. The noise part σ2n = E[|wH z

(ν)|2] is in-versely proportional to the SNR and is given by

σ2n =

wHwγp

= 1Gnγ0

·1 + NTΩp(Sp − 1

)

Sp, (10)

where Gn = 1/(wHw) defines the estimator gain. Accordingto (8) the variance of the interpolation error is determinedby

σ2i = E

[∣∣H(μ,ν)n,� −wH h

(μ,ν)∣∣2]

= E[∣∣H(μ,ν)

n,�

∣∣2]− 2R

{wHr

(μ,ν)

hH

}+ wHR

(μ,ν)

hhw.

(11)

3.2. Equivalent system model and effective SNR

In order to derive a model taking into account channel esti-mation errors, we assume a receiver that processes the chan-

nel estimates H(μ,ν)n,� as if these were the true CTF. The effect

of channel estimation errors on the received signal in (1) isdescribed by the equivalent system model:

Υ(μ,ν)n,� =

NT∑μ=1

√Ed

NTX

(μ)n,�

�H

(μ,ν)n,� +

NT∑

μ=1

√Ed

NTX

(μ)n,� ε

(μ,ν)n,� + Z(ν)

n,�

︸︷︷︸η

(μ,ν)n,�

,

(12)

where η(μ,ν)n,� denotes the effective noise term with zero mean

and variance σ2η = N0 +Edσ

2ε . Apart from the increased noise

term η(μ,ν)n,� , channel estimation impairments affect the equiv-

alent system model (12) by distortions in the signal part of�H

(μ,ν)n,� = wH h

(μ,ν). The useful signal energy observed at the

receiver is given by

σ2�H= E

[∣∣wH h(μ,ν)∣∣2

]= wHRhhw. (13)

Now the effective SNR including channel estimation of theequivalent system model (12) yields

γ = Ed

σ2�H

σ2η

=Edσ

2�H

N0 + Edσ2ε

. (14)

An important difference to the equivalent system modeldevised by [7] is the definition of σ

2�H

in (13). In [7] the es-

timate H(μ,ν)n,� = wH y

(μ,ν) replaces�H

(μ,ν)n,� = wH h

(μ,ν)in (12),

thus containing contributions from the CTF and noise, sothat σ2

H= σ2

�H

+ N0/Ed. Instead, our model with σ2�H

in (13)

exclusively captures the signal part of the channel estimate.As the model of [7] was tailored for an MMSE estimator withσ2H= 1− σ

2ε , meaningful results are produced. However, the

model of [7] implies that the noise term contained in σ2H

con-tributes to the useful signal energy of the effective SNR γ in(14). This becomes problematic at low SNR, when the equiv-

10−4

10−3

10−2

10−1

100

MSE

5 10 15 20 25 30 35

SNR (dB)

σ2n = N0/Ep

GnSp

σ2i

σ2n = N0/(EpGn)

Figure 3: Parametrization of an MSE estimation curve.

alent system model is to be applied to other than MMSE es-timators. For instance, consider an unbiased estimator withwHRhhw = 1 and unitary estimator gain wHw = 1. Thenwith the model of [7]: σ2

H= 1 + N0/Ed, so for low SNR,

N0→∞, the effective SNR in (14) approaches γ = 1/2, whichis clearly a contradiction. On the other hand, by using (13)we get σ

2�H= 1 and γ→0 as N0→∞, so our model in (12)

produces meaningful results for low SNR. In any case, in thehigh SNR regime with N0 � Ed both models converge andwe get σ

2�H≈ σ

2H .

Inserting (7) into the effective SNR γ in (14) and com-puting the ratio γ0/γ quantifies the SNR degradation due tochannel estimation errors, given by

Δγ = γ0

γ= 1

σ2�H

(1 + NTΩp

(Sp − 1

)+ σ

2ε γ0

). (15)

Substituting the MSE σ2ε = σ

2n + σ

2i into (15), with σ

2n being

expressed in the parametrized form of (10), the loss in SNRdue to channel estimation can be transformed to

Δγ = 1

σ2�H

(1 + NTΩp

(Sp − 1

))·(

1 +1

GnSp

)+

σ2i

σ2�H

γ0.

(16)

In the following, a fixed estimator w is considered wherethe estimator coefficients are computed once and are notadapted for changing channel conditions. Then, accordingto (16) the performance penalty due to channel estimationis fully determined by two SNR independent parameters,the estimator gain Gn and the interpolation error σ

2i . On

the other hand, allowing for an SNR dependent estimator,w = w(γ), the parameters Gn and σ

2i would be strictly speak-

ing only valid for one particular SNR value γ. A prominentexample for an SNR dependent estimator is the MMSE esti-mator, known as Wiener filter [27]. In this case, the SNR forwhich Gn and σ

2i lead to a maximum Δγ in (16) should be

used, so to maintain a certain performance under worst caseconditions.

The MSE of a fixed estimator w is plotted in Figure 3.At low SNR, the MSE is dominated by the noise error σ

2n.


Hence, the MSE linearly decreases with the SNR. At highSNR the MSE experiences an error floor caused by the SNRindependent interpolation error. A pilot boost is only effec-tive to reduce the noise part of the MSE in (10), while theinterpolation error, σ

2i in (11), remains unaffected. This is

shown in Figure 3, where a pilot boost shifts the MSE Sp dBto the left.

4. CAPACITY ANALYSIS

The ergodic channel capacity that includes channel estima-tion and pilot insertion losses when the channel is not knownat transmitter can be lower bounded by [7]

C ≥ (1−NTΩp)E[

log 2 det(

INR +Hn,�HH

n,�

NT

γ0

Δγ

)], (17)

where INR is the NR × NR identity matrix and the CTF is de-fined by

Hn,� =

⎛⎜⎜⎜⎝

H(1,1)n,� . . . H(NT,1)

n,�...

. . ....

H(1,NR)n,� . . . H(NT,NR)

n,�

⎞⎟⎟⎟⎠ . (18)

In (17), the expectation is taken over the frequency and timedimension of Hn,� , that is, over indices n and �. The capacitypenalty due to the pilot-aided channel estimation is charac-terized by two factors: the SNR loss due to estimation errors,Δγ from (15) or (16), and the loss in spectral efficiency due toresources consumed by pilot symbols, NTΩp. Inserting (16)into (17) we obtain

C≥(1−NTΩp)E[log 2 det

(INR+

Hn,�HHn,�

NT

·γ0σ

2�H(

1+NTΩp(Sp−1

))·(1+1/(GnSp

))+σ

2i γ0

)].

(19)

An important requirement for the capacity lower boundto become tight is that the signal and noise terms in theequivalent system model of (12) are uncorrelated [7]. Un-fortunately, for arbitrary linear estimators w this may not

be the case, as the interpolation error H(μ,ν)n,� −

�H

(μ,ν)n,� , with

�H

(μ,ν)n,� = wH h

(μ,ν), introduces correlations between the effec-

tive noise term η(μ,ν)n,� and the CTF H

(μ,ν)n,� . On the other hand,

the noise part of the estimation error wH z(ν) is statistically

independent of both H(μ,ν)n,� and

�H

(μ,ν)n,� . Therefore, for perfect

interpolation (�H

(μ,ν)n,� = H

(μ,ν)n,� and σ

2i = 0), the signal part

�H

(μ,ν)n,� and the effective noise term η

(μ,ν)n,� in (12) become sta-

tistically independent. To this end, one condition for a tight

capacity bound is σ2n σ

2i , so to ensure that

�H

(μ,ν)n,� and η

(μ,ν)n,�

are sufficiently decorrelated.In the following, we focus on the problem of capacity

maximization. By doing so, we consider

(i) pilot boost Sp,

(ii) pilot overhead Ωp,(iii) number of transmit antennas NT

as optimization parameters such that the capacity is maxi-mized.

4.1. Optimum pilot boost

The effect of a pilot boost is twofold: first, the estimation er-ror decreases; second, the energy dedicated to pilot symbolsincreases. So, there clearly exists an optimum pilot boost Sp

which minimizes the loss in SNR due to channel estimation,Δγ in (16), and thus maximizes the system capacity in (19).

The optimum pilot boost for the parametrized estima-tion error model is obtained by differentiating Δγ from (16)or C from (19) with respect to Sp and setting the result tozero. This results in

Sp,opt =√√√1−NTΩp

NTΩpGn. (20)

The optimum pilot boost Sp,opt is seen to increase if less pi-lots are used and/or less transmit antennas are used, but de-creases with growing estimator gain, Gn. In any case, a pilotboost is only effective to reduce the noise part of the MSE σ

2n

in (10), while the interpolation error σ2i in (11) remains un-

affected. Hence, the attainable gains of a pilot boost diminishwith growing σ

2i and SNR γ0, as deduced from Δγ in (16) or

C in (19), although the optimum pilot boost Sp,opt in (20) is

independent of both SNR and σ2i .

The loss in SNR for the optimally chosen pilot boost (20)yields

Δγ∣∣Sp=Sp, opt

= 1

σ2�H

(√1−NTΩp +

√NTΩp

Gn

)2

+σ

2i

σ2�H

γ0,

(21)

whereas the capacity becomes

C∣∣Sp=Sp, opt

=(1−NTΩp)E

[log 2 det

(INR+

Hn,�HHn,�

NT

·γ0σ

2�H(√

1−NTΩp+√NTΩp/Gn

)2

+σ2i γ0

)].

(22)

As both Gn and σ2i depend on Ωp, an analytical solution for

Ωp that maximizes (22) is a task of formidable complexitywhich is not pursued here. Instead, for the special case ofperfect interpolation (σ

2i = 0), we derive the optimum pilot

overhead Ωp in closed form, whereas we propose a semiana-lytical procedure for the general case.

4.2. Ideal lowpass interpolation filter (LPIF)

Motivated by previous work where it was shown that equidis-tant placement of pilot symbols minimizes the MSE [13],


−T/Df0 T/Df

τ

1/(2TsymDt)

0τ

−1/(2TsymDt)

fD

2 fD,wτw

Figure 4: Filter transfer function of an ideal 2D low-pass interpolation filter.

as well as maximizes the capacity [8], we focus on channelestimation by interpolation with equispaced pilots in timeand frequency. By using a scattered pilot grid the receivedOFDM frame is sampled in two dimensions, with rate Df/Tand DtTsym in frequency and time, respectively.

An ideal lowpass interpolation filter (LPIF) is character-ized by the 2D rectangular shaped filter transfer function

W(τ, fD

) =∞∑

n=−∞

∞∑

�=−∞wn,� e

− j2π(nτ/T+� fDTsym)

=

⎧⎪⎨⎪⎩

1 , 0 ≤ τ ≤ τw,∣∣ fD

∣∣ ≤ fD,w,

0 , τw < τ ≤ T

Df, fD,w <

∣∣ fD∣∣ ≤ 1

2TsymDt,

(23)

where wn,� denotes the filter coefficient of pilot subcar-rier n and OFDM symbol �. The filter parameters τw andfD,w specify the cut-off region of the filter. The transfor-mation to the time (delay) and Doppler domains is de-scribed by a discrete time Fourier transform (DTFT) [14],between the variable pairs n→τ and �→ fD. Due to sampling,W(τ, fD) is periodically repeated at intervals [0,T/Df] and[−1/(2TsymDt), 1/(2TsymDt)]. This is illustrated in Figure 4where the filter transfer function of an ideal LPIF is drawnin the 2D plane.

Applied to PACE the LPIF is to be designed so that the

spectral components of the CTF H(μ,ν)n,� which are nonzero

within the range [0, τmax] and [− fD,max, fD,max] pass the fil-ter undistorted, while spectral components outside this rangeare blocked. Furthermore, the pilot spacings Df and Dt mustbe sufficiently small to prevent spectral overlap between thefilter passband and its aliases. Hence, in order to reconstructthe signal the sampling theorem requires that [18]

τmax ≤ τw <T

Df, fD,max ≤ fD,w <

12DtTsym

. (24)

The filter parameters τw and fD,w represent the maximumassumed delay of the channel and the maximum assumedDoppler frequency, according to worst case channel condi-tions, as indicated in Figure 4.

Applied to the MSE analysis in Section 3.1, an ideal LPIFhas some appealing properties as follows.

(i) Provided that (24) is satisfied, the interpolation errordiminishes, σ2

i = 0; that is, the LPIF resembles perfectinterpolation. Consequently, the MSE is equivalent to

σ2ε = σ2

n = wHw/γp in (10). Furthermore, the MSE be-comes independent of the subcarrier and OFDM sym-bol indices n and �. Hence, no deviation over n and �is observed, σ2

ε [n, �] = σ2ε . As opposed to the general

case of linear estimators in Section 3.1, averaging overn and � is not required.

(ii) Due to perfect interpolation, the useful signal power in(13) becomes σ2

�H= σ2

H = 1, that is, the LPIF produces

unbiased estimates.(iii) As the number of filter coefficients in frequency and

time approach infinity, {Mf,Mt}→∞, the pilot over-head becomes Ωp = 1/(DfDt).

(iv) In the high SNR regime, the performance of an idealLPIF asymptotically approaches the MMSE [28]. Ingeneral, however, the MSE of the ideal LPIF is strictlylarger than the MMSE.

By invoking Parseval’s theorem [14], the MSE can be trans-formed to

σ2ε =

wHwγp

= DfDtTsym

γpT

∫ T/Df

0

∫ 1/(2DtTsym)

−1/(2DtTsym)

∣∣W(τ, fD

)∣∣2dτdfD.

(25)

The MSE, σ2ε , is determined by the fraction of the AWGN

suppressed by the filter. Inserting (23) and solving (25) yields

σ2ε =

1γpGn

= 1γpβfβt

= c2w

γpΩp(26)

with cw =√

2τw fD,wTsym/T . The factors βf = T/(Dfτw) andβt = 1/(2Dt fD,wTsym) are a measure for the amount of over-sampling in frequency and time, with respect to minimumsampling rates T/Df and 1/(2DtTsym), required by the sam-pling theorem in (24). The MSE is inversely proportional tothe oversampling factors βf and βt, as well as the pilot over-head Ωp = 1/(DfDt). Hence, increasing the pilot overheaddirectly improves the MSE.

4.2.1. Capacity of PACE with perfect 2D interpolation

The expression for the MSE in (26) establishes a relationbetween estimator gain and pilot overhead, Gn = Ωp/c2

w,that allows to maximize the channel capacity in closed form.Moreover, the effective signal and noise terms in the equiva-lent system model (12) become statistically independent, en-suring a tight capacity bound in (22).


For an ideal LPIF, the optimum pilot boost (20) can beconveniently expressed as

Sp,opt = cw

√1−NTΩp√NTΩp

. (27)

Inserting Gn = Ωp/c2w into (21) and after some algebraic

transformations the SNR loss for Sp = Sp,opt becomes

Δγ|Sp=Sp,opt =(√

1−NTΩp + cw

√NT

)2

. (28)

This means that Δγ|Sp=Sp,opt is minimized by the maximumpilot overhead Ωp, that is, all transmitted symbols are dedi-cated to pilots Ωp = 1/NT. However, in this case the capacitybecomes zero. In fact, the capacity is maximized by select-ing the smallest pilot overhead Ωp,min which still satisfies thesampling theorem

Cmax=(1−NTΩp,min)E

[log 2 det

(INR +

Hn,�HHn,�

NT

· γ0(√1−NTΩp,min + cw

√NT

)2

)],

(29)

where Ωp,min = 1/(Df,maxDt,max) is attained by the maxi-mum pilot spacings which satisfy (24), Df,max = T/τw� andDt,max = 1/(2 fD,wTsym)�, where x� is the largest integerequal or smaller than x. To prove (29) it can be easily checkedthat Cmax is a monotonically decreasing function with respectto Ωp, with the global maximum at Ωp = 0. Hence, (29) ismaximized by Ωp,min, since Δγ|Sp=Sp,opt is only valid for pilotgrids which satisfy the sampling theorem in (24).

By ignoring the rounding effects and thus approximatingDf,maxDt,max ≈ c2

w, we obtainΩp,min ≈ c2w, that is, the effects of

channel estimation errors for PACE are completely describedby Ωp,min. Interestingly, the estimator gain now approachesunity, Gn = 1 and the SNR loss becomes

Δγmin =(√

1−NTΩp,min +√NTΩp,min

)2

. (30)

Furthermore, (27) and (29) can be approximated by

Sp,opt =√

1−NTΩp√NTΩp

, (31)

Cmax≈(1−NTΩp,min)E

[log 2 det

(INR +

Hn,�HHn,�

NT

· γ0(√1−NTΩp,min +

√NTΩp,min

)2

)].

(32)

Finally, it turns out that the fraction of energy dedicated todata relative to the overall transmit energy becomes

EdNd

E0NcL= Ed

E0

(1−NTΩp,min

)

≈√

1−NTΩp,min√

1−NTΩp,min +√NTΩp,min

,

(33)

where the ratio Ed/E0 is defined in (6). Interestingly, (30)and (33) are equivalent to the results obtained for theblock fading channel (see [7, equation (34)]). Applied toMIMO-OFDM, the block fading assumption translates to atime/frequency area the channel is assumed constant. Then,the same results apply given that the interval, the channel isconstant for the block fading assumption in [7], is replacedby the maximum pilot spacing that satisfies the sampling the-orem, Df,maxDt,max = 1/Ωp,min. This means that the capacitylower bound of [7] is extended to the more general case oftime-variant frequency selective channels.

An interesting observation can be made by setting

NTΩp,min = 12. (34)

By devoting half of the resources to pilot symbols we obtainfrom (31) and (33), respectively,

Sp,opt = 1,EdNd

E0NcL= 1

2. (35)

In case half of the frame is devoted to training purposes, inorder to maximize capacity, pilots should not be boosted, andconsequently half of the transmit energy is invested on pilots.A similar conclusion is also provided in [7] assuming a blockfading channel.

4.2.2. Number of transmit antennas

Suppose that N ′T out of the NT transmit antennas are used

for communication. Inserting N ′T for NT in the capacity ex-

pression for MIMO-OFDM with optimum pilot grid,Cmax in(32), the number of transmit antennas that maximizes chan-nel capacity Cmax is given by [7, 29]

N ′T = min

{NT, NR,

12Ωp,min

}. (36)

Several important conclusions with respect to the capacitymaximization in MIMO-OFDM can be drawn from (36).

(i) If NT = NR = 1/(2Ωp,min), from (34) and (35) it fol-lows that pilots should not be boosted, that is, theyshould be of equal energy as the data symbols..

(ii) The amount of training should not exceed half of theOFDM frame.

(iii) The number of transmit and receive antennas shouldbe equal.

4.3. Semianalytical approach for pilot grid design

The optimal pilot grid that maximizes the channel capac-ity is derived for an ideal LPIF in Section 4.2. For realizable


estimators, we propose the following procedure to obtain theoptimum pilot grid.

(i) Specify the filter parameters τw and fD,w so that therelation in (24) is satisfied.

(ii) Choose maximum possible pilot spacings and estima-tor dimensions Mf and Mt, that maintain a certain in-terpolation error σ

2i . This determines the minimum

pilot overhead Ωp,min, and the estimator gain Gn = 1/(wHw).

(iii) Determine the optimum pilot boost Sp,opt using (20).(iv) Calculate the optimum number of transmit antennas

using (36).

Considering (i), in a well-designed OFDM system the maxi-mum channel delay τmax should not exceed the cyclic prefix.Therefore, it is reasonable to assume τw = TCP. In addition,fD,w is set according to the maximum Doppler frequency ex-pected in a certain propagation scenario.

Considering (ii), this condition is imposed to keep σ2i suf-

ficiently low. The impact of σ2i on the SNR penalty in (21)

becomes negligible if σ2i < εth/γw, where εth is a small posi-

tive constant and γw denotes the largest expected SNR. Thiscondition effectively enforces a sufficient degree of oversam-pling. That is, Ωp,min is required to be larger than the theo-retical minimum.

The condition on the interpolation error σ2i in step (ii)

serves another important requirement. The fact that σ2i is

negligible in the SNR region of interest ensures that the use-ful signal part and the estimation error in the equivalent sig-nal model in (12) become uncorrelated, so that the capacitybound in (22) becomes tight.

Considering (iii), this step ensures that the capacity in(22) is maximized, given that in step (ii) Ωp,min is appropri-

ately chosen, so that the interpolation error σ2i is sufficiently

small. As a formal proof appears difficult, we verify througha numerical example in Section 5 that the proposed semian-alytical procedure indeed maximizes capacity.


An OFDM system with Nc = 512 subcarriers, and a cyclicprefix of duration TCP = 64·Tspl, is employed. One frameconsists of L = 65 OFDM symbols, and it is assumed thatchannel estimation is carried out, after all pilots of one framehave been received. The signal bandwidth is 20 MHz, whichtranslates to a sampling duration of Tspl = 50 ns. This resultsin an OFDM symbol duration of Tsym = 35.97 μs of whichthe cyclic prefix is TCP = 3.2 μs. A high mobility scenario isconsidered with velocities up to 300 km/h. At 5 GHz carrierfrequency this translates to a normalized maximum Dopplerfrequency of fD,maxTsym ≤ 0.04.

Channel estimation unit

Since pilots belonging to different transmit antennas are or-thogonally separated in time and/or frequency, the problemof MIMO-OFDM channel estimation breaks down to esti-mating the channel of a single antenna OFDM system. A cas-

Table 1: Power delay profile of WINNER channel model C2 [31].

Delay (ns) 0 5 135 160 215 260 385

Power (dB) −0.5 0 −3.4 −2.8 −4.6 −0.9 −6.7

Delay (ns) 400 530 540 650 670 720 750

Power (dB) −4.5 −9.0 −7.8 −7.4 −8.4 −11 −9.0

Delay (ns) 800 945 1035 1185 1390 1470

Power (dB) −5.1 −6.7 −12.1 −13.2 −13.7 −19.8

caded channel estimator consisting of two one-dimensional(1D) estimators termed 2×1D PACE is implemented. 2×1DPACE performs only slightly worse than optimal 2D PACE,while being significantly less complex [16].

The estimator was implemented by a Wiener interpo-lation filter (WIF) with model mismatch [16]. The filtercoefficients in frequency and time are generated assuminga uniformly distributed power delay profile and Dopplerpower spectrum, nonzero within the range [0, τw] and[− fD,w, fD,w]. Furthermore, the average SNR at the filter in-put, γw, is required, which should be equal or larger than ac-tual average SNR, so γw ≥ γ0. To generate the filter coeffi-cients, we set τw = TCP, fD,w = 0.04Tspl and γw = 30 dB.With these parameters, the sampling theorem in (24) re-quires for the pilot spacings in frequency and time Df < 8and Dt ≤ 12.

The WIF with model mismatch is closely related to anLPIF, and therefore inherits many of its properties—signalswith spectral components within the range [0, τw] and[0, fD,max ] pass the filter undistorted, while spectral compo-nents outside this range are blocked. In fact, it was shown in[30] that for an infinite number of coefficients, {Mf,Mt}→∞,the mismatched WIF approaches an ideal LPIF.

Results

The performance of a channel estimation unit generally de-pends on the chosen channel model. On the other hand,the optimum pilot grid and the associated channel estima-tion unit is expected to operated in a wide variety of channelconditions. Hence, it is important to test the performanceof the considered estimators for various channel models. InFigure 5 the channel estimation MSE determined by (8) isplotted for the pilot grid Df = 6, Dt = 8, and filter ordersMf = 16, Mt = 9. The following channel models are consid-ered:

Chn A: IST-WINNER channel model C2 for typical urbanpropagation environments [31]; the power delay pro-file (PDP) is shown in Table 1;

Chn B: flat fading channel with PDP ρ(τ) = δ(τ − TCP/2);Chn C: 2-tap channel with PDP ρ(τ) = (δ(τ)+δ(τ−TCP))/2;Chn D: uniformly distributed PDP nonzero within the range

[0,TCP]. This is the channel used to generate the WIFcoefficients, that is, the WIF is matched to Chn D.

For all models, the independent fading taps are generated us-ing Jakes’ model [32] with fD,maxTsym = 0.033, correspond-ing to a velocity of 250 km/h at 5 GHz carrier frequency. Itis seen in Figure 5 that the MSE is virtually independent


10−4

10−3

10−2

10−1

MSE

10 20 30 40 50

SNR (dB)

Chn A: Df = 6,Dt = 8,Mf = 16,Mt = 9Chn B: Df = 6,Dt = 8,Mf = 16,Mt = 9Chn C: Df = 6,Dt = 8,Mf = 16,Mt = 9Chn D: Df = 6,Dt = 8,Mf = 16,Mt = 9

Figure 5: MSE versus SNR of 2 × 1D-PACE for various channelmodels, Df = 6, Dt = 8, Mf = 16, Mt = 9.

of the particular channel model, although the PDPs of theconsidered channels cover an extensive range of possiblepropagation scenarios. Note that Chn C is the worst-casechannel, since its two taps are placed at the closest positionwith respect to the cutoff regions of the WIF (compare withFigure 4). Likewise, Chn B is the best-case channel, as its sin-gle tap is located right in the center of the filter passband.

The optimum pilot grid for the considered MIMO-OFDM system is assembled in the following as discussed inSection 4.3. All results are plotted for Chn A with normalizedmaximum Doppler fD,maxTsym = 0.033. We note that resultsin Figure 5 suggest that the identified optimum pilot grid isvalid for any channel model with τmax ≤ τw, fD,max ≤ fD,w

and γ0 ≤ γw.From the set of allowable Df and Dt, the following candi-

date grids are selected: Df = {4, 6} and Dt = {4, 8}, whichtranslates to the oversampling factors βf = {2, 1.25} andβt = {3, 1.5}. The filter order in time direction was set equalto the number of pilots per frame, so Mt = {17, 9}. In fre-quency direction, on the other hand, the number of pilots isNc/Df = 85 and 128, respectively, allowing for much higherfilter orders Mf.

Figure 6 shows the interpolation error σ2i (in Figure 6(a))

by computing (11) and the estimator gain Gn = 1/(wHw)(in Figure 6(b)) against the filter order in frequency Mf forMt = {9, 17} and various pilot grids (parameters Df and Dt).Provided that Mf ≥ 12 we observe that for all pilot grids,γwσ

2i < 0.1 with γ0 ≤ γw = 30 dB. Therefore, the impact of

σ2i on Δγ is negligible (less than 0.5 dB). By setting Mf = 16

in the following, none of the considered grids can be ruledout at this point.

In Figure 7, the channel capacity versus pilot boost Sp ofan 8 × 8 MIMO-OFDM system with different pilot grids isdepicted at SNR γ0 = 10 dB. The plots are obtained by in-

serting σ2i and Gn = 1/(wHw) obtained in Figure 6 into the

capacity expression (19) assuming different pilot grids. It is

10−5

10−4

10−3

10−2

σ2 i

2 4 8 12 16 20

Mf

Df = 6,Dt = 8,Mt = 9Df = 4,Dt = 8,Mt = 9

Df = 6,Dt = 4,Mt = 17Df = 4,Dt = 4,Mt = 17

(a)

1

2

3

Gn

2 4 8 12 16 20

Mf

Df = 6,Dt = 4,Mt = 9Df = 4,Dt = 8,Mt = 9

Df = 6,Dt = 4,Mt = 17Df = 4,Dt = 4,Mt = 17

(b)

Figure 6: Interpolation error σ2i and estimator gain Gn at an SNR

of γ0 = 30 dB against the filter order in frequency Mf.

4

6

8

10

12

14

16

18

20

22

24

Ch

ann

elca

paci

ty(b

it/s

/Hz)

−10 −5 0 5 10

Sp (dB)

Df = 6,Dt = 8,Mf = 16,Mt = 9Df = 4,Dt = 8,Mf = 16,Mt = 9Df = 6,Dt = 4,Mf = 16,Mt = 17Df = 4,Dt = 4,Mf = 16,Mt = 17Ideal LPIF, Ωp = c2

w

Figure 7: Capacity versus pilot boost for 8× 8 MIMO-OFDM sys-tem with different pilot grids at an SNR of γ0 = 10 dB.


5

10

15

20

25

30

Ch

ann

elca

paci

ty(b

it/s

/Hz)

−10 −5 0 5 10

Sp (dB)

NT = NR = 1NT = NR = 4NT = NR = 8

NT = NR = 16NT = NR = 21NT = NR = 32

Figure 8: Capacity versus pilot boost for N × N MIMO-OFDMsystem for different number of antennas N , SNR γ0 = 10 dB, Df =6, Dt = 8, Mf = 16, Mt = 9.

0

5

10

15

20

25

30

Ch

ann

elca

paci

ty(b

it/s

/Hz)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Number of transmit antennas, NT

Reference, ideal caseReal channel estimation

Maximum capacity

Figure 9: Capacity versus number of transmit antennas NT forMIMO-OFDM system with NR = 8 receive antennas, SNR γ0 =10 dB, Df = 6, Dt = 8, Mf = 16, Mt = 9.

seen that the most bandwidth efficient grid (Df = 6,Dt = 8)maximizes capacity. Furthermore, maximum capacity Cmax

for all grids is achieved for those Sp that satisfy (20). This plotconfirms the proposed semianalytical procedure described inSection 4.3. As reference, the capacity assuming an ideal LPIFis also plotted in Figure 7. A significant gap in capacity be-tween the ideal LPIF relative to the realizable estimators isvisible. This is mainly due to the fact that a realizable filterdoes not exhibit a rectangular filter transfer function. Thisinevitably requires a higher pilot overhead and also reducesthe attainable estimator gain.

The channel capacity versus pilot boost, Sp, of an N ×NMIMO-OFDM system with pilot grid Df = 6, Dt = 8, filterorders Mf = 16, Mt = 9 and for different number of trans-

mit/receive antennas, N = NT = NR, is depicted in Figure 8.Again, the SNR is set to γ0 = 10 dB. Maximum capacity isachieved for NT = 21 yielding NTΩp ≈ 1/2 and Sp = 0 dB.This confirms the analytical results from Section 4.2.2 as wellas the proposed semianalytical procedure from Section 4.3.The significance of these results lie in the fact that results of[7] are generalized to MIMO-OFDM operating in dynamicchannels and to arbitrary linear estimators.

The channel capacity versus the number of transmit an-tennas NT of an NT × 8 MIMO-OFDM system for gridDf = 6, Dt = 8, filter orders Mf = 16, Mt = 9, at SNRγ0 = 10 dB is shown in Figure 9. The plots were generatedusing the capacity expression in (22) including the optimumpilot boost Sp,opt according to (20). As a reference, capacityof the corresponding system assuming perfect channel esti-mation and no loss due to pilots is shown. It can be observedthat for NT ≈ 8 maximum capacity is achieved, correspond-ing to the conclusion from Section 4.2.2. For higher valuesthe reduction in available bandwidth due to the pilot inser-tion dominates, lowering the achievable capacity.

6. CONCLUSIONS

In this paper, a framework for pilot grid design in MIMO-OFDM was developed and used to determine the pilot spac-ing and boost, so to maximize the capacity of the targetMIMO-OFDM system, including channel estimation errorsand pilot overhead. The analysis show that the previouslyderived capacity lower bound for a block fading channel isalso valid for MIMO-OFDM over time-varying frequency-selective channels. The derived bound applies to perfect in-terpolation, which essentially requires infinitely long pilot se-quences and filter coefficients. Furthermore, a semianalyticalprocedure was proposed to maximize the capacity for realiz-able and possibly suboptimum channel estimation schemes.

ACKNOWLEDGMENTS

This work has been performed in the framework of the ISTproject IST-4-027756 WINNER (World Wireless InitiativeNew Radio), which is partly funded by the European Union.This paper was presented in part at the IEEE InternationalConference on Communications (ICC-07), Glasgow, UK,June 2007.

REFERENCES

[1] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eu-ropean Transactions on Telecommunications, vol. 10, no. 6, pp.585–595, 1999.


[3] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity ofOFDM-based spatial multiplexing systems,” IEEE Transactionson Communications, vol. 50, no. 2, pp. 225–234, 2002.


[4] R. Van Nee, V. K. Jones, G. Awater, A. Van Zelst, J. Gard-ner, and G. Steele, “The 802.11n MIMO-OFDM standard forwireless LAN and beyond,” Wireless Personal Communications,vol. 37, no. 3-4, pp. 445–453, 2006.

[5] 3rd Generation Partnership Project; Technical SpecificationGroup Radio Access Network, “Physical layer aspects forevolved Universal Terrestrial Radio Access (UTRA),” June2006.

[6] G. Auer, “Analysis of pilot-symbol aided channel estimationfor OFDM systems with multiple transmit antennas,” in Pro-ceedings of IEEE International Conference on Communications(ICC ’04), vol. 6, pp. 3221–3225, Paris, France, June 2004.

[7] B. Hassibi and B. M. Hochwald, “How much training is neededin multiple-antenna wireless links?” IEEE Transactions on In-formation Theory, vol. 49, no. 4, pp. 951–963, 2003.

[8] S. Adireddy, L. Tong, and H. Viswanathan, “Optimal place-ment of training for frequency-selective block-fading chan-nels,” IEEE Transactions on Information Theory, vol. 48, no. 8,pp. 2338–2353, 2002.

[9] S. Adireddy and L. Tong, “Optimal placement of known sym-bols for slowly varying frequency-selective channels,” IEEETransactions on Wireless Communications, vol. 4, no. 4, pp.1292–1296, 2005.

[10] H. Vikalo, B. Hassibi, B. Hochwald, and T. Kailath, “Onthe capacity of frequency-selective channels in training-basedtransmission schemes,” IEEE Transactions on Signal Processing,vol. 52, no. 9, pp. 2572–2583, 2004.

[11] O. Simeone and U. Spagnolini, “Lower bound on training-based channel estimation error for frequency-selective block-fading Rayleigh MIMO channels,” IEEE Transactions on SignalProcessing, vol. 52, no. 11, pp. 3265–3277, 2004.

[12] S. Ohno and G. B. Giannakis, “Capacity maximizing MMSE-optimal pilots for wireless OFDM over frequency-selectiveblock Rayleigh-fading channels,” IEEE Transactions on Infor-mation Theory, vol. 50, no. 9, pp. 2138–2145, 2004.

[13] R. Negi and J. Cioffi, “Pilot tone selection for channel estima-tion in a mobile OFDM system,” IEEE Transactions on Con-sumer Electronics, vol. 44, no. 3, pp. 1122–1128, 1998.

[14] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Pro-cessing, Prentice Hall, Englewood Cliffs, NJ, USA, 2nd edition,1999.

[15] J.-J. van de Beek, O. Edfors, M. Sandell, S. Wilson, and P.Borjesson, “On channel estimation in OFDM systems,” inProceedings of the 45th IEEE Vehicular Technology Conference(VTC ’95), vol. 2, pp. 815–819, Chicago, Ill, USA, July 1995.

[16] P. Hoher, S. Kaiser, and P. Robertson, “Pilot-symbol-aided channel estimation in time and frequency,” in Pro-ceedings of the IEEE Global Telecommunications Conference(Globecom ’97), vol. 4, pp. 90–96, Phoenix, Ariz, USA, Novem-ber 1997.

[17] P. Hoher, “TCM on frequency selective land-mobile radiochannels,” in Proceedings of the 5th Tirrenia InternationalWorkshop on Digital Communications, pp. 317–328, Tirrenia,Italy, September 1991.

[18] F. Sanzi and J. Speidel, “An adaptive two-dimensional channelestimator for wireless OFDM with application to mobile DVB-T,” IEEE Transactions on Broadcasting, vol. 46, no. 2, pp. 128–133, 2000.

[19] Y. Li, “Pilot-symbol-aided channel estimation for OFDM inwireless systems,” IEEE Transactions on Vehicular Technology,vol. 49, no. 4, pp. 1207–1215, 2000.

[20] Y.-H. Yeh and S.-G. Chen, “DCT-based channel estimationfor OFDM systems,” in Proceedings of IEEE International Con-ference on Communications (ICC ’04), vol. 4, pp. 2442–2446,Paris, France, June 2004.

[21] M.-H. Hsieh and C.-H. Wei, “Channel estimation for OFDMsystems based on comb-type pilot arrangement in frequencyselective fading channels,” IEEE Transactions on ConsumerElectronics, vol. 44, no. 1, pp. 217–225, 1998.

[22] Y. Li, N. Seshadri, and S. Ariyavisitakul, “Channel estimationfor OFDM systems with transmitter diversity in mobile wire-less channels,” IEEE Journal on Selected Areas in Communica-tions, vol. 17, no. 3, pp. 461–471, 1999.

[23] Y. Li, “Simplified channel estimation for OFDM systems withmultiple transmit antennas,” IEEE Transactions on WirelessCommunications, vol. 1, no. 1, pp. 67–75, 2002.

[24] W. G. Jeon, K. H. Paik, and Y. S. Cho, “Two-dimensionalMMSE channel estimation for OFDM systems with transmit-ter diversity,” in Proceedings of the 54th IEEE Vehicular Technol-ogy Conference (VTC ’01 fall), vol. 3, pp. 1682–1685, AtlanticCity, NJ, USA, October 2001.

[25] M. Speth, “LMMSE channel estimation for MIMO OFDM,”in Proceedings of the 8th International OFDM Workshop(InOWo ’03), Hamburg, Germany, September 2003.

[26] J.-W. Choi and Y.-H. Lee, “Optimum pilot pattern for channelestimation in OFDM systems,” IEEE Transactions on WirelessCommunications, vol. 4, no. 5, pp. 2083–2088, 2005.

[27] S. M. Kay, Fundamentals of Statistical Signal Processing: Estima-tion Theory, Prentice Hall, Englewood Cliffs, NJ, USA, 1993.

[28] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Commu-nication Receivers, John Wiley & Sons, NewYork, NY, USA,2nd edition, 1998.

[29] L. Zheng and D. N. C. Tse, “Communication on the Grass-mann manifold: a geometric approach to the noncoherentmultiple-antenna channel,” IEEE Transactions on InformationTheory, vol. 48, no. 2, pp. 359–383, 2002.

[30] G. Auer and E. Karipidis, “Pilot aided channel estimation forOFDM: a separated approach for smoothing and interpola-tion,” in Proceedings of IEEE International Conference on Com-munications (ICC ’05), vol. 4, pp. 2173–2178, Seoul, Korea,May 2005.

[31] IST-4-027756 WINNER II, “D1.1.2 WINNER II channel mod-els,” September 2007.

[32] W. C. Jakes, Microwave Mobile Communications, John Wiley &Sons, New York, NY, USA, 1974.


Research ArticleDistributed Antenna Channels with Regenerative Relaying:Relay Selection and Asymptotic Capacity

Aitor del Coso and Christian Ibars

Centre Tecnologic de Telecomunicacions de Catalunya (CTTC), Av. Canal Olımpic, Castelldefels, Spain

Received 15 November 2006; Accepted 3 September 2007


Multiple-input-multiple-output (MIMO) techniques have been widely proposed as a means to improve capacity and reliabilityof wireless channels, and have become the most promising technology for next generation networks. However, their practicaldeployment in current wireless devices is severely affected by antenna correlation, which reduces their impact on performance.One approach to solve this limitation is relaying diversity. In relay channels, a set of N wireless nodes aids a source-destinationcommunication by relaying the source data, thus creating a distributed antenna array with uncorrelated path gains. In this paper,we study this multiple relay channel (MRC) following a decode-and-forward (D&F) strategy (i.e., regenerative forwarding), andderive its achievable rate under AWGN. A half-duplex constraint on relays is assumed, as well as distributed channel knowledgeat both transmitter and receiver sides of the communication. For this channel, we obtain the optimum relay selection algorithmand the optimum power allocation within the network so that the transmission rate is maximized. Likewise, we bound the ergodicperformance of the achievable rate and derive its asymptotic behavior in the number of relays. Results show that the achievable rateof regenerative MRC grows as the logarithm of the Lambert W function of the total number of relays, that is, C = log2(W0(N)).Therefore, D&F relaying, cannot achieve the capacity of actual MISO channels.

Copyright © 2007 A. del Coso and C. Ibars. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Current wireless applications demand an ever-increasingtransmission capacity and highly reliable communications.Voice transmission, video broadcasting, and web brows-ing require wire-like channel conditions that the wirelessmedium still cannot support. In particular, channel impair-ments, namely, path loss and multipath fading do not al-low wireless channels to reach the necessary rate and ro-bustness expected for next generation systems. Recently, awide range of multiple antenna techniques have been pro-posed to overcome these channel limitations [1–4]; however,the deployment of multiple transmit and/or receive antennason the wireless nodes is not always possible or worthwhile.For these cases, the most suitable technique to take advan-tage of spatial diversity is node cooperation and relay channels[5, 6].

Relay channels consist of single source-destination pairsaided in their communications by a set of wireless relay nodesthat creates a distributed antenna array (see Figure 1). Therelay nodes can be either infrastructure nodes, placed by the

service provider in order to enhance coverage and rate [7], ora set of network users that cooperate with the source, whilehaving own data to transmit [8]. Relay-based architectureshave been shown to improve capacity, diversity, and delayof wireless channels when properly allocating network re-sources, and have become a key technique for the evolutionof wireless communications [9].

Background

The use of relays to increase the achievable rate of point-to-point transmissions was initially proposed by Cover and ElGamal in [10]. Motivated by this work, many relaying tech-niques have been recently studied, which can be classified,based on their forwarding strategy and required processing atthe relay nodes, as regenerative relaying and nonregenerativerelaying [5, 11]. The former assumes that relay nodes decodethe source information, prior to reencoding and sending it todestination [12, 13]. On the other hand, with the latter, relaynodes transform and retransmit their received signals but donot decode them [14–16].


Dec./enc.Relay 1

Z11

Y11 X2

1 (w)

b1 c1

X1s (w)

X2s (w)

a

...

...

Z1d Z2

d

Y1d

Y2d

Decoder w

Destination

Encoder

Source

w

bN cNZ1N

Y1N

Dec./enc.

Relay NX2N (w)

Time slot 1: s −→ N ,d Time slot 2: s, N −→ d

t

Figure 1: Half-duplex regenerative multiple relay channel with N parallel relays.

Regenerative relaying was initially presented in [10, The-orem 1] for a single-relay channel, and consists of relay nodesdecoding the source data and transmitting it to destination,ideally without errors. Such signal regeneration allows for co-operative coherent transmissions. Therefore, source and re-lays can operate as a distributed antenna array and imple-ment multiple-input single-output (MISO) beamforming.We distinguish two techniques: decode-and-forward (D&F),presented in [10], and partial decoding (PD), analyzed in[17]. D&F requires the relay nodes to fully decode thesource message before retransmitting it. Thus, it penalizesthe achievable rate when poor source-to-relay channel con-ditions occur. Nevertheless, for poor source-to-destinationchannels (e.g., degraded relay channels), it was shown to bethe capacity achieving technique [10]. On the other hand,with PD the relay nodes only partially decode the source mes-sage. Part of the transmitted message is sent directly to thedestination without being relayed [18]. PD is specifically ap-propriate when the source node can adapt the amount of in-formation transmitted through relays to the network channelconditions; otherwise it does not improve the D&F scheme[19]. The diversity analysis of regenerative multiple relay net-works was carried out by Laneman and Wornell in [20],showing that signal regeneration achieves full transmit diver-sity of the system. However, regenerative relaying has somedrawbacks as well: first, decoding errors at the relay nodesgenerate error propagation; second, synchronization amongrelays (specifically in the low SNR regime) may complicate itsimplementation, and finally, the processing capabilities re-quired at the relays increase their cost [5].

The two previously mentioned techniques are wellknown for the single-relay channel. However, the only sig-nificant extensions to the multiple relay setup are found in[6, 21, 22]. In these works, they were applied to physical-layer multihop networks and to the multiple relay channelwith orthogonal components, respectively.

Contributions

This paper studies the point-to-point Gaussian channel withN parallel relays that use decode-and-forward relaying. On

the relays, a half duplex constraint is considered, that is,the relay nodes cannot transmit and receive simultaneouslyin the same frequency band. The communication is ar-ranged into two consecutive, identical time slots, as shownin Figure 1. The source uses the first time slot to transmitthe message to the set of relays and to the destination. Then,during time slot 2, the set of nodes who have successfully de-coded the message, and the source, transmit extra parity bitsto the destination node, which uses its received signal dur-ing the two slots to decode the message. Transmit and re-ceive channel state information (CSI) are available at bothtransmitter and receiver sides, and channel conditions areassumed not to vary during the two slots of the communi-cation. Additionally, we consider that the source knows allrelay-to-destination channels, so that it can implement a re-lay selection algorithm. Finally, the overall transmitted powerduring the two time slots is constrained to a constant, andwe maximize the achievable rate through power allocationon the two slots of the communication, and on the usefulrelays.

The contributions of this paper are as follows.

(i) First, the instantaneous achievable rate of the pro-posed communication is derived in Proposition 1;then the optimum power allocation on the two slotsis obtained in Proposition 2. Results show that theachievable rate is maximized through an optimum re-lay selection algorithm and through power allocationon the two slots, referred to as constrained temporalwaterfilling.

(ii) Second, we analyze the ergodic performance of the in-stantaneous achievable rate derived in Proposition 2,assuming independent, identically distributed (i.i.d.)random channel fading and i.i.d. random relay po-sitions. We assume that the source node transmitsover several concatenated two-slot transmissions. Thechannel is invariant during the two slots, and uncorre-lated from one two-slot transmission to the next (seeFigure 2). Thus, the source transmits with an effectiverate equal to the ergodic achievable rate of the link,which is lower- and upper-bounded in this paper.

A. del Coso and C. Ibars 3

CR

Ea,b,c{CR}

Concatenation of two-slot MRC

Two-slot MRC

s −→ N ,d s, N −→ d s −→ N ,d s, N −→ d s −→ N ,d s, N −→ d s −→ N ,d s, N −→ d s −→ N ,d s, N −→ dTime

· · ·

Figure 2: Ergodic capacity: concatenation in time of half-duplex multiple relay channels.

(iii) Finally, we study the asymptotic performance (in thenumber of relays) of the instantaneous achievable rate,and we show that it grows asymptotically with the log-arithm of the branch 0 of the Lambert W function1 ofthe total number of relays, that is, C = log 2(W0(N)).

The remainder of the paper is organized as follows: inSection 2, we introduce the channel and signal model; inSection 3, the instantaneous achievable of the D&F MRCis derived and the optimum relay selection and power al-location are obtained. In Section 4, the ergodic achievablerate is upper- and lower-bounded, and Section 5 analyzes theasymptotic achievable rate of the channel. Finally, Section 6contains simulation results and Section 7 summarizes con-clusions.

Notation

We define X(2)1:n = [X (2)

1 , . . . ,X (2)n ]T with n ∈ {1, . . . ,N}.

Moreover, in the paper, I (A;B) denotes mutual informationbetween random variables A and B, C(x) = log 2(1 + x), b†

denotes the conjugate transpose of vector b, and b∗ denotesthe conjugate of b.

2. CHANNEL MODEL

We consider a wireless multiple-relay channel (MRC) witha source node s, a destination node d, and a set of par-allel relays N = {1, . . . ,N} (see Figure 1). Wireless chan-nels among network nodes are frequency-flat, memoryless,and modelled with a complex, Gaussian-distributed coeffi-cient; a ∼ CN (0, 1) denotes the unitary power, Rayleigh dis-tributed channel between source and destination, and ci ∼CN (0, 1) the complex channel from relay i to destination.In the system, bi is modelled as a superposition of path loss(with exponent α) and Rayleigh distributed fading, in orderto account for the different transmission distances from thesource to relays, di, i = 1, . . . ,N , and from source to destina-

1 The branch 0 of the Lambert W function, W0(N), is defined as the func-tion satisfying W0(N)eW0(N) = N , with W0(N) ∈ R+ [23].

tion do (used as reference), that is,

bi∼CN

(0,(dodi

)α). (1)

We assume invariant channels during the two-slot commu-nication.

As mentioned, the communication is arranged in twoconsecutive time slots of equal duration (see Figure 1). Dur-ing the first slot, a single-input multiple-output (SIMO)transmission from the source node to the set of relays anddestination takes place. The second slot is then used by relaysand source to retransmit data to destination via a distributedMISO channel. In both slots, the transmitted signals are re-ceived under additive white Gaussian noise (AWGN), anddestination attemps to decode making use of the signal re-ceived during the two phases. The complex signals transmit-ted by the source during slot t = {1, 2}, and by relay i during

phase 2, are denoted by X (t)s and X (2)

i , respectively. Therefore,considering memoryless channels, the received signal at therelay nodes during time slot 1 is given by

Y (1)i = bi·X (1)

s + Z(1)i for i ∈ N , (2)

where Z(1)i ∼CN (0, 1) is normalized AWGN at relay i. Like-

wise, considering the channel definition in Figure 1, the re-ceived signal at the destination node d during time slots 1and 2 is written as

Y (1)d = a·X (1)

s + Z(1)d ,

Y (2)d = a·X (2)

s +N∑i=1

ci·X (2)i + Z(2)

d ,(3)

where, as previously said, Z(t)d ∼CN (0, 1) is AWGN. Notice

that, due to half-duplex limitations, the relay nodes do nottransmit during time slot 1 and do not receive during timeslot 2. The overall transmitted power during the two time

slots is constrained to 2P; thus, defining γ1=E{X (1)s (X (1)

s )∗}

and γ2 = E{X (2)s (X (2)

s )∗} +

∑ Ni=1E{X (2)

i (X (2)i )

∗} as the


transmitted power2 during slots 1 and 2, respectively, we en-force the following two-slot power constraint:

γ1 + γ2 = 2P. (4)

3. ACHIEVABLE RATE IN AWGN

In order to determine the achievable rate of the channel,we consider updated transmitter and receiver channel stateinformation (CSI) at all nodes, and assume symbol andphase synchronization among transmitters. The achievablerate with D&F is given in the following proposition.

Proposition 1. In a half-duplex multiple-relay channel withdecode-and-forward relaying and N parallel relays, the rate

CD&F = max1≤n≤N

{max

p(Xs,X(2)1:n):γ1+γ2=2P

12·I(X (1)s ;Y (1)

d

)

+12·I(X (2)s , X(2)

1:n;Y (2)d

)}

s.t. I(X (1)s ;Y (1)

n

)≥ I

(X (1)s ;Y (1)

d

)+ I(X (2)s , X(2)

1:n;Y (2)d

)(5)

is achievable. Source-relay path gains have been ordered as∣∣b1

∣∣ ≥ · · · ≥ ∣∣bn∣∣ ≥ · · · ≥ ∣∣bN∣∣. (6)

Remark 1. Factor 1/2 comes from time division signalling.Variable n in the maximization represents the number of ac-tive relays; hence, the relay selection is carried out throughthe maximization in (5), considering (6).

Proof. Let the N relays in Figure 1 be ordered as in (6),and assume that only the subset Rn = {1, . . . ,n} ⊆ Nis active, with n ≤ N . The source node selects messageω ∈ [1, . . . , 2mR] for transmission (with m the total num-ber of transmitted symbols during the two slots, and Rthe transmission rate) and maps it into two codebooksX1, X2 ∈ Cm/2, using two independent encoding functions,3

x1 : {1, . . . , 2mR}→X1 and x2 : {1, . . . , 2mR}→X2. The code-word x1(ω) is then transmitted by the source during time

slot 1, that is, X (i)s = x1(ω). At the end of this slot, all re-

lay nodes belonging to Rn are able to decode the transmittedmessage with arbitrarily small error probability if and only ifthe transmission rate satisfies [24]:

R ≤ 12·mini∈Rn

{I(X (1)s ;Y (1)

i

)}

= 12·I(X (1)s ;Y (1)

n

),

(7)

where equality follows from (6), taking into account that allnoises are i.i.d. Later, once decoded ω and knowing the code-book X2 and its associated encoding function, nodes in Rn

2 E{·} denotes expectation.3 Codewords in X1, X2 have length m/2 since each one is transmitted in

one time slot, respectively.

(and also the source) calculate x2(ω) and transmit it duringphase 2. Hence, considering memoryless time-division chan-nels with uncorrelated signalling between the two phases, thedestination is able to decode ω if

R ≤ 12·I(X (1)s ;Y (1)

d

)+

12·I(X (2)s , X(2)

1:n;Y (2)d

). (8)

Therefore, the maximum source-to-destination transmissionrate for the MRC is given by (8) with equality, subject to(7) being satisfied. Finally, noting that the set of active re-lay nodes Rn can be chosen out of {R1, . . . , RN} concludesthe proof.

As previously mentioned, we consider all receivernodes under unitary power AWGN. The evaluation ofProposition 1 for faded Gaussian channels is established inProposition 2. Previously, from an intuitive view of (5), someconclusions can be inferred: first, we note that the relay nodeswhich have successfully decoded during phase 1 transmitduring phase 2 using a distributed MISO channel to desti-nation. Assuming transmit CSI and phase synchronizationamong them, the performance of such a distributed MISO isequal to that of the actual MISO channel. Therefore, the opti-mum power allocation on the relays will also be the optimumbeamforming [1]. For the power allocation over the two timeslots, we also notice the following tradeoff: the higher thepower allocated during time slot 1 is, the more the relays be-long to the decoding set, but the less power they have duringtime slot 2 to transmit. Both considerations are discussed inProposition 2.

Proposition 2. In a Gaussian, half-duplex, multiple relaychannel with decode-and-forward relaying and N parallel re-lays, the rate

CD&F = max1≤n≤N

12·C(γ1nλ1

)+

12·C(γ2nλ2n

)(9)

is achievable, where

λ1 = |a|2, λ2n = |a|2 +n∑i=1

∣∣ci∣∣2(10)

are the beamforming gains during time slots 1 and 2, respec-tively, and the power allocation is computed from

γ1n = max{(

1μn− 1λ1

), γcn

},

γ2n = min{(

1μn− 1λ2n

), 2P − γcn

} (11)

subject to (μ−1n − λ−1

1 ) + (μ−1n − λ−1

2n ) = 2P, and

γcn = φn +

√φ2n +

2Pλ1

,

φn =(

1μn− 1λ1

)− |bn|2

2λ1λ2n.

(12)

Source-relay path gains have been ordered as∣∣b1∣∣ ≥ · · · ≥ ∣∣bn∣∣ ≥ · · · ≥ ∣∣bN∣∣, (13)


Remark 2. As previously, maximization over n selects the op-timum number of relays. The optimum power allocation γ1n,γ2n results in a constrained temporal water-filling over thetwo slots of the communication. Furthermore, γcn is the min-imum power allocation during time slot 1 that satisfies si-multaneously, for a given set of active relays Rn = {1, . . . ,n},the power constraint (4) and the constraint in (5).

Proof. To derive expression (9), we independently solve theoptimization problems in (5):

maxp(Xs,X

(2)1:n):γ1+γ2=2P

12·I(X (1)s ;Y (1)

d

)+

12·I(X (2)s , X(2)

1:n;Y (2)d

)

s.t. I(X (1)s ;Y (1)

n

)≥ I

(X (1)s ;Y (1)

d

)+ I(X (2)s , X(2)

1:n;Y (2)d

)(14)

for every n ∈ {1, . . . ,N}. First, we notice that for AWGNand memoryless channels, the optimum input signal duringthe two slots is i.i.d. with Gaussian distribution. Hence, themutual information in (14) are given by

I(X (1)s ;Y (1)

d

)= C

(γ1λ1

),

I(X (2)s , X(2)

1:n;Y (2)d

)= C

(γ2λ2n

),

I(X (1)s ;Y (1)

n

)= C

(γ1

∣∣bn∣∣2)

,

(15)

with λ1 and λ2n defined in (10), and γ1 and γ2 the transmit-ted powers during time slot 1 and 2, respectively. Then max-imization (14) reduces to

maxγ1,γ2:γ1+γ2=2P

12·C(γ1λ1

)+

12·C(γ2λ2n

)

s.t. C(γ1

∣∣bn∣∣2)≥ C

(γ1λ1

)+ C

(γ2λ2n

).

(16)

The optimization above is solved in Appendix A yielding (9),with γ1n and γ2n the optimum power allocation on each slotfor a given value n. Maximization over n results in the opti-mum relay selection.

4. ERGODIC ACHIEVABLE RATE

In this section, we analyze the ergodic behavior of the in-stantaneous achievable rate obtained in Proposition 2. Weassume that the source transmits over several, concate-nated two-slot multiple relay transmissions, with uncorre-lated channel conditions (see Figure 2). Thus, it achieves aneffective rate equal to the expectation (on the channel dis-tribution) of the achievable rate defined in Proposition 2,that is, it achieves a rate equal to the ergodic achievable rate.Throughout the paper, we assume random channel fadingand random i.i.d. relay positions, invariant during the two-phase transmission but independent between transmissions.

Accordingly, considering the result in (9), we define theergodic achievable rate4 of the half-duplex MRC as

CeD&F = Ea,b,c

{CD&F

}= Ea,b,c

{max

1≤n≤NCn

},

(17)

where a = |a|2 is the source-to-destination channel; c =[|c1|2, . . . , |cN |2] the relay-to-destination channels, and b =[|b1|2, . . . , |bN |2] the source-to-relay channels ordered as (6).Notice that all elements in c are i.i.d. while, due to ordering,elements in b are mutually dependent. Finally, Cn in (17) isdefined from Proposition 2 as

Cn = 12·C(γ1nλ1

)+

12·C(γ2nλ2n

). (18)

There is no closed-form expression for the ergodiccapacity of the multiple-relay channel in (17); capacitiesC1, . . . , CN are mutually dependent, therefore closed-formexpression for the cumulative density function (cdf) ofmax 1≤n≤NCn cannot be obtained. Hence, we turn our atten-tion to obtaining upper and lower bounds.

4.1. Lower bound

A lower bound can be derived using Jensen’s inequality, tak-ing into account the convexity of the pointwise maximumfunction:

CeD&F = Ea,b,c

{max

1≤n≤NCn

}

≥ max1≤n≤N

Ea,b,c{Cn}.

(19)

The interpretation of such bound is as follows: the inequal-ity shows that the ergodic capacities achieved assuming afixed number of active relays are, obviously, always lowerthan the ergodic capacity achieved with instantaneous op-timal relay selection. Analyzing (19) carefully, we notice thatCn does not depend upon entire vector b but only upon |bn|2.Furthermore, we have seen that Cn depends on fading be-tween source and destination, and between relays and des-tination just in terms of beamforming gains λ1 = |a|2 andλ2n = |a|2 +

∑ ni=1|ci|2; therefore, renaming δ = |a|2 and

βn =∑ n

i=1|ci|2, expression (19) simplifies to

CeD&F ≥ max

1≤n≤NEδ,βn,|bn|2

{Cn}

, (20)

where δ is a unitary-mean, exponential random variable de-scribing the square of the fading coefficient between sourceand destination. Likewise, βn describes the relay beamform-ing gain assuming only the set of relays Rn = {1, . . . ,n} tobe active. It is obtained as the sum of n exponentially dis-tributed, unitary mean random variables, and hence it is dis-tributed as a chi-squared random variable with 2n degrees

4 Notice that, due to the power constraint (4), the ergodic achievable rate isdirectly computed as the expectation of the instantaneous achievable rateof the link.


of freedom. Both variables are described by their probabilitydensity functions (pdf) as

fδ(δ) = e−δ ,

fβn(β) = β(n−1)e−β

(n− 1)!.

(21)

The study of |bn|2 is more involved; bn, as defined previously,is the nth better channel from source to relays, following theordering in (13). As stated earlier, source-to-relay channels in(1) are i.i.d. with complex Gaussian distribution and power(do/d)α; d is the random source-to-relay distance, assumedi.i.d. for all relays and with a generic pdf fd(d), d ∈ [0,d+].Hence, defining ξ∼CN (0, (do/d)α), we make use of orderedstatistics to obtain the pdf of |bn|2 as [25]

f|bn|2 (b) = N !(N − n)!1!(n− 1)!

f|ξ|2 (b)P[|ξ|2 ≤ b

]N−n× P[|ξ|2 ≥ b

]n−1,

(22)

where cumulative density function P[|ξ|2 ≤ b] may be de-rived as

P[|ξ|2 ≤ b

] = 1−∫ d+

0e−b(x/do)α fd(x)dx, (23)

and probability density function f|ξ|2 (b) is computed as thefirst derivative of (23) respect to b:

f|ξ|2 (b) =∫ d+

0

(x

do

)αe−b(x/do)α fd(x)dx. (24)

Therefore, proceeding from (20),

CeD&F ≥ max

1≤n≤N

∫∫ ∞

0E|bn|2

{Cn | δ,βn

}fδ(δ) fβn(β)db dβ,

(25)

where E|bn|2{Cn | δ,β} is the mean of Cn over |bn|2 condi-tioned on beamforming gains δ and βn = β. This mean maybe readily obtained using the pdf (22) and power allocationdefined in (10):

E|bn|2{Cn | δ,β

} = 12

∫ ∞

0

(C(γ1nδ

)+ C

(γ2n(δ + β)

))× f|bn|2 (b)db.

(26)

Notice that

[γ1n, γ2n

] =⎧⎪⎨⎪⎩[(

1μn− 1δ

),(

1μn− 1δ + β

)], b ≥ ψ(δ,β),[

γcn, 2P − γcn], b < ψ(δ,β),

(27)

where

ψ(δ,β) =[(

1μn− 1δ

)+

2Pδ

(1μn− 1δ

)−1]δ(δ + β). (28)

4.2. Upper bound

To upper bound the ergodic achievable rate use, once again,Jensen’s inequality. Nevertheless, in this case, we focus on theconcavity of functions Cn in (18). As previously mentioned,the capacity Cn only depends on 3 variables: the randomsource-to-user channel |a|2, the relays-to-destination beam-forming gain

∑ ni=1|ci|2, and the random path gain |bn|2.

Obviously, it also depends on the power allocation and thepower constraint, but notice that power allocation is a di-rect function of those three variables and that the power con-straint is assumed constant.

The concavity of Cn over the three random variables isshown in Appendix B, and obtained applying properties ofthe composition of concave functions [26]. This result allowsus to conclude that CD&F, being defined as the maximum ofa set concave functions (9), is also concave over the variablesthat define Cn. Therefore, the capacity of regenerative MRCis concave over variable a and vectors b and c, and thus wemay define the following upper bound:

CeD&F = Ea,b,c

{max

1≤n≤NCn

}

≤ max1≤n≤N

Cn(a, b, c),(29)

where a = Ea{a} = 1, c = Ec{c} = [1, . . . , 1], and b =Eb{b} = [|b1|2, . . . , |bn|2, . . . , |bN |2] are the mean squaredsource-to-destination, relay-to-destination, and source-to-relay channels, respectively. Notice that |bn|2=

∫∞0 b f|bn|2 (b)db

is computed by using the pdf in (22). Therefore, consideringthe capacity derivation in Proposition 2, we obtain

Cn(a, b, c) = 12

log 2

(1 + ρ1n

)+

12

log 2

(1 + ρ2n·

(n + 1

)),

(30)

where

ρ1n = max{(

1μn− 1

), γcn

}

ρ2n = min{(

1μn− 1n + 1

),(2P − γcn

)},

γcn =((

1μn− 1

)−

∣∣bn∣∣2

2(n + 1)

)

+

√√√√(( 1μn− 1

)−

∣∣b1∣∣2

2(n + 1)

)2

+ 2P.

(31)

Hence, the upper bound on the ergodic capacity of MRC is

CeD&F ≤ max

1≤n≤N12

log 2

(1 + ρ1n

)+

12

log 2

(1 + ρ2n·(n + 1)

).

(32)

The interpretation of this upper bound leads to the com-parison of faded and nonfaded channels: from (29) we con-clude that the capacity of the MRC with nonfaded channelsis always higher than the ergodic capacity of the MRC withunitary-mean Rayleigh-faded channels.


100 101 102 103

Number of relays

1.5

2

2.5

3

3.5

4

4.5

(bps

/Hz)

Ergodic upper bound, SNR = 5 dBErgodic achievable rateErgodic lower bound, SNR = 5 dBDirect link ergodic capacity, SNR = 5 dBDirect link ergodic capacity, SNR = 10 dBDirect link ergodic capacity, SNR = 15 dB

Figure 3: Ergodic achievable rate in [bps/Hz] of a Gaussian multi-ple relay channel with transmit SNR= 5 dB, under Rayleigh fading.The upper and lower bounds proposed in the paper are shown, andthe ergodic capacity of a direct link plotted as reference.

5. ASYMPTOTIC ACHIEVABLE RATE

In previous sections, we analyzed the instantaneous and er-godic achievable rate of multiple-relay channels with full CSI,assuming a finite number of potential relays N . Results sug-gest (as it can be shown in Figure 3) a growth of the spectralefficiency with the total number relays. Nevertheless, neitherthe result in Proposition 2 nor the bounds (25) and (32) aretractable enough to infer the asymptotic behavior. In this sec-tion, we introduce the necessary approximations to simplifythe problem and to analyze the asymptotic achievable rate ofthe MRC. We show that capacity grows with the logarithmof the branch zero of the Lambert W function of the totalnumber of parallel relays.

Prior to the analysis, in the asymptotic domain (N→∞),we rename variable n in maximization (9) as n = κ·N withκ ∈ [0, 1] (see [25, page 71]), and we introduce four key ap-proximations.

(1) For a large number of network nodes, we consider ca-pacities Cn in (18) defined only by the second slot mu-tual information,5 that is,

Cκ·N= 12C(γ1κ·Nλ1

)+

12C(γ2κ·Nλ2κ·N

)≈ 12C(γ2κ·Nλ2κ·N

).

(33)

5 The proposed approximation is also a lower bound. Thus, the asymptoticperformance of the lower bound is valid to lower bound the asymptoticperformance of the achievable rate.

The proposed approximation is justified by the largebeamforming gain obtained during time slot 2 whenthe number of relays grows to ∞ (as shown in ap-proximation 2). As a consequence, γcκ·N computed inAppendix A is recalculated as

γcκ·N = 2Pλ2κ·N

|bκ·N |2 + λ2κ·N. (34)

To derive (34), we recall that γcκ·N is defined in (A.5)as the power allocation during slot 1 that simulta-neously satisfies

∑ 2i=1γi = 2P and C(γ1|bκ·N |2) =

C(γ1λ1)+C(γ2λ2κ·N ) (i.e., γcκ·N = {γ1 : C(γ1|bκ·N |2) =C(γ1λ1) + C((2P − γ1)λ2κ·N )}). Hence, neglecting thefactor C(γ1λ1), then (34) is obtained.

(2) From the Law of Large Numbers, λ2κ·N in (10) is ap-proximated as λ2κ·N ≈ κ·N .

(3) From [25, pages 255–258], the pdf of the or-dered random variable |bκ·N |2 asymptotically satis-fies pdf|bκ·N |2 = N (Q(1 − κ), ε·N−1) as N→∞ (withε a fixed constant). Q(κ) : [0, 1]→R+ is the inversefunction of the cdf of the squared modulus of thenonordered source-to-relay channel defined in (1),

that is , Q(Pr{|b|2 < b}) = b with b∼CN (0, (do/d)α)and d the source-to-relay random distance. From theasymptotic pdf, the following convergence in probabil-ity holds:

∣∣bκ·N∣∣2 P−→ Q(1− κ). (35)

(4) We consider high-transmitted power, so that μκ·N ≈P−1 is in the power allocation (11).

Making use of those four approximations, we may apply (9)to define the asymptotic instantaneous capacity as

CaD&F =

12

limN→∞

maxκ∈[0,1]

Cκ·N

≈ 12

limN→∞

maxκ∈[0,1]

C(γ2κ·Nλ2κ·N

)

= 12

limN→∞

maxκ∈[0,1]

min{C((

1μκ·N

− 1κ·N

)κ·N

),

C((

2P − γcκ·N)κ·N)}

= 12

limN→∞

maxκ∈[0,1]

min{C(P·κ·N − 1),

C(

2PQ(1− κ)κ·NQ(1− κ) + κ·N

)},

(36)

where first equality follows from Proposition 2, and secondequality from approximation 1; third equality comes fromthe power allocation γ2κ·N in (11) and considering λ2κ·N =2κ·N as approximation 2. Finally, forth equality is obtainedmaking use of approximation 4, and introducing the asymp-totic convergence of |bκ·N |2 in (34).

Let us focus now on the last equality in (36). We noticethat (i) C(P·κ·N − 1) is an increasing function in κ ∈ [0, 1],


(ii)Q(1−κ) is a decreasing function in the same interval, (iii)therefore, C(2P(Q(1− κ)κ·N)/(Q(1− κ) + κ·N)) is asymp-totically a decreasing function in κ ∈ [0, 1]. Hence, in thelimit, the maximum in κ of the minimum of an increasingand a decreasing functions would be given at the intersectionof the two curves. As derived in Appendix C, the intersectionpoint6 κo(N) satisfies

κo(N) ≥ W0(ρN)ρN

(37)

with ρ a fixed constant in (0, 1), and with equality when-ever the relay positions are not random but deterministic. Asmentioned earlier, W0(N) is the branch zero of the LambertW function evaluated at N [23].

Finally, applying the forth equality in (36), we derive

CaD&F =

12

limN→∞

C(P·κo(N)·N − 1

)

≥ 12

limN→∞

log 2

(P·W0(ρN)

ρ

).

(38)

This result shows that, for any random distribution of relays,the capacity of MRC with channel knowledge grows asymp-totically with the logarithm of the Lambert W function ofthe total number relays. However, due to approximations 2and 3, our proof only demonstrates asymptotic performancein probability.


In this section, we evaluate the lower and upper bounds de-scribed in (25) and (32), respectively, and compare themwith the ergodic achievable rate of the link, obtained throughMonte Carlo simulation.

As previously pointed out, we assume i.i.d., unitarymean, Rayleigh-distributed fading from all transmitter nodesto destination, while source-to-relay channels are modelledas a superposition of path loss and unitary mean Rayleighfading. Likewise, source and destination are fixed nodes,while the position of the N relays is i.i.d. throughout asquare, limited at its diagonal by the point-to-point source-to-destination link. As mentioned earlier, the position of re-lays is invariant during the two-slot communication but vari-ant and uncorrelated from one transmission to the other.To deal with propagation effects, we defined a simplified ex-ponential indoor propagation model with path loss expo-nent α = 4. Finally, we consider normalized distances, defin-ing distance between source and destination equal to 1, andsource-relay random distance di ∈ [0, 1].

Taking into account the considerations above, we focusthe analysis on the number of relay nodes and the transmit-ted SNR, that is, P/σ2

o. Figure 3 depicts the ergodic boundscomputed for transmit SNR equal to 5 dB for an MRC withthe number of relay nodes ranging from 5 to 200. Likewise,

6 For a fixed number of relays N , a fixed intersection point κo is derived.Thus, κo = κo(N).

100 101 102 103

Number of relays

2.5

3

3.5

4

4.5

5

5.5

6

(bps

/Hz)


Figure 4: Ergodic achievable rate in [bps/Hz] of a Gaussian multi-ple relay channel with transmit SNR = 10 dB, under Rayleigh fad-ing. The upper and lower bounds proposed in the paper are shown,and the ergodic capacity of a direct link plotted as reference.

Figures 4 and 5 plot results for transmit SNR equal to 10 dBand 20 dB, respectively. Firstly, we clearly note that, for allplots, ergodic bounds and simulated result increase with thenumber of users, as we have previously demonstrated in theasymptotic capacity section.

Moreover, the comparison of the three plots shows thatthe advantage of relaying diminishes as the transmittedpower increases. In such a way, it can be seen that for trans-mit SNR = 5 dB only N = 20 parallel relay nodes are neededto double the noncooperative capacity, while for SNR =10 dB more than N = 200 nodes would be necessary to ob-tain twice the spectral efficiency. Furthermore, we may seethat for SNR = 5 dB with only 10 relays, it is possible to ob-tain the same ergodic capacity as a Rayleigh-faded direct linkwith SNR = 10 dB, while to obtain the same power savingfor MRC with SNR = 20 dB, 50 nodes are needed. Finally,plots show that the accuracy of the presented bounds growsas the transmit SNR diminishes, which may be interpreted interms of the meaning of such bounds: for decreasing trans-mitted power, the effect of instantaneous relay selection andthe effect of Rayleigh fading over the cooperative links losesignificance.

Figures 6–8 show results on the mean number of activerelays versus the total number of relay nodes. Recall that theoptimum number of relay nodes is calculated from maxi-mization over n in Proposition 2. Specifically, Figure 6 de-picts results for SNR = 5 dB while Figures 7 and 8 showcooperating nodes for SNR = 10 dB and SNR = 20 dB.In all three, the number of active nodes n that maximizesthe lower and upper bounds, (25) and (32), respectively, is


100 101 102 103

Number of relays

5.5

6

6.5

7

7.5

8

8.5

9

9.5

(bps

/Hz)


Figure 5: Ergodic achievable rate in [bps/Hz] of a Gaussian multi-ple relay channel with transmit SNR = 15 dB, under Rayleigh fad-ing. The upper and lower bounds proposed in the paper are shown,and the ergodic capacity of a direct link plotted as reference.

also plotted; hence, it allows for comparison between themean number of relays with capacity achieving relaying andthe optimum number of relays with no instantaneous re-lay selection (25) and with no fading channels (32), respec-tively. Firstly, results show that the simulated mean num-ber of relays is close to the number of relays maximizingthe upper and lower bounds, being closer for the low SNRregime. Finally, we notice that, as the transmit SNR in-creases, the percentage of relays cooperating with the sourcedecreases. Therefore, we conclude that regenerative relayingis, as previously mentioned, more powerful in the low SNRregime.

7. CONCLUSIONS

In this paper, we examined the achievable rate of a decode-and-forward (D&F) multiple-relay channel with half-duplexconstraint and transmitter and receiver channel state infor-mation. The transmission was arranged in two phases: dur-ing the first phase, the source transmits its message to re-lays and destination. During the second phase, the relaysand the source are configured as a distributed antenna ar-ray to transmit extra parity bits. The instantaneous achiev-able rate for the optimum relay selection and power allo-cation was obtained. Furthermore, we studied and boundedthe ergodic performance of the achievable rate for Rayleigh-faded channels. We also found the asymptotic performanceof the achievable rate in number of relays. Results show that

0 20 40 60 80 100 120 140 160 180 200

Total number of relays

10

15

20

25

30

35

40

45

50

55

60

Perc

enta

geof

acti

vere

lays

(%)

Active relays with the upper bound, SNR = 5 dBActive relays, SNR = 5 dBActive relays with the lower bound, SNR = 5 dB

Figure 6: Expected number of active relays (in %) of a multiplerelay channel with transmit SNR = 5 dB, under Rayleigh fading.The number of relays that optimizes the upper and lower boundsare shown for comparison.

0 20 40 60 80 100 120 140 160 180 200


10

15

20

25

30

35

40

45

Perc

enta

geof

acti

vere

lays

(%)

Active relays with the upper boundActive relaysActive relays with the lower bound


(i) CD&F ∝ log (W0(N)) as N→∞; (ii) with regenerative re-laying, higher capacity is obtained for low signal-to-noise ra-tio, (iii) the percentage of active relays (i.e., the number ofnodes who can decode the source message) decreases for in-creasing N , and (iv) this percentage is low, even at low SNR,due to the regenerative constraint.


0 20 40 60 80 100 120 140 160 180 200


4

6

8

10

12

14

16

18

20

22

24

Perc

enta

geof

acti

vere

lays

(%)

Active relays with the upper bound, SNR = 15 dBActive relays, SNR = 15 dBActive relays with the lower bound, SNR = 15 dB


APPENDICES

A. OPTIMIZATION PROBLEM

For completeness of explanation, in the appendix we solveoptimization problem (16), which can be recast as fol-lows:

C = maxγ1,γ2

12

2∑i=1

log 2

(1 + γiλi

)

s.t.2∑i=1

γi = 2P,

γi ≥Π2i=1

(1 + γiλi

)− 1∣∣bn∣∣2 ,

(A.1)

which is convex in both γ1 ∈ R+ and γ2 ∈ R+. The Lagrangedual function of the problem is

L(γ1, γ2,μ, ν

) = 2∑i=1

log(1 + γiλi

)− μ( 2∑i=1

γi − 2P

)

+ ν(γ1 −

Π2i=1(1 + γiλi)− 1

|bn|2)

,

(A.2)

where μ and ν are the Lagrange multipliers for first andsecond constraints, respectively. The three KKT conditions

(necessary and sufficient for optimality) of the dual problemare

(i)λi

1 + γiλi− μ + ν

d

dγi

(γi −

Π2i=1

(1 + γiλi

)− 1∣∣bn∣∣2

)= 0

for i ∈ {1, 2},

(ii) μ

( 2∑i=1

γi − 2P

)= 0,

(iii) ν(γ1 −

Π2i=1(1 + γiλi)− 1∣∣bn∣∣2

)= 0.

(A.3)

Notice that the set (ν∗, γ∗1 , γ∗2 ,μ∗):

ν∗ = 0, γ∗i =(

1μ∗− 1λi

)+

,1μ∗

= P +12

2∑i=1

1λi

,

(A.4)

satisfies KKT conditions hence yielding the optimum so-lution.7 However, taking into account that optimal primalpoints must satisfy the two constraints in (A.1), and that

2∑i=1

γi = 2P

γ1 ≥Π2i=1

(1+γiλi

)−1∣∣bn∣∣2

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭−→ γ1≥γc=φ+

√φ2 +

2Pλ1∈ R+

(A.5)

with φ = (1/μ∗ − 1/λi) − |bn|2/2λ1λ2. Then, the result inoptimum power allocation is

γ∗1 = max{(

1μ∗− 1λi

), γc}

,

γ∗2 = 2P − γ∗1 ,

1μ∗

= P +12

2∑i=1

1λi.

(A.6)

B. CONCAVITY OF CN

In the appendix, we prove the concavity of capacity Cn (de-fined in (18) based on (9)) over random variables |a|2,∑ n

i=1|ci|2, and |bn|2. To do so, we first rewrite the functionunder study as a composition of functions:

Cn = C(

max(Γ1(x),Γ2(x)

))+ C

(min

(Ψ1(x),Ψ2(x)

)),

(B.7)

7 Using standard notation, we define (A)+ = max {A, 0}.


where x = [|a|2,∑ n

i=1|ci|2, |bn|2] and

Γ1(x) =(

1μn− 1|a|2

)|a|2, Γ1 : R3+ −→ R,

Γ2(x) = γcn(x)|a|2, Γ2 : R3+ −→ R,

Ψ1(x) =(

1μn− 1

|a|2 +∑ n

i=1

∣∣ci∣∣2

)

×(|a|2 +

n∑i=1

∣∣ci∣∣2)

, Ψ1 : R3+ −→ R,

Ψ2(x) = (2P − γcn(x)

)(|a|2 +n∑i=1

∣∣ci∣∣2)

, Ψ2 : R3+ −→ R.

(B.8)

First, we notice that pointwise maximum and pointwiseminimum functions are nondecreasing functions with Hes-sian equal to zero. Next, computing the Hessian of Γ1(x)and Γ2(x) (respect to x), it is shown that both are con-cave functions. Therefore, from [26, pages 86-87], we derivethat max (Γ1(x),Γ2(x)) is concave on x. Accordingly, we mayshow that Ψ1(x) and Ψ2(x) are also concave functions, andso is min (Ψ1(x),Ψ2(x)). Hence, considering that the sum ofconcave functions is always concave, and that C(x) is a con-cave nondecreasing function, we derive that Cn is concaveon x.

C. INTERSECTION OF CAPACITY CURVES

In this appendix, we analyze the intersection point κo ofcurves f1(κ) = log 2(P·κN) and f2(κ) = log 2(1 + 2P(Q(1 −κ)κ·N)/(Q(1−κ) +κ·N)) for a given number of relays N . Todo so, we set f1(κo) = f2(κo) to obtain8

Q(1− κo

) ≈ κo·N. (C.9)

From approximation 3 in Section 5, equality above is equiv-alent to

Pr{|b|2 ≤ κo·N

} = 1− κo (C.10)

with b∼CN (0, (do/d)α) and d the source-to-relay randomdistance. Furthermore, making use of the cdf in (23), we ob-tain

κo = 1− Pr{|b|2 ≤ κo·N

}=∫ d+

0e−(x/do)ακo·N fd(x)dx.

(C.11)

We can now apply Jensen’s inequality for convex functions,in order to lower bound the integral as

κo ≥ e−(E{x}/do)ακo·N (C.12)

with E{x} = ∫ d+

0 x fd(x)dx. Equality is satisfied wheneverthe relays position are not random but deterministic, that is,

8 Approximation (C.9) is obtained neglecting the effect of 1 within the log-arithm in f2(κ), assuming sufficiently large transmitted power P.

fd(x) = δ(x−dr). Next, from [23], we directly solve inequal-ity (C.12) over κo as

κo(N) ≥ W0(ρN)ρN

(C.13)

with ρ = −(E{x}/do)α a fixed constant in (0, 1), and W0(·)the branch zero of the Lambert W function.

This solution is applicable for every possible random dis-tribution of relays.

ACKNOWLEDGMENTS

The material of this paper was partially presented at the 39thAsilomar Conference on Signals, Systems and Computers,Pacific Grove, Calif, November 2005 and at the IEEE WirelessCommunications and Networking Conference (WCNC), LasVegas, Nev, March 2006. This work was partially supportedby the Spanish Ministry of Science and Education grantTEC2005-08122-C03-02/TCM (ULTRARED) and TEC2006-10459/TCM (PERSEO), by the European Comission un-der project IST-2005-27402 (WIP) and by Generalitat deCatalunya under Grant SGR-2005-00690.

REFERENCES

[1] I. Telatar, “Capacity of multi-antenna Gaussian channels,” Eu-ropean Transactions on Telecommunications, vol. 10, no. 6, pp.585–595, 1999.

[2] S. M. Alamouti, “A simple transmit diversity technique forwireless communications,” IEEE Journal on Selected Areas inCommunications, vol. 16, no. 8, pp. 1451–1458, 1998.

[3] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achiev-able rates, and sum-rate capacity of Gaussian MIMO broad-cast channels,” IEEE Transactions on Information Theory,vol. 49, no. 10, pp. 2658–2668, 2003.

[4] D. Tse and P. Viswanath, Fundamentals of Wireless Communi-cations, Cambridge University Press, Cambridge, UK, 1st edi-tion, 2005.

[5] E. Zimmermann, P. Herhold, and G. Fettweis, “On the perfor-mance of cooperative relaying protocols in wireless networks,”European Transactions on Telecommunications, vol. 16, no. 1,pp. 5–16, 2005.

[6] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategiesand capacity theorems for relay networks,” IEEE Transactionson Information Theory, vol. 51, no. 9, pp. 3037–3063, 2005.

[7] D. Chen and J. N. Laneman, “The diversity-multiplexingtradeoff for the multiaccess relay channel,” in Proceedings ofthe 40th Annual Conference on Information Sciences and Sys-tems, pp. 1324–1328, Princeton, NJ, USA, March 2006.

[8] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperationdiversity—part I: system description,” IEEE Transactions onCommunications, vol. 51, no. 11, pp. 1927–1938, 2003.

[9] A. Høst-Madsen and J. Zhang, “Capacity bounds and powerallocation for wireless relay channels,” IEEE Transactions onInformation Theory, vol. 51, no. 6, pp. 2020–2040, 2005.

[10] T. Cover and A. El Gamal, “Capacity theorems for the relaychannel,” IEEE Transactions on Information Theory, vol. 25,no. 5, pp. 572–584, 1979.

[11] J. N. Laneman, “Cooperative diversity in wireless net-works: algorithms and architectures,” Ph.D. Dissertation, Mas-sachusetts Institute of Technology, Cambridge, Mass, USA,2002.


[12] J. N. Laneman, D. Tse, and G. W. Wornell, “Cooperative diver-sity in wireless networks: efficient protocols and outage behav-ior,” IEEE Transactions on Information Theory, vol. 50, no. 12,pp. 3062–3080, 2004.

[13] A. F. Dana, M. Sharif, R. Gowaikar, B. Hassibi, and M. Effros,“Is broadcast plus multiaccess optimal for Gaussian wirelessnetworks?” in Proceedings of the 37th Asilomar Conference onSignals, Systems, and Computers, vol. 2, pp. 1748–1752, PacificGrove, Calif, USA, November 2003.

[14] R. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relaychannels: performance limits and space-time signal design,”IEEE Journal on Selected Areas in Communications, vol. 22,no. 6, pp. 1099–1109, 2004.

[15] A. del Coso and C. Ibars, “Achievable rate for Gaussian multi-ple relay channels with linear relaying functions,” in Proceed-ings of IEEE International Conference on Acoustics, Speech, andSignal Processing (ICASSP ’07), vol. 3, pp. 505–508, Honolulu,Hawaii, USA, April 2007.

[16] A. El Gamal, M. Mohseni, and S. Zahedi, “Bounds on capacityand minimum energy-per-bit for AWGN relay channels,” IEEETransactions on Information Theory, vol. 52, no. 4, pp. 1545–1561, 2006.

[17] A. del Coso and C. Ibars, “Partial decoding for synchronousand asynchronous Gaussian multiple relay channels,” in Pro-ceedings of the International Conference on Communications(ICC ’07), pp. 713–718, Glasgow, Scotland, UK, June 2007.

[18] A. Høst-Madsen, “On the capacity of wireless relaying,” inProceedings of the 56th IEEE Vehicular Technology Conference(VTC ’02), vol. 3, pp. 1333–1337, Vancouver, BC, Canada,September 2002.

[19] A. El Gamal, “Capacity theorems for relay channels,” in Pro-ceedings of MSRI Workshop on Mathematics of Relaying and Co-operation in Communication Networks, Berkeley, Calif, USA,April 2006.

[20] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wirelessnetworks,” IEEE Transactions on Information Theory, vol. 49,no. 10, pp. 2415–2425, 2003.

[21] M. Dohler, Virtual antenna arrays, Ph.D. thesis, King’s CollegeLondon, London, UK, 2003.

[22] I. Maric and R. Yates, “Bandwidth and power allocation forcooperative strategies in Gaussian relay networks,” in Proceed-ings of the 38th Asilomar Conference on Signals, Systems andComputers, vol. 2, pp. 1907–1911, Pacific Grove, Calif, USA,November 2004.

[23] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, andD. E. Knuth, “On the Lambert W function,” Advances in Com-putational Mathematics, vol. 5, no. 4, pp. 329–359, 1996.

[24] T. Cover and J. Thomas, Elements of Information Theory, WileySeries in Telecommunications, Wiley-Interscience, New York,NY, USA, 1991.

[25] H. David, Order Statistics, John Wiley & Sons, New York, NY,USA, 2rd edition, 1981.

[26] S. Boyd and L. Vandenberghe, Convex Optimization, Cam-bridge University Press, Cambridge, UK, 1st edition, 2004.


Research ArticleA Simplified Constant Modulus Algorithm forBlind Recovery of MIMO QAM and PSK Signals:A Criterion with Convergence Analysis

Aissa Ikhlef and Daniel Le Guennec

IETR/SUPELEC, Campus de Rennes, Avenue de la Boulaie, CS 47601, 35576 Cesson-Sevigne, France

Received 31 October 2006; Revised 18 June 2007; Accepted 3 September 2007


The problem of blind recovery of QAM and PSK signals for multiple-input multiple-output (MIMO) communication systemsis investigated. We propose a simplified version of the well-known constant modulus algorithm (CMA), named simplified CMA(SCMA). The SCMA cost function consists in projection of the MIMO equalizer outputs on one dimension (either real or imag-inary part). A study of stationary points of SCMA reveals the absence of any undesirable local stationary points, which ensures aperfect recovery of all signals and a global convergence of the algorithm. Taking advantage of the phase ambiguity in the solutionof the new cost function for QAM constellations, we propose a modified cross-correlation term. It is shown that the proposedalgorithm presents a lower computational complexity compared to the constant modulus algorithm (CMA) without loss in per-formances. Some numerical simulations are provided to illustrate the effectiveness of the proposed algorithm.

Copyright © 2007 A. Ikhlef and D. Le Guennec. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

In the last decade, the interest in blind source separa-tion (BSS) techniques has been important. The problemof blind recovery of multiple independent and identicallydistributed (i.i.d.) signals from their linear mixture in amultiple-input multiple-output (MIMO) system arises inmany applications such as spatial division multiple access(SDMA), multiuser communications (such as CDMA forcode division multiple access), and more recently Bell Labslayered space-time (BLAST) [1–3]. The aim of blind sig-nals separation is to retrieve source signals without the useof a training sequence, which can be expensive or impos-sible in some practical situations. Another interesting classof blind methods is blind identification. Unlike blind sourceseparation, the aim of blind identification is to find an es-timate of the MIMO channel matrix [4–6]. Once this es-timate has been obtained, the source signals can be effi-ciently recovered using MIMO detection methods, such asmaximum likelihood (ML) [7] and BLAST [8] detectionmethods. The main difference between blind source sepa-ration and blind identification is that in the first case, thesource signals are recovered directly from the observations,whereas in the second case, a MIMO detection algorithm

is needed, which may increase complexity (complexity de-pends on used methods). Note that, unlike BSS techniques,ML detector is nonlinear but optimum and suffers fromhigh complexity. Sphere decoding [7] allows to reduce con-siderably ML detector complexity. On the other hand, theperformance of MIMO detection methods depends stronglyon the quality of the channel estimate which results fromblind identification. In this paper, we consider the problemof blind source separation of MIMO instantaneous chan-nel.

In literature, the constant modulus of many communi-cation signals, such as PSK and 4-QAM signals, is a widelyused property in blind source separation and blind equaliza-tion. The initial idea can be traced back to Sato [9], Godard[10], and Treichler et al. [11, 12]. The algorithms are knownas CMAs. The first application after blind equalization wasblind beamforming [13, 14] and more recently blind signalsseparation [15, 16]. In the case of constant modulus signals,CMA has proved reasonable performances and desired con-vergence requirements. On the other hand, the CMA yieldsa degraded performance for nonconstant modulus signalssuch as the quadrature amplitude modulation (QAM) sig-nals, because the CMA projects all signal points onto a singlemodulus.


In order to improve the performance of the CMA forQAM signals, the so-called modified constant modulus algo-rithm (MCMA) [17], known as MMA for multimodulus al-gorithm, has been proposed [18–20]. This algorithm, insteadof minimizing the dispersion of the magnitude of the equal-izer output, minimizes the dispersion of the real and imagi-nary parts separately; hence the MMA cost function can beconsidered as a sum of two one-dimensional cost functions.The MMA provides much more flexibility than the CMA andis better suited to take advantage of the symbol statistics re-lated to certain types of signal constellations, such as non-square and very dense constellations [18]. Please notice thatboth CMA and MMA are two-dimensional (i.e., employ bothreal and imaginary part of the equalizer outputs). Anotherclass of algorithm has been proposed recently and namedconstant norm algorithm (CNA), whose CMA represents aparticular case [21, 22].

In this paper, we propose a simplified version of the CMAcost function named simplified CMA (SCMA) and basedonly on one dimension (either real or imaginary part), asopposed to CMA. The major advantage of SCMA is its lowcomplexity compared to that of CMA and MMA. Because,instead of using both real and imaginary parts as in CMAand MMA, only one dimension, the real or imaginary part,is considered in SCMA, which makes it very attractive forpractical implementation especially when complexity issuearises such as in user’s side. We will demonstrate that only theexisting stationary points of the SCMA cost function corre-spond to a perfect recovery of all source signals except for thephase and permutation indeterminacy. We will show that thephase rotation is not the same for QAM, 4-PSK, and P-PSK(P ≥ 8). Moreover, in order to reduce the complexity further,we will introduce a modified cross-correlation term by tak-ing advantage of the phase ambiguity of the SCMA cost func-tion for QAM constellations. An adaptive implementation bymeans of the stochastic gradient algorithm (SGA) will be de-scribed. A part of the results presented in this paper (QAMcase with its convergence analysis) was previously reportedin [23].

The remainder of the paper is organized as follows. InSection 2, the problem formulation and assumptions are in-troduced. In Section 3, we describe the SCMA criterion. Theconvergence analysis of the proposed cost function is car-ried out in Section 4. Section 5 introduces a modified cross-correlation constraint for QAM constellations. In Section 6,we present an adaptive implementation of the algorithm. Fi-nally, Section 7 presents some numerical results.


We consider a linear data model which takes the followingform:

y(n) = Ha(n) + b(n), (1)

where a(n) = [a1(n), . . . , aM(n)]T is the (M × 1) vector ofthe source signals, H is the (N ×M) MIMO linear memory-less channel, y(n) = [y1(n), . . . , yN (n)]T is the (N ×1) vectorof the received signals, and b(n) = [b1(n), . . . , bN (n)]T is the

(N×1) noise vector.M andN represent the number of trans-mit and receive antennas, respectively.

In the case of the MIMO frequency selective channel(convolutive model), the system can be reduced to the modelin (1) tanks to the linear prediction method presented in [5].Afterwards, blind source separation methods can be applied.The following assumptions are considered:

(1) H has full column rank M,(2) the noise is additive white Gaussian independent from

the source signals,(3) the source signals are independent and identically dis-

tributed (i.i.d), mutually independent E[aaH] = σ2aIM ,

and drawn from QAM or PSK constellations.

Please notice that these assumptions are not very restrictiveand satisfied in BLAST scheme whose corresponding modelis given in (1). Moreover, throughout this paper by QAMconstellation we mean only square QAM constellation. In or-der to recover the source signals, the received signal y(n) isprocessed by an (N ×M) receiver matrix W = [w1, . . . , wM].Then, the receiver output can be written as

z(n) = WTy(n) = WTHa(n) + WTb(n)

= GTa(n) + ˜b(n),(2)

where z(n) = [z1(n), . . . , zM(n)]T is the (M × 1) vector ofthe receiver output, G = [g1, . . . , gM] = HTW is the (M ×M) global system matrix, and ˜b(n) is the filtered noise at thereceiver output.

The purpose of blind source separation is to find the ma-trix W such that z(n) = a(n) is an estimate of the sourcesignals.

Please note that in blind signals separation, the best thatcan be done is to determine W up to a permutation and scalarmultiple [3]. In other words, W is said to be a separationmatrix if and only if

GT = WTH = PΛ, (3)

where P is a permutation matrix and Λ a nonsingular diago-nal matrix.

Throughout this paper, we use small and capital boldfaceletters to denote vectors and matrices, respectively. The sym-bols (·)∗ and (·)T denote the complex conjugate and trans-pose, respectively, (·)H is the Hermitian transpose, and Ip isthe (p × p) identity matrix.

3. THE PROPOSED CRITERION

Unlike the CMA algorithm [10], whose aim consists in con-straining the modulus of the equalizer outputs to be on a cir-cle (projection onto a circle), we suggest to project the equal-izer outputs onto one dimension (either real or imaginarypart). To do so, we suggest to penalize the deviation of thesquare of the real (imaginary) part of the equalizer outputsfrom a constant.

A. Ikhlef and D. Le Guennec 3

43210−1−2−3−4

Real

−4−3−2−1

01234

Imag

inar

ySource signal 1

(a)

6420−2−4−6

Real

−6

−4

−2

0

2

4

6

Imag

inar

y

Mixture 1

(b)

43210−1−2−3−4

Real

−4−3−2−1

01234

Imag

inar

y

Equalizer output 1

(c)

43210−1−2−3−4

Real

−4−3−2−1

01234

Imag

inar

y

Source signal 2

(d)

6420−2−4−6

Real

−6

−4

−2

0

2

4

6

Imag

inar

y

Mixture 2

(e)

43210−1−2−3−4

Real

−4−3−2−1

01234

Imag

inar

y

Equalizer output 2

(f)

Figure 1: 16-QAM constellation. Left column: the constellations of the transmitted signals, middle column: the constellations of the receivedsignals (mixtures), right column: the constellations of the recovered signals.

For the �th equalizer, we suggest to optimize the follow-ing criterion:

minw�

J(

w�) = E

[

(

zR,�(n)2 − R)2]

, � = 1, . . . ,M, (4)

where zR,�(n) denotes the real part of the �th equalizer out-put z�(n) = wT

� y(n) and R is the dispersion constant fixedby assuming a perfect equalization with respect to the zeroforcing (ZF) solution, and is defined as

R = E[

aR(n)4]

E[

aR(n)2] , (5)

where aR(n) denotes the real part of the source signal a(n).The term on the right side of the equality (4) prevents

the deviation of the square of the real part of the equalizeroutputs from a constant. The minimization of (4) allowsthe recovery of only one signal at each equalizer output (seeproof in Section 4). But the algorithm minimization (4) doesnot ensure the recovery of all source signals because it mayconverge in order to recover the same source signal at manyoutputs. In order to avoid this problem, we suggest to usea cross-correlation term due to its computational simplicity.Then (4) becomes

minw�

J(

w�) = E

[

(

zR,�(n)2 − R)2]

+ α�−1∑

i=1

∣

∣r�i(n)∣

∣

2, � = 1, . . . ,M,

(6)

where α ∈ R+ is the mixing parameter and r�i(n) =E[z�(n)z∗i (n)] is the cross-correlation between the �th and

the ith equalizer outputs and prevents the extraction of thesame signal at many outputs. Then the first term in (6) en-sures the recovery of only one signal at each equalizer out-put and the cross-correlation term ensures that each equal-izer output is different from the other ones; this results in therecovery of all source signals (see Section 4). In the followingsections, we name (6) the cross-correlation simplified CMA(CC-SCMA) criterion. In (6) we could also use the imaginarypart thanks to the symmetry of the QAM and PSK constel-lations. Since the analysis is the same for the imaginary part,throughout this paper, we only consider the real part.

4. CONVERGENCE ANALYSIS

Theorem 1. Let M be i.i.d. and mutually independent signalsai(n), i = 1, . . . ,M, which share the same statistical proper-ties, are drawn from QAM or PSK constellations and are trans-mitted via an (M × N) MIMO linear memoryless channeland without the presence of noise. Provided that the weight-ing factor α is chosen to satisfy α ≥ 2E[a4

R]/σ4ad

2 (whered = 1 for QAM and P-PSK (P ≥ 8) constellations andd = √

2 for 4-PSK constellation), the algorithm in (6) willconverge to a setting that corresponds, in the absence of anynoise, to a perfect recovery of all transmitted signals, and theonly stable minima are the Dirac-type vector taking the fol-lowing form: g� = [0, . . . , 0, d�e jφ� , 0, . . . , 0]T , where g� isthe �th column vector of G, d� is the amplitude, and φ� isthe phase rotation of the nonzero element. The pair (d� ,φ�) isgiven by {1, modulo (π/2)}, {√2, modulo [(2k + 1)π/4]}, and{1, arbitrary in [0, 2π]} for QAM, 4-PSK, and P-PSK (P ≥ 8)constellations, respectively.


1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

ySource signal 1

(a)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Mixture 1

(b)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Equalizer output 1

(c)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Source signal 2

(d)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Mixture 2

(e)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Equalizer output 2

(f)

Figure 2: 8-PSK constellation. Left column: the constellations of the transmitted signals, middle column: the constellations of the receivedsignals (mixtures), right column: the constellations of the recovered signals.

Proof. For simplicity, the analysis is restricted to noise-freecase, that is,

z(n) = GTa(n). (7)

Note that due to the assumed full column rank of H, all re-sults in the G domain will translate to the W domain as well.For convenience, the stationary points study will be carriedout in the G domain [24].

Considering (7), the cross-correlation term in (6) can besimplified as

E[

z�(n)z∗i (n)] = E

[

wT� Ha(n)wH

i H∗a(n)∗]

= gT� E[

a(n)a(n)H]

g∗i = σ2agT� g∗i = σ2

agHi g� ,(8)

where we use the fact that wTi H = gTi .

Using (8) in (6), we get

J(g�) = E[

(

zR,�(n)2 − R)2]

+ ασ4a

�−1∑

i=1

|gHi g�|2, � = 1, . . . ,M.(9)

From (9), we first notice that the adaptation of each g� de-pends only on g1, . . . , g�−1. Then, we can begin by the firstoutput, because g1 is optimized independently from all theother vectors g2, . . . , gM . Hence, for the first equalizer, g1, wehave

ming1

J(g1) = E[

(

zR,1(n)2 − R)2]

. (10)

By developing (10), we get (for notation convenience, in thefollowing, we will omit the time index n)

J(g1) = E[

z4R,1

]− 2RE[

z2R,1

]

+ R2. (11)

Because the development is not the same for QAM, 4-PSK,and P-PSK (P ≥ 8) constellations, we will enumerate theproof of each case separately.

4.1. QAM case

After a straightforward development of the terms in (11)with respect to statistical properties of QAM signals (seeAppendix A), (11) can be written as

J(g1) = E[

a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2 − 1

)2

+ β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

+ 2βM∑

k=1

g2R,k1g

2I ,k1 − E

[

a4R

]

+ R2,

(12)

where

g1 =[

g11, . . . , gM1]T

,

gk1 = gR,k1 + jgI ,k1,

β = 3E2[a2R

]− E[a4R

] = −κaR > 0,

(13)

and κaR = E[a4R]− 3E2[a2

R] represents the kurtosis of the realparts of the symbols. It is always negative in the case of PSKand QAM signals.


1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

ySource signal 1

(a)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Mixture 1

(b)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Equalizer output 1

(c)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Source signal 2

(d)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Mixture 2

(e)

1.510.50−0.5−1−1.5

Real

−1.5

−1

−0.5

0

0.5

1

1.5

Imag

inar

y

Equalizer output 2

(f)

Figure 3: 4-PSK constellation. Left column: the constellations of the transmitted signals, middle column: the constellations of the receivedsignals (mixtures), right column: the constellations of the recovered signals.

80006000400020000

Iteration

6

10

14

18

22

26

SIN

R(d

B)

CC-SCMACC-CMALMS (supervised)

Figure 4: Performance comparison in term of SINR of the pro-posed algorithm (CC-SCMA) with CC-CMA and supervised LMS.

The minimum of (11) can be found easily by replacingthe equalizer output in (12) by any of the transmitted signals,it is given by

Jmin = E[

(

a2R − R

)2]

= E[

a4R

]− 2RE[

a2R

]

+ R2. (14)

From (5), we have

RE[

a2R

] = E[

a4R

]

. (15)

Then

Jmin = −E[

a4R

]

+ R2. (16)

Comparing (12) and (16), we can write

J(

g1) = Jmin + β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

+ E[

a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2 − 1

)2

+ 2βM∑

k=1

g2R,k1g

2I ,k1.

(17)

Since β > 0 and that

( M∑

k=1

∣

∣gk1∣

∣

2)2

≥M∑

k=1

∣

∣gk1∣

∣

4, (18)

we have

β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

≥ 0. (19)

According to (19), J(g1) is composed only of positive terms.Then, minimizing J(g1) is equivalent to finding g1, whichminimizes all terms simultaneously. One way to find theminimum of (17) is to look for a solution that cancels thegradients of each term separately. From (16), we know that


Jmin is a constant (∂Jmin/∂g∗�1 = 0), hence we only deal with

the reminder terms. For that purpose, let us have

J(

g1) = Jmin + J1

(

g1)

+ J2(

g1)

+ J3(

g1)

, (20)

where

J1(

g1) = β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

,

J2(

g1) = E

[

a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2 − 1

)2

,

J3(

g1) = 2β

M∑

k=1

g2R, k1g

2I ,k1.

(21)

When we compute the derivatives of J1(g1), J2(g1), and J3(g1)with respect to g∗�1, we find

∂J1(

g1)

∂g∗�1= 2βg�1

( M∑

k=1

∣

∣gk1∣

∣

2 − ∣∣g�1∣

∣

2)

= 0

=⇒M∑

k=1,k�=�

∣

∣gk1∣

∣

2 = 0,

(22)

∂J2(

g1)

∂g∗�1= 2E

[

a4R

]

g�1

( M∑

k=1

∣

∣gk1∣

∣

2 − 1

)

= 0

=⇒M∑

k=1

∣

∣gk1∣

∣

2 = 1,

(23)

∂J3(

g1)

∂g∗�1= 2β

(

gR,�1g2I ,�1 + jg2

R,�1gI ,�1) = 0

=⇒ gR,�1=0 or gI ,�1=0 or gR,�1=gI ,�1=0.(24)

Equation (22) implies that only one entry, g�1, of g1 isnonzero and the others are zeros. Equation (23) indicatesthat the modulus of this entry must be equal to one (|g�1|2 =1). Finally, from (24) either the real part or the imaginarypart must be equal to zero. As a result of (23), the squaredmodulus of the nonzero part is equal to one, that is, eitherg2R,�1 = 1 and g2

I ,�1 = 0 or g2R,�1 = 0 and g2

I ,�1 = 1. Therefore,the solution g�1 is either a pure real or a pure imaginary withmodulus equal to one, which corresponds to

g�1 = e jm1(π/2), (25)

where m1 is an arbitrary integer.This solution shows that the minimization of J(g1)

forces the equalizer output to form a constellation that corre-sponds to the source constellation with a modulo π/2 phaserotation.

From (22), (23), (24), and (25), we can conclude thatthe only stable minima for g1 take the following form:g1 = [0, . . . , 0, e jm1(π/2), 0, . . . , 0]T , that is, only one entry isnonzero, pure-real, or pure-imaginary with modulus equalto one, which can be at any of the M positions and all theother ones are zeros. This solution corresponds to the recov-

ery of only one source signal and cancels the others. For thesecond equalizer g2, from (9), we have

J(

g2) = E

[

(

zR,2(n)2 − R)2]

+ ασ4a

∣

∣gH1 g2∣

∣

2, (26)

this means that the adaptation of g2 depends on g1.We examine the convergence of g2 once g1 has converged

to one signal, because the adaptation of g1 is realized inde-pendently from the other gi. For the sake of simplicity, andwithout loss of generality, we consider that g1 has convergedto the first signal, that is,

g1 =[

de jϕ | 0, . . . , 0]T

, d = 1, ϕ = m1π

2, (27)

then∣

∣gH1 g2∣

∣

2 = d2∣

∣g12∣

∣

2. (28)

Using this and the result in Appendix A, J(g2) in (26) can beexpressed as follows:

J(

g2) = E

[

a4R

]

( M∑

k=1

∣

∣gk2∣

∣

2 − 1

)2

+ β

[( M∑

k=1

∣

∣gk2∣

∣

2)2

−M∑

k=1

∣

∣gk2∣

∣

4]

+ 2βM∑

k=1

g2R,k2g

2I ,k2

− E[a4R

]

+ R2 + ασ4ad

2∣

∣g12∣

∣

2.

(29)

If we differentiate (29) directly, with respect to g∗12, and thencancel the operation result, we get

∂J(

g2)

∂g∗12= 2E

[

a4R

]

g12

( M∑

k=1

∣

∣gk2∣

∣

2 − 1

)

+ 2βg12

M∑

k=2

∣

∣gk2∣

∣

2

+ 2β(

gR,12g2I ,12 + jg2

R,12gI ,12)

+ ασ4ad

2g12 = 0.(30)

By canceling both real and imaginary parts of (30), we have

gR,12 = 0 or χ + 2βg2I ,12 = 2E

[

a4R

]− αd2σ4a,

gI ,12 = 0 or χ + 2βg2R,12 = 2E

[

a4R

]− αd2σ4a,

(31)

where χ = 2E[a4R]∑M

k=1|gk2|2 + 2β∑M

k=2|gk2|2 ≥ 0.However, since χ + 2βg2

I ,12 ≥ 0 and χ + 2βg2I ,12 ≥ 0, the

theorem’s condition 2E[a4R]−ασ4

ad2 ≤ 0 requires that gR,12 =

0 and gI ,12 = 0, that is, g12 = 0.Hence, g2 will take the form

g2 =[

0 | gT2]T

, (32)

which results in

gH1 g2 = 0. (33)

Therefore, (26) is reduced to

ming2

J(

g2

) = E[

(

z2R,2(n)− R)2

]

, (34)


where the second equalizer output z2 = gT2 a = gT2 a, witha = [a2, . . . , aM]T .

Equation (34) has the same form as (10). Hence theanalysis is exactly the same as described previously. Con-sequently, the stationary points of (34) will take the formg2 = [0, . . . , 0, e jm2(π/2), 0, . . . , 0]T , which corresponds to g2 =[0 | 0, . . . , 0, e jm2(π/2), 0, . . . , 0]T . Hence g2 will recover per-fectly a different signal than the one already recovered by g1.Without loss of generality, again, we assume that the sin-gle nonzero element of g2 is in its second position, that is,g2 = [0, e jm2(π/2), 0, . . . , 0]T .

If we continue in the same manner for each gi, we can seethat each gi converges to a setting, in which zeros have thepositions of the already recovered signals and its remainingentries contain only one nonzero element; this correspondsto the recovery of a different signal, and this process contin-ues until all signals have been recovered.

On the basis of this analysis, we can conclude that theminimization of the suggested cost function in the case ofQAM signals ensures a perfect recovery of all source signalsand that the recovered signals correspond to the source sig-nals with a possible permutation and a modulo π/2 phaserotation.

4.2. P-PSK case (P ≥ 8)

On the basis of the results in Appendix B, we have

J(

g1) = E

[

a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2 − 1

)2

+ β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

− E[a4R

]

+ R2.

(35)

And from (16), Jmin = −E[a4R] +R2 is a constant. If we cancel

the derivatives of the first and second terms on the right sideof (35), we obtain

M∑

k=1

∣

∣gk1∣

∣

2 = 1, (36)

M∑

k=1,k�=�

∣

∣gk1∣

∣

2 = 0. (37)

Therefore, (36) and (37) dictate that the solution must takethe form

g1 =[

0, . . . , 0, e jφ� , 0, . . . , 0]T

, (38)

where φ� ∈ [0, 2π] is an arbitrary phase in the �th positionof g1 which can be at any of the M possible positions.

The solution g1 has only one nonzero entry with a mod-ulus equal to one, and all the other ones are zeros. This so-lution corresponds to the recovery of only one source signaland cancels the other ones. With regard to the other vectors,the analysis is exactly the same as the one in the case of QAMsignals.

Then, we can say that the minimization of the SCMA cri-terion, in the case of P-PSK (P ≥ 8) signals, ensures the re-covery of all signals except for an arbitrary phase rotation foreach recovered signal.

4.3. 4-PSK case

On the basis of the results found in Appendix C, J(g1) can bewritten as

J(

g1) = E2[a2

R

]

[

3

( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4

− 4M∑

k=1

∣

∣gk1∣

∣

2 − 4M∑

k=1

g2R,k1g

2I ,k1

]

+ R2.

(39)

In order to find the stationary points of (39), we cancel itsderivative

∂J(

g1)

∂g∗�1= E2[a2

R

]

[

6g�1

M∑

k=1

∣

∣gk1∣

∣

2 − 2g�1∣

∣g�1∣

∣

2 − 4g�1

− 4(

gR,�1g2I ,�1 + jg2

R,�1gI ,�1)

]

= 0.

(40)

By canceling both real and imaginary parts of (40), we have

3gR,�1

M∑

k=1

∣

∣gk1∣

∣

2 − gR,�1∣

∣g�1∣

∣

2 − 2gR,�1 − 2gR,�1g2I ,�1 = 0,

3gI ,�1

M∑

k=1

∣

∣gk1∣

∣

2 − gI ,�1∣

∣g�1∣

∣

2 − 2gI ,�1 − 2gI ,�1g2R,�1 = 0.

(41)

According to (41),

gR,�1 = gI ,�1. (42)

Then (41) can be reduced to

6M∑

k=1,k�=�g2R,k1 + 2g2

R,�1 − 2 = 0. (43)

Thus

g2R,�1 = −3

M∑

k=1,k�=�g2R,k1 + 1. (44)

Finally, we find

g2R,�1 = −3(p − 1)g2

R,�1 + 1, (45)

where 1 ≤ p ≤ M is the number of nonzero elements in g1,which gives

g2R,�1 = g2

I ,�1 =

⎧

⎪

⎨

⎪

⎩

1(3p − 2)

, if � ∈ Fp,

0, otherwise,∀p = 1, . . . ,M,

(46)

where Fp is any p-element subset of {1, . . . ,M}.


Now, we study separately the stationary points for eachvalue of p.

(i) p = 1: in this case, g1 has only one non zero entry,with

gR,�1 = ±1, gI ,�1 = ±1, (47)

that is,

g�1 = c�ejφ� , (48)

where c�=√

2 and φ�=(2q + 1)(π/4), with q is an arbi-trary integer. Therefore, g1=[0, . . . , 0, c�e jφ� , 0, . . . , 0]T

is the global minimum.

(ii) p ≥ 2: in this case, the solutions have at least twononzero elements in some positions of g1. All nonzeroelements have the same squared amplitude of 2/(3p −2).

Let us consider the following perturbation:

g′1 = g1 + e, (49)

where e = [e1, . . . , eM]T is an (M × 1) vector whose norm‖e‖2 = eHe can be made arbitrarily small and is chosen sothat its nonzero elements are only in positions where the cor-responding elements of g1 are nonzero:

e��=0 ⇐⇒ � ∈ Fp. (50)

Let this perturbation be orthogonal with g1, that is, eHg1 = 0.Then, we have

∑

�∈Fp

∣

∣g′�1

∣

∣

2 =∑

�∈Fp

∣

∣g�1∣

∣

2+∑

�∈Fp

∣

∣e�∣

∣

2. (51)

We now define, as ε� , the difference between the squaredmagnitudes of g′�1 and g�1, that is,

∣

∣g′�1

∣

∣

2 = ∣∣g�1∣

∣

2+ ε� , ε� ∈ R, ε��=0 ⇐⇒ � ∈ Fp, (52)

where

∑

�∈Fpε� =

∑

�∈Fp

∣

∣e�∣

∣

2. (53)

We assume that

(

g′R,�1

)2 = g2R,�1 +

ε�2

,(

g′I ,�1

)2 = g2I ,�1 +

ε�2. (54)

By evaluating J(g′1), we find

J(

g′1) = E2[a2

R

]

[

3

(

∑

�∈Fp

∣

∣g�1∣

∣

2+ ε�

)2

−∑

�∈Fp

(∣

∣g�1∣

∣

2+ ε�

)2

− 4∑

�∈Fp

(∣

∣g�1∣

∣

2+ ε�

)

− 4∑

�∈Fp

(

g2R,�1 +

ε�2

)

×(

g2I ,�1 +

ε�2

)

]

+ R2,

J(

g′1) = E2[a2

R

]

[

3

(

∑

�∈Fp

∣

∣g�1∣

∣

2)2

−∑

�∈Fp

∣

∣g�1∣

∣

4

− 4∑

�∈Fp

∣

∣g�1∣

∣

2 − 4∑

�∈Fpg2R,�1g

2I ,�1

]

+ R2

+ E2[a2R

]

[

6∑

�∈Fp

∣

∣g�1∣

∣

2 ∑

�∈Fpε� − 4

∑

�∈Fp

∣

∣g�1∣

∣

2ε�

− 4∑

�∈Fpε� + 3

(

∑

�∈Fpε�

)2

− 2∑

�∈Fpε2�

]

.

(55)

Using (46) in (55), and after some simplifications, we get

J(

g′1) = J

(

g1)

+ E2[a2R

]

[

3

(

∑

�∈Fpε�

)2

− 2∑

�∈Fpε2�

]

. (56)

It exists ε� ∈ R, (� ∈ Fp) so that

3

(

∑

�∈Fpε�

)2

− 2∑

�∈Fpε2� < 0. (57)

Then, ∃ε� ∈ R, (� ∈ Fp) so that

J(

g′1)

< J(

g1)

. (58)

Hence, J(g1) cannot be a local minimum.Now we consider another perturbation which takes the

form

g′�1 =⎧

⎨

⎩

√

1 + ξg�1 if � ∈ Fp,

0 otherwise,(59)

where ξ is a small positive constant.By evaluating J(g′1), we obtain

J(

g′1) = E2[a2

R

]

[

3

(

∑

�∈Fp(1 + ξ)

∣

∣g�1∣

∣

2)2

−∑

�∈Fp(1 + ξ)2∣

∣g�1∣

∣

4 − 4∑

�∈Fp(1 + ξ)

∣

∣g�1∣

∣

2

− 4∑

�∈Fp(1 + ξ)2g2

R,�1g2I ,�1

]

+ R2.

(60)


30252015105

SNR (dB)

−30

−25

−20

−15

−10

−5

MSE

(dB

)

MMSE (supervised)CC-SCMAMCC-SCMA

Figure 5: MSE versus SNR of CC-SCMA, MCC-SCMA, and super-vised MMSE.

Using (36) and after some simplifications, we get

J(

g′1) = J

(

g1)

+4ξ2p

3p − 2E2[a2

R

]

. (61)

Therefore, we always have

J(

g′1)

> J(

g1)

, ∀p ∈ N+∗. (62)

Hence, g1 cannot be a local maximum.Then, on the basis of (58) and (62), g1 is a saddle point

for p ≥ 2.Therefore, the only stable minima correspond to p = 1.We conclude that the only stable minima take the form

g1 = [0, . . . , 0, c�e jφ� , 0, . . . , 0]T , which ensure the extractionof only one source signal and cancel the other ones. For theremainder of the analysis, we proceed exactly as we did forQAM signals.

Finally, in order to conclude this section, we can say thatthe minimization of the cost function in (6) ensures the re-covery of all source signals in the case of source signals drawnfrom QAM or PSK constellations.

5. MODIFIED CROSS-CORRELATION TERM

In the previous section, we have seen that, in the case ofQAM constellation, the signals are recovered with moduloπ/2 phase rotation. By taking advantage of this result, wesuggest to use, instead of cross-correlation term in (6), thefollowing term:

�−1∑

m=1

E2[

zR,�(n)zR,m(n)]

+ E2[

zR,�(n)zI ,m(n)]

, � = 1, . . . ,M.

(63)

Using (63) in (6), instead of the classical cross-correlationterm, the criterion becomes

J�(n) = E[(

z2R,�(n)− R)2]

+ α�−1∑

m=1

(

E2[zR,�(n)zR,m(n)]

+ E2[zR,�(n)zI ,m(n)])

,

� = 1, . . . ,M.(64)

Please note that in (64) the multiplications in cross-correlation terms are not complexes, as opposed to (6) whichreduces complexity.

The cost function in (64) is named modified cross-correlation SCMA (MCC-SCMA).

Remark 1. The new cross-correlation term could be also usedby the MMA algorithm, because it recovers QAM signalswith modulo π/2 phase rotation.

In the following section, the complexity of the modifiedcross-correlation term will be discussed and compared withthe classical cross-correlation term.

6. IMPLEMENTATION AND COMPUTATIONALCOMPLEXITY

6.1. Implementation

In order to implement (6) and (64), we suggest to use theclassical stochastic gradient algorithm (SGA) [25]. The gen-eral form of the SGA is given by

W(n + 1) = W(n)− 12μ∇W(J), (65)

where∇W(J) is the gradient of J with respect to W.

6.1.1. For CC-SCMA

The equalizer update equation at the nth iteration is writtenas

w�(n + 1) = w�(n)− μe�(n)y∗(n), � = 1, . . . ,M, (66)

where the constant which arises from the differentiation of(6) is absorbed within the step size μ. e�(n) is the instanta-neous error e�(n) for the �th equalizer given by

e�(n) = (

z2R,�(n)− R)zR,�(n) +

α

2

�−1∑

m=1

r�m(n)zm(n), (67)

where the scalar quantity r�m represents the estimate of r�m,it can be recursively computed as [25]

r�m(n + 1) = λr�m(n) + (1− λ)z�(n)z∗m(n), (68)

where λ ∈ [0, 1] is a parameter that controls the length of theeffective data window in the estimation.

Please note that since E[r�m(n)] = E[z�(n)z∗m(n)], thenthe estimator r�m(n) is unbiased.


Table 1: Comparison of the algorithms complexity against weightupdate.

Algorithm Multiplications Additions

CC-CMA 2M(3M + 4N + 1)− 2 4M(M + 2N − 1)

CC-SCMA 6M(M +N)− 2 M(4M + 6N − 5) + 1

MCC-SCMA 4M(M +N)− 1 2M(M + 2N − 1)

6.1.2. For MCC-SCMA

We have exactly the same equation as (66), but the instanta-neous error signal of the �th equalizer is given by

e�(n) = (z2R,�(n)− R)zR,�(n)

+α

2

�−1∑

m=1

[

rRR,�m(n)zR,m(n) + rRI ,�m(n)zI ,m(n)]

,(69)

where

rRR,�m(n + 1) = λrRR,�m(n) + (1− λ)zR,�(n)z∗R,m(n),

rRI ,�m(n + 1) = λrRI ,�m(n) + (1− λ)zR,�(n)z∗I ,m(n).(70)

6.2. Complexity

We consider the computational complexity of (66) for oneiteration and for all equalizer outputs. With

(i) for CC-SCMA,

e�(n) = (zR,�(n)2 − R)zR,�(n) +α

2

�−1∑

m=1

r�m(n)zm(n); (71)

(ii) for MCC-SCMA (modified cross-correlation SCMA),

e�(n) = (zR,�(n)2 − R)zR,�(n)

+α

2

�−1∑

m=1

[

rRR,�m(n)zR,m(n) + rRI ,�m(n)zI ,m(n)]

;

(72)

(iii) for CC-CMA (cross-correlation CMA)

e�(n) = (∣

∣z�(n)∣

∣

2 − R)z�(n) +α

2

�−1∑

m=1

r�m(n)zm(n). (73)

According to Table 1, the CC-SCMA presents a low com-plexity compared to that of CC-CMA. On a more interest-ing note, the results in Table 1 show that the use of modi-fied cross-correlation term reduce significantly the complex-ity. Please note that the number of operations is per iteration.


Some numerical results are now presented in order to con-firm the theoretical analysis derived in the previously sec-

tions. For that purpose, we use the signal to interference andnoise ratio (SINR) defined as

SINRk =∣

∣gkk∣

∣

2

∑

�,��=k∣

∣g�k∣

∣

2+ wT

k Rbw∗k

,

SINR = 1M

M∑

k=1

SINRk,

(74)

where SINRk is the signal-to-interference and noise ratio atthe kth output. gi j = hTi w j , where w j and hi are the jth andith column vector of matrices W and H, respectively. Rb =E[bbH] = σ2

bIN is the noise covariance matrix. The sourcesignals are assumed to be of unit variance.

The SINR is estimated via the average of 1000 indepen-dent trials. Each estimation is based on the following model.The system inputs are independent, uniformly distributedand drawn from 16-QAM, 4-PSK, and 8-PSK constellations.We considered M transmit and N receive spatially decor-related antennas. The channel matrix H is modeled by an(N×M) matrix with independent and identically distributed(i.i.d.), complex, zero-mean, Gaussian entries. We consid-ered α = 1 (this value satisfy the theorem condition) andλ = 0.97. The variance of noise is determined according tothe desired Signal-to-Noise Ratio (SNR).

Figures 1, 2, and 3 show the constellations of the sourcesignals, the received signals, and the receiver outputs (afterconvergence) using the proposed algorithm for 16-QAM, 8-PSK, and 4-PSK constellations, respectively. We have consid-ered that SNR = 30 dB,M = 2,N = 2, and that μ = 5×10−3.Please note that the constellations on Figures 1, 2, and 3 aregiven before the phase ambiguity is removed (this ambiguitycan be solved easily by using differential decoding).

In Figure 1, we see that the algorithm recovers the 16-QAM source signals, but up to a modulo π/2 phase rotationwhich may be different for each output. Figure 2 shows thatthe 8-PSK signals are recovered with an arbitrary phase ro-tation. In Figure 3, the 4-PSK signals are recovered with a(2k+ 1)(π/4) phase rotation and an amplitude of

√2. Theses

results are in accordance with the theoretical analysis givenin Section 4.

In order to compare the performances of CC-SCMA andCC-CMA, the same implementation is considered for bothalgorithms (see Section 6). We have considered M = 2, N =3, SNR = 25 dB, and the step sizes were chosen so that the al-gorithms have sensibly the same steady-state performances.We have also used the supervised least-mean square algo-rithm (LMS) as a reference.

Figure 4 represents the SINR performance plots for theproposed approach and the CC-CMA algorithm. We observethat the speed of convergence of the proposed approach isvery close to that of the CC-CMA. Hence, it represents a goodcompromise between performance and complexity.

Figure 5 represents the mean-square error (MSE) ver-sus SNR. In order to verify the effectiveness of the modifiedcross-correlation term, we have considered M = 2, N = 3,16-QAM, and μ = 0.02 for both algorithms. In this figure, thesupervised minimum mean square (MMSE) receiver servesas reference. We observe that CC-SCMA and MCC-SCMA


have almost the same behavior. So, in the case of QAM sig-nals, it is preferable to use a modified cross-correlation termbecause of its low complexity and of its similar performancescompared to the one of the classical cross-correlation term.

8. CONCLUSION

In this paper, we have proposed a new globally convergentalgorithm for the multiple-input multiple-output (MIMO)adaptive blind separation of QAM and PSK signals. The cri-terion is based on one dimension (either real or imaginary)and consists in penalizing the deviation of the real (or theimaginary) part from a constant. It was demonstrated thatthe proposed approach is globally convergent to a setting thatrecovers perfectly, in the absence of noise, all the source sig-nals. A modification for the cross-correlation constraint inthe case of QAM constellation has been suggested. Our al-gorithm has shown a low computational complexity com-pared to that of CMA, especially when the modified cross-correlation constraint is used, which makes it attractive forimplementation in practical applications. Simulation resultshave shown that the suggested algorithm has a good perfor-mance despite its lower complexity.

APPENDICES

A. QAM CASE

From (11),

J(g1) = E[

z4R,1

]− 2RE[

z2R,1

]

+ R2. (A.1)

We have

z1(n) = gT1 a(n), (A.2)

where

g1 =[

g11, . . . , gM1]T

, gk1 = gR,k1 + jgI ,k1,

a = [a1, . . . , aM]T

, ak = aR,k + jaI ,k.(A.3)

Then

zR,1(n) =M∑

k=1

(

gR,k1aR,k − gI ,k1aI ,k)

. (A.4)

For i.i.d. and mutually independent source signals thatdrawn from square QAM constellation, we have

E[

aR] = E

[

aI] = 0,

E[

amR] = E

[

amI]

, ∀m,

E[

amR,kanI ,�

] = E[

amR,k

]

E[

anI ,�]

, ∀k, �,m,n,

E[

amR,kanR,�

] = E[

amI ,kanI ,�

] =⎧

⎨

⎩

E[

am+nR

]

, if k = �,

E[

amR]

E[

anR]

, otherwise.(A.5)

We have

E[

z2R,1

] = E

{[ M∑

k=1

(


]2}

=M∑

k=1

M∑

�=1

E[(

gR,k1aR,k − gI ,k1aI ,k)(

gR,�1aR,� − gI ,�1aI ,�)]

=M∑

k=1

M∑

�=1

(

gR,k1gR,�1E[

aR,kaR,�]− gR,k1gI ,�1E

[

aR,kaI ,�]

−gI ,k1gR,�1E[

aI ,kaR,�])

+gI ,k1gI ,�1E[

aI ,kaI ,�]

.(A.6)

Using (A.5), we obtain

E[

z2R,1

] = E[

a2R

]

M∑

k=1

∣

∣gk1∣

∣

2. (A.7)

Similarly

E[

z4R,1

] = E

{[ M∑

k=1

(


]4}

=M∑

k=1

M∑

�=1

M∑

m=1

M∑

n=1

E[(


× (gR,�1aR,� − gI ,�1aI ,�)

× (gR,m1aR,m − gI ,m1aI ,m)

× (gR,n1aR,n − gI ,n1aI ,n)]

.

(A.8)

Developing (A.8) and using (A.5), we have the followingthree cases:

(i) k = � = m = n:

E[

z4R,1

] = E[

a4R

]

M∑

k=1

(

g4R,k1 + g4

I ,k1

)

+ 6E2[a2R

]

M∑

k=1

(

g2R,k1g

2I ,k1

)

;

(A.9)

(ii) k = ��=m = n:

E[

z4R,1

] = 3E2[a2R

]

M∑

k=1

M∑

�=1,��=k

∣

∣gk1∣

∣

2∣∣g�1

∣

∣

2; (A.10)

(iii) otherwise: (A.8) is equal to zero.

From (A.9) and (A.10), (A.8) becomes

E[

z4R,1

] = E[

a4R

]

M∑

k=1

(

g4R,k1 + g4

I ,k1

)

+ 6E2[a2R

]

M∑

k=1

(

g2R,k1g

2I ,k1

)

+ 3E2[a2R

]

M∑

k=1

M∑

�=1,��=k

∣

∣gk1∣

∣

2∣∣g�1

∣

∣

2.

(A.11)


Then

E[

z4R,1

] = E[

a4R

]

M∑

k=1

∣

∣gk1∣

∣

4 − 2E[

a4R

]

M∑

k=1

g2R,k1g

2I ,k1

+ 6E2[a2R

]

M∑

k=1

g2R,k1g

2I ,k1

+ 3E2[a2R

]

M∑

k=1

M∑

�=1,��=k

∣

∣gk1∣

∣

2∣∣g�1

∣

∣

2.

(A.12)

On the other hand, we have

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

=M∑

k=1

M∑

�=1,��=k

∣

∣gk1∣

∣

2∣∣g�1

∣

∣

2.

(A.13)

Substituting (A.13) into (A.12), we get

E[

z4R,1

] = E[

a4R

]

M∑

k=1

∣

∣gk1∣

∣

4

+ 3E2[a2R

]

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

+ 2βM∑

k=1

g2R,k1g

2I ,k1,

(A.14)

where β = 3E2[a2R]− E[a4

R].Using (A.6) and (A.14) in (A.1), we obtain

J(

g1)

= E[

a4R

]

M∑

k=1

∣

∣gk1∣

∣

4+ 3E2[a2

R

]

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

+ 2βM∑

k=1

g2R,k1g

2I ,k1 − 2E

[

a2R

]

M∑

k=1

∣

∣gk1∣

∣

2+ R2

= E[

a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2)2

− E[a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2)2

+ E[

a4R

]

M∑

k=1

∣

∣gk1∣

∣

4+ 3E2[a2

R

]

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

+2βM∑

k=1

g2R,k1g

2I ,k1−2E

[

a2R

]

M∑

k=1

∣

∣gk1∣

∣

2+R2 +E

[

a4R

]−E[a4R

]

.

(A.15)

Rearranging terms, we get

J(

g1) = E

[

a4R

]

( M∑

k=1

∣

∣gk1∣

∣

2 − 1

)2

+ β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

+ 2βM∑

k=1

g2R,k1g

2I ,k1

− E[a4R

]

+ R2.(A.16)

Finally, we get (12).

B. P-PSK CASE (P ≥ 8)

In the case of P-PSK (P ≥ 8), (A.5) hold and moreover wehave

E[

a2R,ka

2I ,�

] =

⎧

⎪

⎨

⎪

⎩

13E[

a4R

]

if k = �,

E2[

a2R

]

otherwise.(B.17)

Using (A.5) in (A.6),

E[

z2R,1

] = E[

a2R

]

M∑

k=1

∣

∣gk1∣

∣

2. (B.18)

Using (A.5), and (B.17) in (A.8), we find

(i) k = � = m = n:

E[

z4R,1

] = E[

a4R

]

M∑

k=1

∣

∣gk1∣

∣

4; (B.19)

(ii) k = ��=m = n:

E[

z4R,1

] = 3E2[

a2R

]

M∑

k=1

M∑

�=1,��=k

∣

∣gk1∣

∣

2∣∣g�1

∣

∣

2; (B.20)

(iii) otherwise: (A.8) is equal to zero.Then

E[

z4R,1

] = E[

a4R

]

M∑

k=1

(

g4R,k1 + g4

I ,k1

)

+ 3E2[a2R

]

M∑

k=1

M∑

�=1,��=k

∣

∣gk1∣

∣

2∣∣g�1

∣

∣

2.

(B.21)

By developing the above equation as in the QAM case, we get

J(

g1)=E[a4

R

]

( M∑

k=1

∣

∣gk1∣

∣

2−1

)2

+β

[( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4]

− E[a4R

]

+ R2.(B.22)

C. 4-PSK CASE

For 4-PSK signals,

E[

a2R,ka

2I ,�

] ={

0 if k = �,

E2[

a2R

]

otherwise.(C.23)

Considering (A.5), and (C.23), and proceeding in the sameway as in the case of P-PSK signals (P ≥ 8), we can easilyfind

J(

g1) = E2[a2

R

]

[

3

( M∑

k=1

∣

∣gk1∣

∣

2)2

−M∑

k=1

∣

∣gk1∣

∣

4 − 4M∑

k=1

∣

∣gk1∣

∣

2

− 4M∑

k=1

g2R,k1g

2I ,k1

]

+ R2.

(C.24)


ACKNOWLEDGMENT

The authors are grateful to the anonymous referees for theirconstructive critics and valuable comments.

REFERENCES

[1] G. J. Foschini, “Layered space-time architecture for wirelesscommunication in a fading environment when using multi-element antennas,” Bell Labs Technical Journal, vol. 1, no. 2,pp. 41–59, 1996.

[2] A. Mansour, A. K. Barros, and N. Ohnishi, “Blind separationof sources: methods, assumptions and applications,” IEICETransactions on Fundamentals of Electronics, Communicationsand Computer Sciences, vol. E83-A, no. 8, pp. 1498–1512, 2000.

[3] A. Cichocki and S.-I. Amari, Adaptive Blind Signal and ImageProcessing: Learning Algorithms and Applications, John Wiley& Sons, New York, NY, USA, 2003.

[4] K. Abed-Meraim, J.-F. Cardoso, A. Y. Gorokhov, P. Loubaton,and E. Moulines, “On subspace methods for blind identifica-tion of single-input multiple-output FIR systems,” IEEE Trans-actions on Signal Processing, vol. 45, no. 1, pp. 42–55, 1997.

[5] A. Gorokhov and P. Loubaton, “Blind identification ofMIMO-FIR systems: a generalized linear prediction ap-proach,” Signal Processing, vol. 73, no. 1-2, pp. 105–124, 1999.

[6] J. K. Tugnait, “Identification and deconvolution of multichan-nel linear non-Gaussian processes using higher order statisticsand inverse filter criteria,” IEEE Transactions on Signal Process-ing, vol. 45, no. 3, pp. 658–672, 1997.

[7] B. Hassibi and H. Vikalo, “On the sphere-decodingalgorithm—II: generalizations, second-order statistics,and applications to communications,” IEEE Transactions onSignal Processing, vol. 53, no. 8, pp. 2819–2834, 2005.

[8] B. Hassibi, “An efficient square-root algorithm for BLAST,” inProceedings of the IEEE International Conference on Acoustics,Speech, and Signal Processing (ICASSP ’00), vol. 2, pp. 737–740,Istanbul, Turkey, June 2000.

[9] Y. Sato, “A method of self-recovering equalization for mul-tilevel amplitude-modulation systems,” IEEE Transactions onCommunications, vol. 23, no. 6, pp. 679–682, 1975.

[10] D. Godard, “Self-recovering equalization and carrier track-ing in two-dimensional data communication systems,” IEEETransactions on Communications, vol. 28, no. 11, pp. 1867–1875, 1980.

[11] J. Treichler and B. Agee, “A new approach to multipath correc-tion of constant modulus signals,” IEEE Transactions on Acous-tics, Speech, and Signal Processing, vol. 31, no. 2, pp. 459–472,1983.

[12] M. G. Larimore and J. Treichler, “Convergence behavior of theconstant modulus algorithm,” in Proceedings of the IEEE In-ternational Conference on Acoustics, Speech, and Signal Process-ing (ICASSP ’83), vol. 1, pp. 13–16, Boston, Mass, USA, April1983.

[13] J. Treichler and M. G. Larimore, “New processing techniquesbased on the constant modulus adaptive algorithm,” IEEETransactions on Acoustics, Speech, and Signal Processing, vol. 33,no. 2, pp. 420–431, 1985.

[14] R. Gooch and J. Lundell, “The CM array: an adaptive beam-former for constant modulus signals,” in Proceedings of theIEEE International Conference on Acoustics, Speech, and SignalProcessing (ICASSP ’86), vol. 11, pp. 2523–2526, Tokyo, Japan,April 1986.

[15] L. Castedo, C. J. Escudero, and A. Dapena, “A blind signal sep-aration method for multiuser communications,” IEEE Trans-actions on Signal Processing, vol. 45, no. 5, pp. 1343–1348,1997.

[16] C. B. Papadias and A. J. Paulraj, “A constant modulus al-gorithm for multiuser signal separation in presence of delayspread using antenna arrays,” IEEE Signal Processing Letters,vol. 4, no. 6, pp. 178–181, 1997.

[17] K. N. Oh and Y. O. Chin, “Modified constant modulus al-gorithm: blind equalization and carrier phase recovery algo-rithm,” in Proceedings of the IEEE International Conference onCommunications (ICC ’95), vol. 1, pp. 498–502, Seattle, Wash,USA, June 1995.

[18] J. Yang, J.-J. Werner, and G. A. Dumont, “The multimodulusblind equalization and its generalized algorithms,” IEEE Jour-nal on Selected Areas in Communications, vol. 20, no. 5, pp.997–1015, 2002.

[19] L. M. Garth, J. Yang, and J.-J. Werner, “Blind equalization al-gorithms for dual-mode CAP-QAM reception,” IEEE Transac-tions on Communications, vol. 49, no. 3, pp. 455–466, 2001.

[20] P. Sansrimahachai, D. B. Ward, and A. G. Constantinides,“Blind source separation for BLAST,” in Proceedings of the 14thInternational Conference on Digital Signal Processing (DSP ’02),vol. 1, pp. 139–142, Santorini, Greece, July 2002.

[21] A. Goupil and J. Palicot, “Constant norm algorithms class,”in proceedings of the 11th European Signal Processing Confer-ence (EUSIPCO ’02), vol. 1, pp. 641–644, Toulouse, France,September 2002.

[22] A. Ikhlef, D. Le Guennec, and J. Palicot, “Constant norm al-gorithms for MIMO communication systems,” in Proceedingsof the 13th European Signal Processing Conference (EUSIPCO’05), Antalya, Turkey, September 2005.

[23] A. Ikhlef and D. Le Guennec, “Blind recovery of MIMO QAMsignals : a criterion with its convergence analysis,” in Proceed-ings of the 14th European Signal Processing Conference (EU-SIPCO ’06), Florence, Italy, September 2006.

[24] C. B. Papadias, “Globally convergent blind source separationbased on a multiuser kurtosis maximization criterion,” IEEETransactions on Signal Processing, vol. 48, no. 12, pp. 3508–3519, 2000.

[25] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Upper SaddleRiver, NJ, USA, 4th edition, 2002.


Research ArticleEmploying Coordinated Transmit and Receive Beamformingin Clustering Double-Directional Radio Channel

Chen Sun,1 Makoto Taromaru,1 and Takashi Ohira2

1 ATR Wave Engineering Laboratories, 2-2-2 Hikaridai, Keihanna Science City, Kyoto 619-0288, Japan2 Department of Information and Computer Sciences, Toyohashi University of Technology, 1-1 Hibariga-oka,Toyohashi 441-8580, Japan

Received 31 October 2006; Accepted 1 August 2007


A novel beamforming (BF) system that employs two switched beam antennas (SBAs) at both ends of the wireless link in an indoordouble-directional radio channel (DDRC) is proposed. The distributed directivity gain (DDG) and beam pattern correlation inDDRC are calculated. The channel capacity of the BF system is obtained from an analytical model. Using the channel capacityand outage capacity as performance measures, we show that the DDG of the BF system directly increases the average signal-to-noise ratio (SNR) of the wireless link, thus achieving a direct increase of the ergodic channel capacity. By jointly switchingbetween different pairs of transmit (Tx) and receive (Rx) directional beam patterns towards different wave clusters, the systemprovides diversity gain to combat against multipath fading, thus reducing the outage probability of the random channel capacity.Furthermore, the performance of the BF system is compared with that of a multiple-input multiple-output (MIMO) system thatis set up using linear antenna arrays. Results show that in a low-SNR environment, the BF system outperforms the MIMO systemin the same clustering DDRC.

Copyright © 2007 Chen Sun et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Along with the evolution of wireless technologies, broadbandInternet access and multimedia services are expected to beavailable for commercial mobile subscribers. This enthusi-asm has created a need for high-data-rate wireless transmis-sion systems. Study on adaptive antenna systems has demon-strated their potential in increasing the spectrum efficiencyof a wireless radio channel. The beamforming (BF) ability ofthe adaptive antennas increases the transmission range, re-duces delay spread, and suppresses interference [1–4]. In amultipath rich environment, they provide diversity gain tocounteract multipath fading [5, 6]. Installing antenna arraysat both transmitter (Tx) and receiver (Rx) sides builds upa multiple-input multiple-output (MIMO) system [7]. TheMIMO system increases the data rate of wireless transmis-sion by sending multiple data streams though multiple Txand Rx antennas over fading channels [8].

The potential increase in the spectrum efficiency of theMIMO systems has been evaluated under various wirelesspropagation channel models. Recent investigation on wire-less channel model has introduced the concept of double-

directional radio channel (DDRC) that incorporates direc-tional information at both Tx and Rx sides of a wirelesslink [9]. The techniques of extracting directional informa-tion at both ends of the wireless link are presented in [10, 11].In [11] Wallace and Jensen extend the Saleh-Valenzuela’sclustering model [12] and the Spencer’s model [13] to in-clude both angle-of-arrival (AOA) information and angle-of-departure (AOD) information to describe an indoor DDRCwith clustering phenomenon, which is here after referredto as clustering DDRC. Incorporating spatial information atboth ends of the link, the statistical model provides a betterdescription of a wireless propagation channel, thus allowinga better evaluation of the system performance in a practicalsituation than other models. The channel capacity that canbe achieved by a MIMO system in a clustering DDRC modelis evaluated in [11]. The predicted capacities match perfectlywith the measured data. This justifies the clustering DDRCmodel being a better description of a practical wireless prop-agation channel independently of specific antenna character-istics.

However, the predicted capacity of a MIMO system in aclustering DDRC model is lower than that predicted under


an ideal statistical channel model, which assumes statisticallyindependent fading at antenna array elements [7]. The rea-son is that the potential capacity of MIMO systems relies onthe richness of scattering waves. The clustering phenomenonof impinging waves in the Wallace’s model [11] leads to anincrease of fading correlation among antenna array elements,thus resulting in a lower-channel capacity.

In addition to the capacity loss of a MIMO system in aclustering channel, the implementation of a MIMO systemposes practical problems, such as the power consumptionlimitation and the size constraints of commercial mobile ter-minals. A MIMO receiver that is equipped with an antennaarray consists of multiple RF channels connected to individ-ual array elements so that spatial processing (such as BF, di-versity combining) is carried out at digital stage. The powerconsumption and fabrication cost of the MIMO system in-crease with the number of array elements.

To circumvent the afore-mentioned limitations of theMIMO system and to increase the channel capacity of wire-less communication in a clustering DDRC, we propose a BFsystem that employs two switched beam antennas (SBAs)with predefined directional beam patterns at both ends of thewireless link. Knowing that the main energy of the signals istransported spatially through wave clusters [14], the direc-tional beam patterns at both Tx and Rx sides are directedtowards the wave clusters. Employing the directional beampattern in the presence of distributed waves, the distributeddirectivity gains (DDGs) at both Tx and Rx antennas directlyincrease the average SNR of the wireless link. By switchingbetween different pairs of Tx and Rx directional beam pat-terns, the system provides a diversity freedom to further im-prove the performance.

A potential application of the proposed technique is anad hoc network. It is a self-configuring network of mo-bile terminals without a wireless backbone, such as ac-cess points [3]. Commercial applications of adaptive an-tenna technology at these battery-powered mobile termi-nals (e.g., laptops, PDAs) require adaptive antennas thathave low-power consumption and low-fabrication cost. Theparasitic array antennas have shown their potential in ful-filling the goal [3]. Therefore, in this paper, the Tx andRx SBAs are realized by two electronically steerable par-asitic array radiator (ESPAR) antennas [15–17]. We donot employ a multibeam antennas as their structures con-sist of a number of individual RF channel that lead toundesirable high-power consumption and fabrication cost[3].

The rest of the paper is organized as follows. In Section 2,we describe the clustering DDRC model, the SBA and theBF system. In Section 3, we calculate the DDG at both endsof the wireless link. The correlation of different wirelesslinks that are set up with different pairs of transmit andreceive beam patterns is obtained. Based on these results,an analytical model of the BF system is built. In Section 4,we calculate the channel capacity of the BF system basedon the analytical model and compare the performance ofthe BF system with a MIMO system in the same cluster-ing DDRC model. Finally, in Section 5, we conclude the pa-per.

2. SYSTEM MODEL

In order to present the system model of the proposed BF sys-tem, we review the clustering DDRC model first. After that,the ESPAR antennas that operate as SBAs in the proposedsystem are described. Finally, the BF system that employs twoESPAR antennas at both ends of the wireless link in a cluster-ing DDRC is given.

2.1. Clustering DDRC

In [12] Saleh and Valenzuela describe the clustering phe-nomenon of impinging waves in an indoor multipath chan-nel, that is, the result of scattering and reflection on smallobjects, rough surface, and so forth, along the paths of wavepropagation In [13]Spencer et al. extend Saleh’s model ac-cording to the measurement results to include spatial infor-mation of the clusters and waves within each cluster. TheSpencer’s model is further extended in [11] to describe a clus-tering DDRC, which takes into account the spatial informa-tion at both Tx and Rx sides.

The spatial channel response (SCR) of a clustering DDRCis written as [11]

h(φr ,φt) = 1√

LK

L−1∑

l=0

K−1∑

k=0

αl,k

× δ(φr −Φr

l − ωrl,k

)δ(φt −Φt

l − ωtl,k

),

(1)

where superscripts (·)rand (·)t denote, respectively, the Rxand Tx sides of a wireless link. Let φrand φt represent theAOA and AOD, respectively. The nominal AOA and AOD ofthe lth cluster are denoted by Φr

l and Φtl , which are uniformly

distributed over the azimuthal plane [13]. Here, L is the to-tal number of wave clusters. In the following discussion, wecall the waves within each cluster as rays and let ωr

l,k and ωtl,k

denote, respectively, the AOA and AOD of the kth ray withinthe lth cluster. Within each wave cluster, there are totally Krays. Furthermore, ωr

l,k and ωtl,k follow a Laplacian distribu-

tion centered at Φrl and Φt

l , respectively, as

fω(ωl,k | Φl

) = 1√2σ

e−|√

2(ωl,k−Φl)/σ|, (2)

where σ is the angular spread of rays within each cluster. In(1), the amplitude of αl,k associated with the kth ray in the lthcluster is assumed to follow a Rayleigh distribution [11, 13]with the mean power following a double-exponential distri-bution. That is,

E[α2l,k] = E

[α2

0,0

]e −Γl /Γe −Γl,k/γ, (3)

where E[·] is the expectation operator. α20,0 is the power decay

of the first ray of the first cluster with E[α20,0] = 1. Γ and γ are

the constant decay time of cluster and rays, respectively. Γl isthe delay of the first arrival ray of the lth cluster and Γ0 = 0nanosecond. It is exponentially distributed and conditionedon Γl−1 as

fΓ(Γl | Γl−1

) = Λe−Λ(Γl−Γl−1), 0 ≤ Γl−1 < Γl <∞. (4)

Chen Sun et al. 3

Antennaground plane

Signal direction

Position circle

Loadcontrol

ReceiverReceived signals

Tunablereactances

X

Y

Z

#0

#1

#2

#3#4

#5

#6

θ

Elev

atio

nan

gle

Azi

mut

han

gleφ

50oh

m

60◦

RF

port

x 1 x 2x 3x 4x 5 x 6

Figure 1: Structure of a 7-element ESPAR antenna. The elementsare monopoles with interelement spacing of a quarter wavelength.

The delay of rays τl,k within each cluster is also exponentiallydistributed and conditioned on the previous delay τl,k−1 as

fτ(τl,k | τl,k−1

) = λe−λ(τl,k−τl,k−1), 0 ≤ τl,k−1 < τl,k <∞.(5)

Here, Λ and λ are the constant arrival rates of clusters andrays, respectively. The statistics of rays and clusters are as-sumed being independent [11, 13], that is,

fτ,ω(τl,k,ωl,k | τl,k−1

) = fτ(τl,k | τl,k−1

)fω(ωl·k

). (6)

In this study, we set Γ = 30 nanoseconds, γ = 20 nanosec-onds, 1/Λ = 30 nanoseconds, 1/λ = 5 nanoseconds, andσ = 20◦ to represent an indoor or urban microscopic en-vironment as in [12].

2.2. ESPAR antenna

In this paper, two parasitic array antennas named as ESPAR[15, 16] are employed at both ends of the wireless link to pro-duce Tx and Rx directional beam patterns. In this subsection,we briefly explain the operation principle of an ESPAR an-tenna.

Figure 1 shows the structure of a seven-element ESPARantenna. There, one active central monopole is surroundedby parasitic elements on a circle of radius of a quarterwavelength on the circular grounded base plate. The centralmonopole is connected to an RF receiver and each parasiticmonopole is loaded with a tunable reactance. Let s(t) be thefar-field impinging wave from direction φ, the output signalat the RF port is written as

y(t) = wTESPα(φ)s(t) + n(t), (7)

where α(φ) is the 7-by-1 dimensional steering vector definedbased upon the array geometry of the ESPAR antenna. It iswritten as

α(φ) = [ A1 A2 A3 A4 A5 A6 A7 ]T

, (8)

where A1 = 1, A2 = e j(π/2) cos(φ), A3 = e j(π/2) cos(φ−π/3), A4 =e j(π/2) cos(φ−2π/3), A5 = e j(π/2) cos(φ−π), A6 = e j(π/2) cos(φ−4π/3),A7 = e j(π/2) cos(φ−5π/3), In (7), n(t) denotes an additivewhite Gaussian noise (AWGN) component. Superscript(·)Tdenotes transpose. wESP is written as

wESP = z0(

Y−1 + X)−1

u1, (9)

where Y is a mutual admittance matrix of array elements.The values of entities of Y are calculated using a methodof moment (MoM) Numerical Electromagnetic Code (NEC)simulator [18] and are given in [16]. Matrix X is a di-agonal matrix given by diag[z0, jx1, . . . , jx6 ], where xi(i =1, 2, . . . , 6) [Ω] is the reactance loaded at the array parasiticelements as shown in Figure 1. The characteristic impedanceat the central RF port is z0 = 50 [Ω]. In (9), u1 is defined as

u1 = [ 1 0 0 0 0 0 0 ]T. From (7), we can see that wESP

is equivalent to an array BF weight vector. Here, we write thenormalized wESP as

w = wESP√wH

ESPwESP

. (10)

The normalized azimuthal directional beam pattern for agiven weight vector w is written as

g(φ∣∣w) =

∣∣wTα(φ)∣∣2

(1/2π)∫ 2π

0 wTα(φ)αH(φ)w∗dφ, (11)

where superscript (·)∗ is the complex-conjugate. In thisstudy we have neglected the effect of load mismatch. Forsimplicity, we only consider the impinging waves that arecopolarized with the array elements in the azimuthal plane.Since w is dependent on the tunable reactances loaded atthose parasitic elements, changing the values of reactance,the beam adjusts the beam pattern of an ESPAR antenna.This model describes the beam pattern control based on tun-ing the loaded reactances at the parasitic elements. It enablesus to examine the system performance at different setting ofreactance values. The validity of this model has been provedby experiments and simulation based on (7)–(11) in [16].

In this paper, we use the ESPAR to generate a few prede-fined directional beam patterns and employ the antenna asan SBA. When the reactance values are set to (−9 000 000)[Ω], the ESPAR antenna forms a directional beam patternpointing to 0◦ in the azimuthal plane. The 3D beam pat-tern that is produced using the NEC simulator is shown inFigure 2. The azimuthal plane of this simulated beam pat-tern using an NEC simulator is compared with the calcu-lated beam pattern with MATLAB using (11) and the mea-sured beam pattern in Figure 3. Since the three beam patternsmatch closely, we use (11) to describe a beam pattern that isgenerated by an ESPAR antenna in the following analysis. Byshifting the reactance values, the antenna produces six con-secutive beam patterns as shown in Figure 4. The reactancevalues for those six directional beam patterns are listed inTable 1. In the following analysis, we use wm, m = 1, 2, . . . , 6,to denote the ESPAR weight vector that is described in (10)for the six directional beam patterns pointing to (m−1)×60◦

in the azimuthal plane. g(φ | wm) is the corresponding nor-malized directional beam pattern as described in (11).


Table 1: Reactance values for different directional beam patterns.

Equivalent weight Beam azimuthal direction x1 x2 x3 x4 x5 x6

w1 0◦ −90 0 0 0 0 0

w2 60◦ 0 −90 0 0 0 0

w3 120◦ 0 0 −90 0 0 0

w4 180◦ 0 0 0 −90 0 0

w5 240◦ 0 0 0 0 −90 0

w6 300◦ 0 0 0 0 0 −90

3D radiation pattern

X

Z

−40

−34.9

−29.7

−24.6

−19.5

−14.4

−9.2

−4.1

1

|Gai

nto

t|

Figure 2: Three-dimensional beam pattern of a 7-element ESPARantenna that is calculated by NEC simulator. X = [−9 000 000]Ω.The directional beam pattern is pointing to 0◦ in the azimuthalplane.

360300240180120600

Azimuthal angle (deg)

−25

−20

−15

−10

−5

0

Rel

ativ

ega

in(d

B)

MeasurementNEC simulatorMATLAB simulation

Figure 3: Comparison of the directional beam patterns of measure-ment and simulation using MATLAB and NEC simulator.

2.3. BF system

Two ESPAR antennas are employed at both ends of the wire-less link. When both the Tx and Rx directional beam pat-terns are steered towards one of the existing clusters, the linkis set up. This prerequisites a cluster identification process.

0 dB

−10 dB

−20 dB

Pattern 1Pattern 2Pattern 3

Pattern 4Pattern 5Pattern 6

0

30

60

90

120

150

180

210

240

270

300

330

Nor

mal

ized

gain

(dB

)

Figure 4: Six consecutive directional beam patterns generated byshifting the reactance values.

In [9, 13, 19], wave clusters are identified in spatial-temporaldomain through visual observation. In this paper, we assumethat the clusters’ spatial information is known at both Tx andRx sides, and the directional beam patterns at both ends ofthe link can be steered towards a common wave cluster. Asshown in Figure 5, there are two clusters of waves at both theTx and Rx sides. Both Tx and Rx can choose one out of sixpredefined directional beam patterns pointing to a commonwave cluster.

3. PERFORMANCE EVALUATION

Given the above model, the question that arises is how muchcapacity this system can provide. In order to calculate thechannel capacity, in this section, we examine the directivitygain and the beam pattern correlation in a clustering DDRCmodel. The channel capacity will be given in Section 4.

3.1. DDG

Firstly, we examine the directivity gain of the beam pat-tern for the system model. Analyzing the directivity gain in

Chen Sun et al. 5

Laplacian

distributedangular spread

Scattering atrough surface and

small objects

Scattering atrough surface and

small objects

Cluster 1

Cluster 2

Transmitter ESPARantenna

Receiver ESPARantenna

BLK

BLK

BLK

Figure 5: System model of employing directional antennas at bothends of the wireless link in a clustering DDRC.

the presence of angular distribution of the impinging wavesinvokes calculating the DDG [20]. Since we only considerwaves in the azimuthal plane, the DDG can be given as

D(Φ, σ , w) = ηant2π∫ 2π

0fω(φ | Φ)g(φ | w)dφ, (12)

where fω(φ | Φ) is the probability density function (pdf) ofthe angular distribution of impinging waves and g(φ | w)is given in (11). Here, ηant is the antenna efficiency that isassumed to be unity. For simplicity, we omit ηant in the fol-lowing analysis.

Figure 6 shows the situation when there is one clustercentered at Φl. The AOAs of rays within the cluster followLaplacian distribution as expressed in (2). As stated previ-ously, the ESPAR antenna can produce six consecutive direc-tional beam patterns by shifting the reactance values. Let usnow calculate the DDG for the directional beam pattern thatpoints to 0◦ in the azimuthal plane, that is, we use w1 in (11),while the nominal direction of the cluster is moving from 0◦

to 180◦. In this case, we can calculate the DDG as

D(Φ, σ , w1

) =√

2π∫ 2π

0

∣∣wT1 α(φ)

∣∣2e−|

√2(φ−Φ)/σ|dφ

σ∫ 2π

0 wT1 α(φ)αH(φ)w∗

1 dφ. (13)

The DDG corresponding to different angular spreads is alsoshown in Figure 7. For the Laplacian distributed scenarios,using a directional beam pattern pointing to a wave clusterprovides a significant DDG. However, the DDG degrades asthe center of the cluster moves away from the direction ofthe beam pattern’s maximum gain. Please note that when thewaves are uniformly distributed over the azimuthal plane, us-ing a directional beam pattern does not provide any DDG.

The DDG calculated in Figure 7 considers only one clus-ter of impinging waves. For multiple clusters (e.g., two clus-ters), we can write the angular distribution according to (2)as

fω(φ | Φ1,Φ2

) = 12√

2σ(e−|

√2(φ−Φ1)/σ1| + e−|

√2(φ−Φ2)/σ2|).

(14)

Rays withincluster

pdf

ωl,k

φ = Φl

ESPAR

y

x

Figure 6: The impinging waves from a cluster with nominal direc-tion Φl on directional beam patterns of the ESPAR antenna.

1801501209060300

Azimuthal angle Φ of cluster (deg)

−12

−10

−8

−6

−4

−2

0

2

4

6

Dis

trib

ute

ddi

rect

ivit

yga

in(d

B)

Lap σ = 20◦

Lap σ = 40◦Lap σ = 60◦

Uniform

Figure 7: DDG for the directional beam pattern by the ESPAR an-tennas in different spatially distributed radio channels.

We also follow the assumption in [11, 13] that the angularspreads of all clusters are the same, that is, σ1 = σ2. Insert-ing (14) into (12), we obtain the DDG when there are twoclusters as

D(Φ1,Φ2, σ , w1

) = 12D(Φ1, σ , w1

)+

12D(Φ2, σ , w1

).

(15)

3.2. Link gain

By now, we have calculated the DDG at one (Tx or Rx) sideof the wireless link. Because both the Tx and Rx employ di-rectional beam patterns, the joint directivity gain incorpo-rates the directivity gains at both sides of the link. This is inaccordance with the concept of DDRC. Given the SRC of a


clustering DDRC in (1) and the beam pattern in (11), thespatial channel model of the BF system is given by

hBF =√g(φr | wr

m

)∗h(φr ,φt)∗√g(φt | wt

m

)

= 1√LK

L−1∑

l=0

K−1∑

k=0

αl,k√g(Φr

l + ωrl,k | wr

m

)√g(Φt

l + ωtl,k | wt

m

),

(16)

where wrm and wt

m are the BF weight vectors at the Rx sideand Tx sides, respectively. Taking the expectation of |hBF|2 in(16), we obtain the link gain as

G = E[hBFh

HBF

] = 1LK

L−1∑

l=0

K−1∑

k=0

E[αl,kα

∗l,k

]

× E[g(Φr

l + ωrl,k | wr

m

)]E[g(Φt

l + ωtl,k | wt

m

)].

(17)

For a given cluster realization, we take the expectation overrays. To simplify the model, we assume that αl,k of each rayfollows a complex normal distribution CN(0, |αl,0|2) withzero mean and a variance that is equal to the mean powerdecay of the first arrival ray within the lth cluster. Thus, wehave

1K

K−1∑

k=0

E(αl,kα

∗l,k

) = e−Γl /Γ, for l = 0, 1, . . . ,L− 1. (18)

Furthermore, according to (4), we assume that Γl = l −1/Λ for l = 0, 1, . . . ,L− 1. Therefore, given a channel clus-ter realization, that is, the locations of clusters, the link gainis given by

G = 1LK

L−1∑

l=0

E(αl,kα

∗l,k

)∫ 2π

0fω(φ | Φr

l

)g(φ | wr

m

)dφ

×∫ 2π

0fω(φ | Φt

l

)g(φ | wt

m

)dφ

= 1L

[D(Φr

0, σ , wrm

)... D

(Φr

L−1, σ , wrm

) ]

× diag[

1 e−1/ΛΓ ... e−(L−1)/ΛΓ]

×[ D(Φt0, σ , wt

m

)... D

(Φt

L−1, σ , wtm

) ]T.

(19)

The physical meanings of (19) are that, firstly, each routethrough a pair of Tx and Rx directional beam patterns point-ing to a common cluster is associated with a mean powerdecay that increases exponentially with the cluster’s delaytime. Therefore, directional beam patterns should be steeredtowards early arrival clusters for a high-average link gain.Secondly, the mean link gain of the BF system decreases asthe number of clusters increases. The channel is closer to amultipath-rich environment where the diversity scheme ismore advantageous over the BF system. Finally, there existsextra diversity freedom by switching among different pairsof Tx and Rx beams to select different clusters as transmis-sion routes in that the fading statistics of the rays within dif-ferent clusters are assumed to be independent according to[11, 13]. The correlation between different routes is induceddue to beam pattern overlapping.

Now we consider the case where there exist two clusters.For the simplicity of analysis, we arbitrarily set the positionsof both clusters at both Tx and Rx sides as Φr

0 = Φt0 = 0◦

and Φr1 = Φt

1 = 120◦. By jointly pointing directional beampatterns towards 0◦ using w1 at both sides of the link, thelink provides a joint gain of

G1 = 12

{D(Φr

0, σ , wr1

)D(Φt

0, σ , wt1

)

+ e−1/ΛΓD(Φr

1, σ , wr1

)D(Φt

1, σ , wt1

)}.

(20)

When both Tx and Rx point towards 120◦ using w3 the linkgain is

G2 = 12

{D(Φr

0, σ , wr3)D

(Φt

0, σ , wt3)

+ e−1/ΛΓD(Φr

1, σ , wr3)D

(Φt

1, σ , wt3)}.

(21)

Therefore, the link gain difference between these twobranches can be obtained as

η =D(Φr

0, σ , wr1)D(Φt

0, σ , wt1)+e−1/ΛΓD(Φr

1, σ , wr1)D(Φt

1, σ , wt1)

D(Φr0, σ , wr

3)D(Φt0, σ , wt

3)+e−1/ΛΓD (Φr1, σ , wr

3)D(Φt1, σ , wt

3).

(22)

The weight vectors wr3 and wt

3 produce directional beam pat-terns at both ends of the link towards 120◦, whereas wr

1 andwt

1 produce directional beam pattern towards 0◦. Therefore,D(Φr

0, σ|wr3) , D(Φt

0, σ|wt3), D(Φr

1, σ|wr1), and D(Φt

1, σ|wt1) in

(22) represent the DDGs when the beam patterns have a 120◦

misalignment. From Figure 7, we know that the DDG valueswith such misalignment are relatively small. Furthermore, weassume that the beam pattern for 0◦ and 120◦ are same. Asstated in the previous section, we assume Γ = 30 nanosec-onds, 1/Λ = 30 nanosecond, σ = 20◦. Thus, (22) is simpli-fied to

η ≈ D(Φr

0, σ , wr1

)D(Φt

0, σ , wt1

)

e−1/ΛΓD(Φr

1, σ , wr3

)D(Φt

1, σ , wt3

) ≈ e = 4.34 dB.

(23)

Therefore, by switching directional beam patterns jointly atboth sides of the channel, we set up two wireless links with a4.34 dB difference of link gains. We approximate the gain ofeach wireless link as

G1 ≈ D(Φr

0, σ , wr1

)D(Φt

0, σ , wt1

),

G2 ≈ e−1/ΛΓD(Φr

1, σ , wr3

)D(Φt

1, σ , wt3

).

(24)

From Figure 7, we know that G1 = 2 × 4.9 = 9.8 dB andG2 = 5.5 dB for the channel with σ = 20◦. Please note thatthe link gain calculated in our model is normalized such thatwhen omnidirectional antennas are installed at both ends ofthe link, the link gain G is 0 dB.

3.3. Channel correlation

As explained previously, by jointly switching different Tx andRx beam patterns, the two wireless links can be exploited to

Chen Sun et al. 7

achieve certain diversity advantage. The correlation betweenthe two wireless links is induced due to beam pattern over-lapping. In this subsection, we examine the beam patterncorrelation.

We assume that there are two clusters in the clusteringDDRC as shown in Figure 5. Furthermore, each cluster’s cen-ter is assumed to be aligned with the direction of a beam pat-tern. Therefore, there is no beam pattern direction misalign-ment. Given the distribution of rays fω(φ|Φ1,Φ2) in (14), thecorrelation for the two directional beam patterns g(φ|wm)and g(φ|wn) can be expressed as

ρm,n =∣∣∫ 2π

0 fω(φ|Φ1,Φ2

)wTmα(φ)αH(φ)w∗

n dφ∣∣

√∫ 2π0 fω(φ|Φ1,Φ2)g(φ|wm)dφ

×∣∣∫ 2π

0 fω(φ|Φ1,Φ2

)wTmα(φ)αH(φ)w∗

n dφ∣∣

√∫ 2π0 fω

(φ|Φ1,Φ2

)g(φ|wn

)dφ

,

(25)

where f (φ) is given in (14). The correlation ρm,n is shownin Figure 8 for different values of σ . The markers in the fig-ure indicate the correlation for six discrete beam patterns,whereas the dashed lines show the correlation for continu-ously rotated beam patterns. We notice that the correlationdecreases as the angular spread of rays increases. The backlobe of the beam patterns as shown in Figure 4 induces thehigh correlation for two beam patterns pointing the oppositedirections (180◦ angular separation).

Knowing the correlation of any given two beam patternsat each side of the wireless link, the complex correlationof two link channels constructed by jointly pointing direc-tional beam patterns towards the same clusters is given asρ = ρrm,nρ

tm,n . For the channel realization with σ = 20◦,

Φr0 = Φr

0 = 0◦, and Φr1 = Φr

1 = 120◦, we obtain the channelcorrelation as ρ = 0.8× 0.8 = 0.64.

Using the DDG and beam pattern correlation, we set upan analytical model of the BF system. In the following sec-tion, we examine the channel capacity of the BF system basedon this analytical model.

4. NUMERICAL RESULT

In Section 3, we have obtained the link gains and the corre-lation of two wireless links. In this section, we examine thelink performance of the BF system and compare it with aMIMO system. To simplify the performance evaluation pro-cess and make the system independent of particular modu-lation schemes and coding techniques, we use the Shannon’stheoretical channel capacity and the outage rate to evaluatethe link performance and investigate the benefits from beampattern diversity gain.

4.1. Outage probability relative to output SNR

In the case of two clusters, after the Tx and Rx have identifiedthe clusters, two link channels can be established by jointlydirecting beam patterns at both Tx and Rx sides towards thecommon cluster. Switching between these two channels pro-

360300240180120600

Pattern azimuthal angular separation (deg)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Com

plex

patt

ern

corr

elat

ion

Lap σ = 20◦

Lap σ = 40◦Lap σ = 60◦

Uniform

Figure 8: Complex correlation of beam patterns in the presence ofa given angular distribution property of the channel.

vides a diversity gain. In this paper, we denote the link fromTx to Rx through the cluster at Φr

0 = Φt0 = 0◦ as channel

no.1, and denote that through the cluster at Φr1 = Φt

1 = 120◦

as channel no.2. In the previous subsections, we have calcu-lated that their complex envelop correlation is 0.8× 0.8, andthe difference of link gains is 4.34 dB. Without selection com-bining (SC), each link experiences a Rayleigh fading process.The outage rate of the output SNR relative to the input SNRat each channel is given by

Pr(r < x) = Fr(x) = 1− exp(− x

G1

), for i = 1, or 2,

(26)

where r is the instantaneous link gain.The outage rate with the SC of these two channels is ob-

tained by calculating

Pr(r < x) = Fr(x) = 1− exp(− x

G1

)Q(b, a|ρ|)

− exp(− x

G2

)(1−Q(b|ρ|, a)

).

(27)

Q(a, b) is the Marcum’s Q-function:

Q(a, b) =∞∫

b

exp{− 1/2

(a2 + x2)}I0(ax)x dx, (28)

and I0 is the modified Bessel function:

a =√

2x

G1(1− |ρ|2) ,

b =√

2x

G2(1− |ρ|2) .

(29)


20100−10−20−30−40−50

SNR (dB), relative to mean SNR of stronger channel

10−5

10−4

10−3

10−2

10−1

100

Pr

(SN

R<

absc

issa

)

Theo., no diversitySimu., no diversityTheo., joint SC, ρ = 0.64, η = 4.3 dBSimu., joint SCTheo., SC, ρ = 0, η = 0 dB

Figure 9: The outage rates of different channels using directionalantennas at both ends of the wireless links.

The outage rates of relative output SNR from the channelswithout and with SC technique are shown in Figure 9, re-spectively. Given the DDGs for the two channels and theircorrelation in the presence of two wave clusters, we calcu-late the theoretical outage rate with the SC of two channelsusing (26) and (27). The curve without a diversity approachshows the typical Rayleigh fading. The curve with the squaremarkers shows the situation when ideal SC of two channelswith equal SNRs and independent fading processes is em-ployed. Compared to the curve without diversity, we can seethat even the two channels have a 4.3 dB difference of linkgains and a correlation of ρ = 0.64, significant diversity gainis still achieved by jointly switching between different pairsof Tx and Rx directional beam patterns.

Simulation is also carried out for the same channel re-alization. Totally, 1000 ray realizations are created. In addi-tion, for each ray realization, 1000 fading channels are ran-domly generated. At each instant channel realization, the Txand Rx ESPAR antennas jointly switch to a common clusterthat gives the maximum instantaneous link gain. The outputlink gain after SC is used to compute the outage rate. Thesimulation results match closely with the theoretical resultsand show significant performance. This verifies our analyti-cal model. In the following analysis, we will use this analyti-cal model to examine the performance of the BF system andcompare it with a MIMO system.

Please note that the difference between the theoreticaland simulated results is due to (a) the simplification of dis-tribution of amplitude of rays within each cluster as a normaldistribution with the variance equal to the mean power de-cay of the first arrival ray in (18): (b) the approximation ofthe difference between the link gains of two channels in (23).

4.2. Channel capacity

Given an instantaneous link gain r and the average SNR, thechannel capacity of the BF system is given by

CBF = log (1 + rSNR). (30)

Using the outage probabilities for the BF system without andwith diversity given in (26) and (27), respectively, we obtainthe outage probably of CBF as

Pr(CBF < Cout

) = FBF(Cout

) = Pr(r <

2Cout − 1SNR

)

= Fr

(2Cout − 1

SNR

).

(31)

In order to compare the performance of the BF system withthat of a MIMO system, we also numerically obtain the chan-nel capacity of a MIMO system that is set up with linear an-tenna arrays of half-wavelength interelement spacing at bothends of the wireless link. The MIMO channel capacity is de-pendent on the array orientation with respect to the posi-tion of wave clusters [21]. To compare the BF system and theMIMO system at a given cluster realization, we assume thatat each side of the link, the directions of the two clusters withrespect to the end-fire direction of the linear array are 0◦ and120◦, respectively. All the elements of the linear arrays havethe same omnidirectional beam pattern. Thus, the DDG ofeach element is 0 dB. The steering vector of the linear arraythat consists of M array elements reads

α(φ) = [ 1 e− jπ cos(φ) ... e− jπ(M−1) cos(φ) ]T . (32)

We employ Wallace’s method [11] to obtain the MIMO chan-nel model in a clustering DDRC. In the following analysis,we refer to this MIMO channel model as Wallace’s MIMOmodel. Given the SRC of a clustering DDRC in (1), the spa-tial channel model for the MIMO system in the clusteringDDRC is given by

H = α(φr)∗h(φr ,φt)∗αT(φt), (33)

with each matrix entry being normalized as E[|H(i, j)|2] =1. In this study, we assume that the linear arrays at bothends of the link have two elements, that is, M = 2. For theMIMO system in an ideal statistical model [7], the entries ofH are given as independent and identically distributed (iid)complex Gaussian random variables (RV), that is, H(i, j) ∼CN(0, 1). We here after refer to this MIMO channel model asFoschini’s MIMO channel. The channel capacity of a 2 × 2MIMO system is given by

CMIMO = log 2 det(

I +SNR

2HHH

), (34)

where superscript (·)Hdenotes conjugate transpose, and I isan identity matrix. Furthermore, det (·) denotes matrix de-terminant.

The channel capacities for both BF and MIMO systemswhen SNR = 10 dB are shown in Figure 10. The line “omni-omni, gain = 0 dB” refers to the situation when one omnidi-rectional antenna (DDG is 0 dB) is installed at each side of

Chen Sun et al. 9

109876543210

Channel capacity (bits/s/Hz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr

(cap

acit

y>

absc

issa

)

SNR = 10 dB

Omni-omni, gain = 0 dBFoschini’s 2× 2 MIMOWallace’s 2× 2 MIMO, σ = 20◦

Ch#1, gain = 9.8 dB, σ = 20◦

Ch#2, gain = 5.5 dB, σ = 20◦Joint SC, Ch#1 & #2, ρ = 0.64

Figure 10: Shannon’s theoretical channel capacities of both BF andMIMO systems in different scenarios.

the link. The dashed line shows the theoretical channel ca-pacity of the Foschini’s 2 × 2 MIMO system as in [7]. Thischannel capacity is predicted based on the assumption of richmultipath. The solid line in Figure 10 shows the channel ca-pacity of the Wallace’s 2 × 2 MIMO system in a clusteringDDRC model [11]. As expected, in a clustering channel, thechannel capacity of a MIMO system is lower than that pre-dicted in ideal statistical model.

When directional beam patterns are jointly employed to-wards the first arrival cluster, the Tx and Rx directional an-tennas jointly provide a DDG of G1 = 9.8 dB to the wirelesschannel, which significantly increases the channel capacity.It is interesting to note that the ergodic capacity achievedby the BF system is even higher than that of the Foschini’s2× 2 MIMO channel [7]. As shown in Figure 10, SC of thesetwo links channels does not provide significant gain in termsof ergodic capacity. This is consistent with the conclusion in[22] that the influence of the diversity gain on the ergodicchannel capacity is negligible. However, as will be shown inthe following analysis, the diversity advantage of SC does sig-nificantly reduce the outage probability, thus improving thestability of wireless transmission.

Figure 11 shows the outage rate for Cout = 3 bits/sec/Hzat different SNR situations. Compared with the curve of ascalar Rayleigh fading channel “omni-omni, gain = 0 dB,”that is, one onmidirectional antenna is installed at each endof the wireless link, the outage curves for both channel #1and channel #2 are moved to the left by the joint DDG. Thisis due to the fact that the DDG obtained from using direc-tional antennas directly increase the average SNR of the wire-less link. When SC of the two channels is employed, the out-age probability is further reduced by the diversity gain. Thus,

302520151050−5−10

SNR (dB)

10−3

10−2

10−1

100

Ou

tage

prob

abili

ty

3 bits/s/Hz

Omni-omni, gain = 0 dBDirec-direc, ch#1, gain = 9.8 dBDirec-direc, ch#2, gain = 5.5 dBDirec-direc, joint SC, ρ = 0.64Wallace’s 2× 2 MIMO, σ = 20◦

Foschini’s 2× 2 MIMO

Figure 11: Outage rates of channel capacity at 3 bits/sec/Hz forboth BF and MIMO systems in different scenarios.

the required SNR to achieve the same outage rate at a givenchannel capacity is significantly reduced.

In the same figure, the outage probabilities at Cout =3 bits/s/Hz of the 2× 2 MIMO system in both ideal statisticalchannel and clustering DDRC are also plotted. It is interest-ing to note that in a low-SNR environment, the BF systemwith SC outperforms the Wallace’s MIMO system in termsof the outage probability. This shows that in a low-SNR en-vironment, the BF system provides a better wireless link interms of stability than that of a MIMO system in the sameclustering environment.

5. CONCLUSION AND DISCUSSION

In this paper, we proposed a BF system that directs both Txand Rx directional beam patterns towards a common wavecluster. In a clustering DDRC, we obtained the DDG andbeam pattern correlation for the BF system. Based on theseresults, we built an analytical model through which the chan-nel capacity of the BF system was obtained. The close matchbetween the simulation and analytical results justifies ourmodel. Using the outage probability of the theoretical chan-nel capacity, we have shown that the DDG of the BF directlyincreases the mean SNR of the wireless link, thus achievinga direct increase of the ergodic channel capacity. By jointlyswitching between different pairs of Tx and Rx directionalbeam patterns towards different wave clusters, the systemalso provides diversity gain to combat against multipath fad-ing, thus reducing the outage probability of the channel ca-pacity.

By comparing the proposed BF system with a MIMO sys-tem, we have shown that for a given cluster realization in a


clustering DDRC, the BF system provides a higher-ergodicchannel capacity. This is because of the significant DDG thatis achieved by employing directional antennas in a cluster-ing channel. Furthermore, in a low-SNR environment, theBF system that employs SC outperforms the MIMO systemin terms of outage probability of the channel capacity. Thus,the BF system provides a wireless link that is of higher stabil-ity than that of the MIMO system.

From the viewpoint of practical implementations ofadaptive antennas, if the beamforming network (BFN) is im-plemented at the RF stage, we can employ a directional an-tenna at a transceiver that has only one RF channel. This sim-plifies the system structure and reduces the DC power con-sumption. As an implementation example, the ESPAR an-tennas are employed in this study. The MIMO systems, onthe other hand, require independent RF channels, increasingsystem complexity and cost. Since performance gains similarto MIMO can be obtained with the less expensive BF systemfor realistic environments with cluster-type scattering, the re-sults suggest that BF is a good candidate for advanced indoorwireless communications networks.

REFERENCES

[1] K. Sheikh, D. Gesbert, D. Gore, and A. Paulraj, “Smart anten-nas for broadband wireless access networks,” IEEE Communi-cations Magazine, vol. 37, no. 11, pp. 100–105, 1999.

[2] J. Litva and T. K.-Y. Lo, Digital Beamforming in Wireless Com-munications, Artech House, Boston, Mass, USA, 1996.

[3] C. Sun and N. C. Karmakar, “Adaptive array antennas,” in En-cyclopedia of RF and Microwave Engineering, K. Chang, Ed.,p. 5832, John Wiley & Sons, New York, NY, USA, 2005.

[4] C. Passerini, M. Missiroli, G. Riva, and M. Frullone, “Adaptiveantenna arrays for reducing the delay spread in indoor radiochannels,” Electronics Letters, vol. 32, no. 4, pp. 280–281, 1996.

[5] D. G. Brennan, “Linear diversity combining techniques,” Pro-ceedings of the IEEE, vol. 91, no. 2, pp. 331–356, 2003.

[6] M. Schwartz, W. R. Bennett, and S. Stein, Communication Sys-tems and Techniques, McGraw-Hill, New York, NY, USA, 1966.


[8] A. F. Naguib, N. Seshadri, and A. R. Calderbank, “Increasingdata rate over wireless channels,” IEEE Signal Processing Mag-azine, vol. 17, no. 3, pp. 76–92, 2000.

[9] M. Steinbauer, A. F. Molisch, and E. Bonek, “The double-directional radio channel,” IEEE Antennas & Propagation Mag-azine, vol. 43, no. 4, pp. 51–63, 2001.

[10] C. Sun and N. C. Karmakar, “Direction of arrival estimationwith a novel single-port smart antenna,” EURASIP Journalon Applied Signal Processing, vol. 2004, no. 9, pp. 1364–1375,2004.

[11] J. W. Wallace and M. A. Jensen, “Modeling the indoor MIMOwireless channel,” IEEE Transactions on Antennas and Propa-gation, vol. 50, no. 5, pp. 591–599, 2002.

[12] A. Saleh and R. Valenzuela, “A statistical model for indoormultipath propagation,” IEEE Journal on Selected Areas inCommunications, vol. 5, no. 2, pp. 128–137, 1987.

[13] Q. H. Spencer, B. D. Jeffs, M. A. Jensen, and A. L. Swindlehurst,“Modeling the statistical time and angle of arrival characteris-tics of an indoor multipath channel,” IEEE Journal on SelectedAreas in Communications, vol. 18, no. 3, pp. 347–360, 2000.

[14] M. A. Jensen and J. W. Wallace, “A review of antennasand propagation for MIMO wireless communications,” IEEETransactions on Antennas and Propagation, vol. 52, no. 11, pp.2810–2824, 2004.

[15] T. Ohira and J. Cheng, “Analog smart antennas,” in AdaptiveAntenna Arrays: Trends and Applications, S. Chandran, Ed., pp.184–204, Springer, New York, NY, USA, 2004.

[16] C. Sun, A. Hirata, T. Ohira, and N. C. Karmakar, “Fast beam-forming of electronically steerable parasitic array radiator an-tennas: theory and experiment,” IEEE Transactions on Anten-nas and Propagation, vol. 52, no. 7, pp. 1819–1832, 2004.

[17] H. Kawakami and T. Ohira, “Electrically steerable passive ar-ray radiator (ESPAR) antennas,” IEEE Antennas & PropagationMagazine, vol. 47, no. 2, pp. 43–50, 2005.

[18] http://www.qsl.net/wb6tpu/swindex.html.

[19] C.-C. Chong, C.-M. Tan, D. I. Laurenson, S. McLaughlin, M.A. Beach, and A. R. Nix, “A new statistical wideband spatio-temporal channel model for 5-GHz band WLAN systems,”IEEE Journal on Selected Areas in Communications, vol. 21,no. 2, pp. 139–150, 2003.

[20] R. Vaugh and J. B. Andersen, Channels, Propagation andAntennas for Mobile Communications, IEE ElectromagneticWaves Series, Institution of Electrical Engineers, Glasgow, UK,2003.

[21] X. Li and Z.-P. Nie, “Effect of array orientation on perfor-mance of MIMO wireless channels,” IEEE Antennas and Wire-less Propagation Letters, vol. 3, no. 1, pp. 368–371, 2004.

[22] C. G. Gunther, “Comment on “estimate of channel capacity inRayleigh fading environment”,” IEEE Transactions on VehicularTechnology, vol. 45, no. 2, pp. 401–403, 1996.


Research ArticleInter- and Intrasite Correlations of Large-Scale Parametersfrom Macrocellular Measurements at 1800 MHz

Niklas Jalden, Per Zetterberg, Bjorn Ottersten, and Laura Garcia

ACCES Linnaeus Center, KTH Signal Processing Lab, Royal Institute of Technology, 100 44 Stockholm, Sweden

Received 15 November 2006; Accepted 31 July 2007


The inter- and intrasite correlation properties of shadow fading and power-weighted angle spread at both the mobile station andthe base station are studied utilizing narrowband multisite MIMO measurements in the 1800 MHz band. The influence of thedistance between two base stations on the correlation is studied in an urban environment. Measurements have been conductedfor two different situations: widely separated as well as closely located base stations. Novel results regarding the correlation of thepower-weighted angle spread between base station sites with different separations are presented. Furthermore, the measurementsand analysis presented herein confirm the autocorrelation and cross-correlation properties of the shadow fading and the anglespread that have been observed in previous studies.

Copyright © 2007 Niklas Jalden et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

As the demand for higher data rates increases faster thanthe available spectrum, more efficient spectrum utilizationmethods are required. Multiple antennas at both the receiverand the transmitter, so-called multiple input multiple output(MIMO) systems, is one technique to achieve high spectralefficiency [1, 2]. Since multiantenna communication systemsexploit the spatial characteristics of the propagation envi-ronment, accurate channel models incorporating spatial pa-rameters are required to conduct realistic performance eval-uations. Since future systems may reuse frequency channelswithin the same cell to increase system capacity, the charac-terization of the communication channel, including corre-lation properties of spatial parameters, becomes more criti-cal. Several measurement campaigns have been conducted todevelop accurate propagation models for the design, analy-sis, and simulation of MIMO wireless systems [3–9]. Most ofthese studies are based on measurements of a single MIMOlink (one mobile and one base station). Thus, these mea-surements may not capture all necessary aspects required formultiuser MIMO systems. From the measurement data col-lected, several parameters describing the channel character-istics can be extracted. This work primarily focusses on ex-tracting some key parameters that capture the most essentialcharacteristics of the environment, and that later can be used

to generate realistic synthetic channels with the purpose oflink level simulations. To evaluate system performance withseveral base stations (BS) and mobile stations (MS), it hasgenerally been assumed that all parameters describing thechannels are independent from one link (single BS to sin-gle MS) to another [3, 10]. However, correlation betweenthe channel parameters of different links may certainly ex-ist, for example, when one BS communicates with two MSsthat are located in the same vicinity, or vice versa. In thiscase, the radio signals propagate over very similar environ-ments and hence, parameters such as shadow fading and/orspread in angle of arrival should be very similar. This hasalso been experimentally observed in some work where theautocorrelation of the so-called large scale (LS) is studied.These LS parameters, such as shadow fading, delay spread,and angle spread, are shown to have autocorrelation that de-creases exponentially with a decorrelation distance of sometenths of meters [11, 12]. High correlation of these parame-ters is expected if the MS moves within a small physical area.We believe that this may also be the case for multiple BSsthat are closely positioned. The assumption that the chan-nel parameters for different links are completely indepen-dent may result in over/under estimation of the performanceof the multiuser systems. Previous studies [13–15] have in-vestigated the shadow fading correlation between two sepa-rate base station sites and found substantial correlation for


closely located base stations. However, the intersite correla-tion of angle spreads has not been studied previously. Herein,multisite MIMO measurements have been conducted to ad-dress this issue. We investigate the existence of correlationbetween LS parameters on separate links using data collectedin two extensive narrow-band measurement campaigns. Theintra- and intersite correlations of the shadow fading and thepower-weighted angle spread at the base and mobile stationsare investigated. The analysis provides unique correlation re-sults for base- and mobile-station angle spreads as well aslog-normal (shadow) fading.

The paper is structured as follows: in Section 2 we givea short introduction to the concept of large-scale parame-ters and in Section 3 some relevant previous research is sum-marized. The two measurement campaigns are presented inSection 4. In Section 5, we state the assumptions on the chan-nel model while Section 6 describes the estimation proce-dure. The results are presented in Section 7 and conclusionsare drawn in Section 8.

2. INTRODUCTION TO LARGE-SCALE PARAMETERS

The wireless channel is very complex and consists of timevarying multipath propagation and scattering. We considerchannel modeling that aims at characterizing the radio mediafor relevant scenarios. One approach is to conduct measure-ments and “condense” the information of typical channelsinto a parameterized model that captures the essential statis-tics of the channel, and later create synthetic data with thesame properties for evaluating link and system-level perfor-mance, and so on. Large-scale parameters are based on thisconcept. The term large-scale parameters was used [3] for acollection of quantities that can be used to describe the char-acteristics of a MIMO channel. This collection of parametersare termed large scale because they are assumed to be con-stant over “large” areas of several wavelengths. Further, theseparameters are assumed to depend on the local environmentof the transmitter and receiver. Some of the possible LS pa-rameters are listed below:

(i) shadow fading,

(ii) angle of arrival (AoA) angle spread,

(iii) angle of departure (AoD) angle spread,

(iv) AoA elevation spread,

(v) AoD elevation spread,

(vi) cross polarization ratio,

(vii) delay spread.

This paper investigates only the shadow fading and theangle spread parameters. Shadow fading describes the varia-tion in the received power around some local mean, whichdepends on the distance between the transmitter and re-ceiver; see Section 6.1. The power-weighted angle spread de-scribes the size of the sector or area from which the majorityof the power is received. The angle spread parameter will bedifferent for the transmitter (Tx) and receiver (Rx) sides ofthe link, since it largely depends on the amount of local scat-

MS

BS1

BS2

αd1

d2

Figure 1: Model of the cross-correlation as a function of the relativedistance and angle separation, also proposed in [16].

tering; see further in Section 5. A description of the other LSparameters may be found in [3].

3. PREVIOUS WORK

An early paper by Graziano [13] investigates the correlationof shadow fading in an urban macrocellular environment be-tween one MS and two BSs. The correlation is found to beapproximately 0.7-0.8 for small angles (α < 10◦), where αis defined as displayed in Figure 1. Later, Weitzen argued in[14] that the correlation for the shadow fading can be muchless than 0.7 even for small angles, in disagreement with theresults presented by Graziano. This was illustrated by ana-lyzing measurement data collected in the downtown Bostonarea using one custom made MS and several pairs of BSsfrom an existing personal communication system. These re-sults are reasonable since in most current systems the BS sitesare widely spread over an area. If the angle α separating thetwo BSs is small, the relative distance is large, and a small rel-ative distance corresponds to a large angle separation. Thus,a more appropriate model for the correlation of the shadowfading parameter is to assume that it is a function of the rel-ative distance d = log10(d1/d2) between the two BSs and theangle α separating them as proposed in [16]. The distances d1

and d2 are defined as in Figure 1. Further studies on the cor-relation of shadow fading between several sites can be foundin, for example, [15, 17–19].

The angular spread parameter has been less studied. In[12], the autocorrelation of the angle spread at a single basestation is studied and found to be well modeled by an expo-nential decay, and the angle spread is further found to be neg-atively correlated with shadow fading. However, to the au-thors’ knowledge, the intersite correlation of the angle spreadat the MS or BS has not been studied previously. Herein, weextend the analysis performed on the 2004 data in [20]. Wealso investigate data collected in 2005 and find substantialcorrelation between the shadow fading but less between theangular spreads. The low correlation of the spatial parame-ters may be important for future propagation modeling. Theangle spread at the mobile station is studied and a distribu-tion proposed. Further, we find that the correlation betweenthe base station and mobile station angular spreads (of thesame link) is significant for elevated base stations but virtu-ally zero for base stations just above rooftop.

Niklas Jalden et al. 3

4. MEASUREMENT CAMPAIGNS

Two multiple-site MIMO measurement campaigns havebeen conducted by KTH in the Stockholm area using cus-tom built multiple antenna transmitters and receivers. Thesemeasurements were carried out in the summer of 2004 andthe autumn of 2005 and will in the following be refereed toas the 2004 and 2005 campaigns.

Because of measurement equipment shortcomings, themeasured MIMO channels have unknown phase rotations.This is due to small unknown frequency offsets. In the 2004campaign, these phase rotations are introduced at the mobileside and therefore the relation between the measured channeland the true channel is given by

Hmeasured, 2004 = Λ f Htrue, (1)

where Λ f = diag(exp( j2π f1t), . . . , exp( j2π fnt)) and f1, . . . ,fn are unknown. Similarly, the campaign of 2005 has un-known phase rotations at the base station side1 resulting inthe following relation:

Hmeasured, 2005 = HtrueΛ f . (2)

The frequencies changed up 5 Hz per second. However,the estimators that will be used are designed with these short-comings in mind.

4.1. Measurement hardware

The hardware used for these measurements is the same asthe hardware described in [21, 22]. The transmitter continu-ously sends a unique tone on each antenna in the 1800 MHzband. The tones are separated 1 kHz from each other. The re-ceiver downconverts the signal to an intermediate frequencyof 10 kHz, samples and stores the data on a disk. This datais later postprocessed to extract the channel matrices. Thesystem bandwidth is 9.6 kHz, which allows narrow-bandchannel measurements with high sensitivity. The offline andnarrow-band features simplify the system operation, sinceneither real-time constrains nor broadband equalization isrequired. For a thorough explanation of the radio frequencyhardware, [23] may be consulted.

4.2. Antennas

In both measurements campaigns, Huber-Suhner dual-polarized planar antennas with slanted linear polarization(±45◦), SPA 1800/85/8/0/DS, were used at both the trans-mitter and the receiver. However, only one of the polariza-tions (+45◦) was actually used in these measurements. Theantennas were mounted in different structures on the mobileand base stations as described below. For more informationon the antenna radiation patterns and so on, see [24].

1 In the 2004 campaign, the phase rotations are due to drifting and un-locked local oscillators in the four mobile transmitters, while in the 2005campaign they are due to drifting sample-rates in the D/A and A/D con-verters.

23

(a)

Ref.

Tx1

Tx2

Tx3

Tx4

(b)

Figure 2: Mobile station box antenna.

4.2.1. Base satation array

At the base station, the antenna elements were mounted ona metal plane to form a uniform linear array with 0.56 wave-length (λ) spacing. In the 2004 campaign, an array of four byfour elements was used at the BS. However, the “columns”were combined using 4 : 1 combiners to produce four ele-ments with higher vertical gain. The base stations in the 2005campaign were only equipped with 2 elements.

4.2.2. Mobile station array

At the mobile side, the four antenna elements were mountedon separate sides of a wooden box as illustrated in Figure 2.This structure is similar to the uniform linear array usingfour elements. A wooden box is used so that the antenna ra-diation patterns are unaffected by the structure.

4.3. 2004 campaign

In this campaign uplink measurements were made using one4-element box-antenna transmitter at the MS, see Figure 2,and three 4-element uniform linear arrays (ULA), with anantenna spacing of 0.56λ, at the receiving BSs. The BSs cov-ered 3 sectors on two different sites. Site 1, Karhuset-A, hadone sector while site 2, Vanadis, had two sectors, B and C,separated some 20 meters and with boresights offset 120-degrees in angle. We define a sector by the area seen fromthe BS boresight±60◦. The environment where the measure-ments where conducted can be characterized as typical Eu-ropean urban with mostly six to eight storey stone buildingsand occasional higher buildings and church towers. Figure 3shows the location of the base station sites and the route cov-ered by the MS. The BS sectors are displayed by the dashedlines in the figure, and the arrow indicates the antenna point-ing direction. Sector A is thus the area seen between thedashed lines to the west of site Karhuset. Sector B and sec-tor C are the areas southeast and northeast of site Vanadis,respectively. A more complete description of the transmitterhardware and measurement conditions can be found in [25].


Karhuset

Vanadis

MS

1

23

4

Figure 3: Measurement geography and travelled route for 2004campaign.

Figure 4: Measurement map and travelled route for the 2005 cam-paign.

4.4. 2005 campaign

In contrast to the previous campaign, the 2005 campaigncollected data in the downlink. Two BSs with two antennaseach were employed (the same type of antenna elements as inthe 2004 campaign was used), each transmitting, simultane-ously, one continuous tone separated 1 kHz in the 1800 MHzband. The two base stations were located on the same roofseparated 50 meters, with identical boresight and thereforecovering almost the same sector. The characteristics of theenvironment in the measured area are the same as 2004. Theroutes were different but with some small overlap. The MSwas equipped with the 4-element box antenna as was used

in 2004, see Figure 2, to get a closer comparison between thetwo campaigns. In Figure 4, we see the location of the twoBSs (in the upper left corner) and the measured trajectorywhich covered a distance of about 10 km. The arrow in thefigure indicates the pointing direction of the base station an-tennas. The campaign measurements were conducted duringtwo days, and the difference in color of the MS routes depictswhich area was measured which day. The setups were identi-cal on these two days.

5. PRELIMINARIES

Assume we have a system with M Tx antennas at the basestation and K Rx antennas at the mobile station. Let hk,m(t)denote the narrow-band MIMO channel between the kthreceiver antenna and the mth transmitter antenna. Thenarrow-band MIMO channel matrix is then defined as

H(t) =

⎛⎜⎜⎜⎜⎜⎝

h1,1(t) h1,2(t) . . . h1,M(t)

h2,1(t). . .

......

. . ....

hK ,1(t) . . . . . . hK ,M(t)

⎞⎟⎟⎟⎟⎟⎠. (3)

The channel is assumed to be composed of N propaga-tion rays. The nth ray has angle of departure θk, angle ofarrival αk, gain gk, and Doppler frequency fk. The steeringvector2 of the transmitter given by aTx(θk) and that of thereceiver is aRx(αk). Thus, the channel is given by

H =N∑

k=1

gkej2π fktaRx(αk

)(aTx(θk

))H. (4)

The ray parameters (θk,αk, gk, and fk) are assumed to beslowly varying and approximately constant for a distance of30λ. Below, we define the shadow fading and the base stationand the mobile station angle spread.

5.1. Shadow fading

The measured channel matrices are normalized so that theyare independent of the transmitted power. The receivedpower, PRx, at the MS is defined as

PRx = E|H|2PTx =N∑

k=1

∣∣gk∣∣2∣∣aBS(θk

)∣∣2∣∣aMS,(αk)∣∣2

PTx,

(5)

where PTx is the transmit power. The ratio of the received andthe transmitted powers is commonly assumed to be related as[26]

PRx

PTx= K

RnSSF, (6)

2 The steering vector a(θ) can be seen a complex-valued vector of lengthequal to the number of antenna elements in the array. The absolute valueof the kth element is the square root of the antenna gain of that elementand the phase shift of the element relative to some common reference

point. That is ak(θ) =√ak(θ)e jφk .


where K is a constant, proportional to the squared norms ofthe steering vectors that depend on the gain at the receiverand transmitter antennas as well as the carrier frequency,base station height, and so on. The distance separating thetransmitter and receiver is denoted by R. The variable SSF de-scribes the slow variation in power, usually termed shadowfading, and is due to obstacles and obstruction in the propa-gation path. Expressing (6) in decibels (dB) and rearrangingthe terms in the path loss, which describe the difference be-tween transmitted and received powers, we have

L = 10log10

(PTx)− 10log10

(PRx)

= n10log10(R)− 10log10(K)− 10log10

(SSF),

(7)

where the logarithm is taken with base ten. Thus, the pathloss is assumed to be linearly decreasing with log-distanceseparating the transmitter and receiver when measured indB.

5.2. Base station power-weighted angle spread

The power-weighted angle spread at the base station, σ2AS,BS,

is defined as

σ2AS,BS =

N∑

k=1

pk(θk − θ

)2, (8)

where pk =∣∣gk∣∣2

is the power of the kth ray and the meanangle θ is given by

θ =N∑

k=1

pkθk. (9)

5.3. Mobile station power-weighted angle spread

The power-weighted angle spread at the mobile station,σ2

AS,MS, is defined as

σ2AS,MS = min

α

{1∑ N

k=1 pk

N∑

k=1

pk(�mod

(αk − α

))2}

, (10)

where �mod is short for modulo and defined as

�mod (α) =

⎧⎪⎪⎨⎪⎪⎩

α + 180, when α < −180,

α, when |α| < 180,

α− 180, when α > 180.

(11)

The definition of the MS angle spread is equivalent to thecircular spread definition in [10, Annex A]. In the following,the power-weighted angle spread will be refereed to as theangle spread.

6. PARAMETER ESTIMATION PROCEDURES

In the measurement equipment, the receiver samples thechannel on all Rx antennas simultaneously at a rate whichprovides approximately 35 channel realizations per wave-length. The first step of estimating the LS parameters is

Table 1: Number of measured 30λ segments from each measure-maent campaign, and number of segments in each BS sector.

All data 2004 SA SB SC All data 2005

2089 1742 1636 453 1637

to segment the data into blocks of length 30λ. This corre-sponds to approximately a 5m trajectory, during which theray-parameters are assumed to be constant, [12], and there-fore the LS parameters are assumed to be constant as well.Then smaller data sets for each BS are constructed such thatthey only contain samples within the given BS’s sector andblocks outside the BS’s sector of coverage are discarded; seedefinition in Section 4.3. Table 1 shows the total number ofmeasured 30λ segments from the campaigns as well as thenumber of segments within each BS sector.

6.1. Estimation of shadow fading

The fast fading due to multipath scattering varies with a dis-tance on the order of a wavelength [26]. Thus, the first stepto estimate the shadow fading is to remove the fast fadingcomponent. This is done by averaging the received powerover the entire 30 λ-segment and over all Tx and Rx anten-nas. The path loss component is estimated by calculating theleast squares fit to the average received powers from all 30 λ-segments against log-distance. The shadow fading, which isthe variation around a local mean, is then estimated by sub-tracting the distant dependent path loss component from theaverage received power for each local area. This estimationmethod for the shadow fading is the same as in, for example,[12].

6.2. Estimation of the base stationpower-weighted angle spread

Although advanced techniques have been developed for es-timating the power-weighted angle spread, [27–29], a sim-ple estimation procedure will be used here. Previously re-ported estimation procedures use information from severalantenna elements where both amplitude and phase informa-tion is available. In [25], the angle spread for the 2004 dataset is estimated using a precalculated look-up table generatedusing the gain from a beam steered towards the angle of ar-rival. However, as explained in Section 4.4, the BSs used in2005 were only equipped with two antenna elements withunknown frequency offsets, and thus a beam-forming ap-proach, or more complex estimation methods, are not appli-cable. Therefore, we have devised another method to obtainreasonable estimates of the angle spread applicable to bothour measurement campaigns. We cannot measure the angleof departure distribution itself, thus we will only consider itssecond-order moment, that is, the angle of departure spread.This method is similar to the previous one [25] in that a look-up table is used for determining the angle spreads. How-ever here the cross-correlation between the signal envelopesis used instead of the beam-forming gain.


The look-up table, which contains the correlation coeffi-cient as a function of the angle spread and the angle of depar-ture, has been precalculated by generating data from a modelwith a Laplacian (power-weighted) AoD distribution, sincethis distribution has been found to have a very good fit tomeasurement data; see, for example, [30]. The details of thelook-up table generation is described in Appendix A. Notethat our method is similar to the method used in [31], wherethe correlation coefficient is studied as a function of the an-gle of arrival and the antenna separation. To estimate the an-gle spread with this approach, only the correlation coefficientbetween the envelopes of the received signals at the BS andthe angle to the MS is calculated, where the latter is derivedusing the GPS information supplied by the measurements.

For the 2005 measurements, which were conducted withtwo antenna elements at the BS and four antennas at the MS,the cross-correlation between the signal envelopes at the BSis averaged over all four mobile antennas as

c1,2 =4∑

k=1

E{(∣∣Hk,1

∣∣−mk,1)(∣∣Hk,2

∣∣−mk,2)}

σk,1σk,2, (12)

where

mk,1 = E{∣∣Hk,1

∣∣},

mk,2 = E{∣∣Hk,2

∣∣},

σ2k,1 = E

{(∣∣Hk,1∣∣−mk,1

)2},

σ2k,2 = E

{(∣∣Hk,2∣∣−mk,2

)2}.

(13)

For the 2004 measurements, where also the BS had 4 anten-nas, the average correlation coefficient over the three antennapairs is used.

The performance of the estimation method presentedabove has been assessed by generating data from the SCMmodel, [10], then calculating the true angle spread (whichis possible on the simulated data since all rays are known)and the estimated angle spread using the method describedabove. The results of this comparison are shown in Figure 5.From the estimates in the figure, it is readily seen that the an-gle spread estimate is reasonably unbiased, with a standarddeviation of 0.1 log-degrees.

6.3. Estimation of the mobile stationpower-weighted angle spread

At the mobile station, an estimate of the power-weighted an-gle spread is extracted from the power levels of the four MSantennas. Accurate estimate cannot be expected, however,the MS angle spread is usually very large due to rich scat-tering at ground level in this environment and reasonable es-timates can still be obtained as will be seen.

A first attempt is to use a four-ray model where the AoAsof the four rays are identical to the boresights of the four MSantennas, that is, αn = 90◦(n − 2.5). The powers of the fourrays p1, . . . , p4 are obtained from the powers of the four an-tennas, that is, the Euclidean norm of the rows of the chan-nel matrices H. These estimates are obtained by averaging thefast fading over the 30λ segments. From the powers the angle

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

log 10

esti

mat

edan

gle

spre

ad

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

log10 true angle spread

Figure 5: Performance of the angle spread estimator on SCM gen-erated data.

spread is calculated using the circular model defined in (11)resulting in

σ2AS,MS-fe

= minα

{1

∑ 4

n=1pn

∑ 4

n=1pn(�mod

(90(n− 2.5

)− α))2}

,

(14)

where (·)fe is short for first-estimate. As explained in [10,Annex A], the angle spread should be invariant to the ori-entation of the antenna, hence, knowledge of the moving di-rection of the MS is not required. The performance of theestimate is first evaluated by simulating a large number ofwidely different cases, using the SCM model, and estimatingthe spread based on four directional antennas as proposedhere. The result is shown in Figure 6. The details of the sim-ulation are described in Appendix B.

The results show that the angle spread is often overesti-mated using the proposed method. However, as indicated byFigure 6, a better second estimate (·)se is obtained by the fol-lowing compensation:

σ2AS,MS-se =

(σ2

AS,MS-fe − 30)100

70. (15)

The performance of this updated estimator is shown inFigure 7. The second estimate is reasonable when σ2

AS,MS-se >

33. When σ2AS,MS-se < 33, the true angle spread may be any-

where from zero and σ2AS,MS-se. For small angle spreads, prob-

lems occur since all rays may fall within the bandwidth of asingle-antenna. The estimated angle spread from our mea-surements at the MS is usually larger than 33◦, thus thisdrawback in the estimation method has little impact on thefinal result. From the estimates in Figure 7, it is readily seenthat the angle spread estimate is unbiased, with a standarddeviation of 6 degrees.


100

90

80

70

60

50

40

30

20

10

0

Tru

ean

gle

spre

ad

30 40 50 60 70 80 90 100

First estimate of MS angle spread

EstimatesFitted line y = (x − 30)∗100/70

Figure 6: Performance of the first mobile station angle spread esti-mate.

100

90

80

70

60

50

40

30

20

10

0

Tru

ean

gle

spre

ad

0 10 20 30 40 50 60 70 80 90 100

Second estimate of MS angle spread

EstimatesLine y = x

Figure 7: Performance of the second mobile station angle spreadestimate.

7. RESULTS

In this section, the results of the analysis are presented inthree parts. First the statistical information of the param-eters is shown followed by their autocorrelation and cross-correlation properties.

7.1. Statistical properties

The first- and the second-order statistics of the LS parametersare estimated and shown in Table 3. The standard deviationof the shadow fading is given in dB while the angle spread atthe BS is given in logarithmic degrees. Further, the MS an-

Table 2: Parmeters α and β for the beta best fit distribution to theangle spread at the mobile.

2004:A 2004:B 2004:C 2005:1 2005:2

α 8.69 5.74 4.22 6.85 7.07

β 2.85 2.36 2.44 2.72 2.77

gle spread is given in degrees. The mean value of the shadowfading component is not tabulated since it is zero by defini-tion. As seen from the histograms in Figure 8, which showsthe statistics of the LS parameters for site 2004:B, the shadowfading and log-angle spread can be well modeled with a nor-mal distribution. This agrees with observations reported in[12, 26]. The angle spread at the mobile on the other hand isbetter modeled by a scaled beta distribution, defined as

f (x,α,β) = 1B(α,β)

(x

η

)α−1(1− x

η

)β−1

, (16)

where η = 360/√

12 is a normalization constant, equal to themaximum possible angle spread. The best fit shape parame-ters α and β for each of the measurement sets are tabulated inTable 2. The parameter B(α,β) is a constant which dependson α and β such that

∫ η0 f (x,α,β)dx = 1. The distributions

of the parameters from all the other measured sites are sim-ilar, with statistics given in Table 3. From the table it is seenthat the angle spread clearly depends on the height of the BS.The highest elevated BS, 2004:B, has the lowest angle spreadand correspondingly, the BS at rooftop level, 2004:A, has thelargest angle spread. The mean angle spreads at the base sta-tion are quite similar to the typical urban sites in [12] (0.74–0.95) and to those of the SCM urban macromodel (0.81–1.18) [10]. Furthermore, the standard deviations of the an-gle spread and the shadow fading found here, see Table 3, aresomewhat smaller than those of [12]. One explanation forthis could be that the measured propagation environmentsin 2004 and 2005 are more uniform than those measured in[12].

7.2. LS autocorrelation

The rate of change of the LS parameters is investigated byestimating the autocorrelation as a function of distance trav-elled by the MS. The autocorrelation functions for the large-scale parameters are shown in Figures 9 and 10, where thecorrelation coefficient between two variables is calculated asexplained in Appendix C. Note that the autocorrelation func-tions can be well approximated by an exponential functionwith decorrelation distances as seen in Table 4. The decorre-lation distance is defined as the distance for which the cor-relation has decreased to e−1. Furthermore, it can be notedthat these distances are very similar for the 2004 and the2005 measurements, which is reasonable since the environ-ments are similar. The exponential model has been proposedbefore, see [12], for the shadow fading and angle spread atthe BS. The results shown herein indicate that this is a goodmodel for the angle spread at the MS as well.


Table 3: Inter-BS correlation for measurement campaign 2004 site A.

2004:A 2004:B 2004:C 2005:1 2005:2

std[SF] 5.6 dB 5.2 dB 5.4 dB 4.9 dB 4.9 dB

E[σAS,BS] 1.2 ld 0.91 ld 0.85 ld 0.96 ld 0.87 ld

std[σAS,BS] 0.25 ld 0.2 ld 0.23 ld 0.19 ld 0.17 ld

E[σAS,MS] 75.1 deg 70.6 deg 65.9 deg 71.6 deg 72.2 deg

std[σAS,MS] 15.7 deg 18.7 deg 19.2 deg 16.1 deg 16.9 deg

Table 4: Average decorrelation distane in maters for the estimatedlarge-scale parameters.

SF σAS,BS σAS,MS

ddecorr(m) 113 88 32

Table 5: Intra-BS correlation of LS parameters for measurementcampaign 2004 site A.

2004:A

SF σAS,MS σAS,BS

SF 1.00 −0.37 −0.46

σAS,MS −0.37 1.00 0.10

σAS,BS −0.46 0.10 1.00

7.3. Intrasite correlation

The intrasite correlation coefficients between different large-scale parameters at the same site are calculated for the twoseparate measurement campaigns. In Tables 5 and 6, the cor-relation coefficients for the two base stations, sectors A andB, from 2004 are shown, respectively. The last sector (C) isnot shown since it is very similar to B and these parame-ters are based on a much smaller set of data, see Table 1. InTable 7, the same results are shown for the 2005 measure-ments. Since sites (2005:1 and 2005:2) show similar resultsand are from similar environments, the average correlationof the two is shown. It follows from mathematics that thesetables are symmetrical, and in fact they only contain threesignificant values. The reason for showing nine values, in-stead of three, is to ease comparison with the intersite cor-relation coefficients shown in Tables 8–10. As seen from thetables, the angle spread is negatively correlated with shadowfading as was earlier found in for example [3, 12]. The cross-correlation coefficient between the shadow fading and basestation angle spread is quite close to that of [12], that is −0.5to −0.7. For the two cases where the BS is at rooftop level,Karhuset, 2004:A, and the 2005 sites, there is no correlationbetween the angle spreads at the MS and the BS. However,for Vanadis, 2004:B, there is a positive correlation of 0.44. Apossible explanation is that the BS is elevated some 10 metersover average rooftop height. Thus, no nearby scatterers existand the objects that influence the angle spread at the BS arethe same as the objects that influence the angle spread at theMS. A BS at rooftop on the other hand may have some nearbyscatterers that will affect the angle of arrival and spread. InFigure 11, this is explained graphically. The stars are some of

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

020 0 20

(dB)

Shadow fading

(a)

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5

log10 (degrees)

Angle spread at BS

(b)

0.025

0.02

0.015

0.01

0.005

00 50 100

(Degrees)

Angle spread at MS

(c)

Figure 8: Histograms of the estimated large-scale parameters forsite 2004:B.

the scatterers and the dark section of the circles depicts thearea from which the main part of the signal power comes,that is the angle spread. In the left half of the picture, we seean elevated BS, without close scatterers, and therefore a largeMS angle spread results in a large BS spread. In the right halfof Figure 11, a BS at rooftop is depicted, with nearby scatter-ers, and we see how a small angle spread at the MS can resultin a large BS angle spread (or the other way around).

7.4. Inter-BS correlation

The correlation coefficients between large-scale parametersat two separate sites are calculated for the data collected fromboth measurement campaigns. Only the data points whichare common to both base station sectors, Si ∩ Sj , are usedfor this evaluation, that is, points that are within the ±60◦

beamwidth of both sites. As seen in section 4, describing themeasurement campaigns, there is no overlap between site2004:B and 2004:C if one considers ±60◦ sectors. For thisspecific case, the sector is defined as the area within ±70◦ ofthe BS’s boresight, thus resulting in a 20◦ sector overlap. Theresults of this analysis are displayed in Tables 8, 9, and 10 for2004:A-B, B-C, and 2005:1-2, respectively. As earlier shownin [20], the average correlation between the two sites 2004:A


1

0.8

0.6

0.4

0.2

0

0.2

0.4

Cor

rela

tion

0 100 200 300 400 500

Distance (m)

Site: A SFSite: A ASBS

Site: B SF

Site: B ASBS

Site: 05 SFSite: 05 ASBS

Figure 9: Autocorrelation of the shadow fading and the anglespread at the base station for both measurements.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Cor

rela

tion

0 20 40 60 80 100

Distance (m)

Site: A ASMS

Site: B ASMS

Site: 05 ASMS

Figure 10: Autocorrelation of the angle spread at the mobile stationfor both measurements.

and 2004:B is close to zero. This is not surprising since theangular separation is quite large and the environments at thetwo separate sites are different. The correlations between sec-tors B and C of 2004 are similar as between sectors 1 and 2of 2005. In both cases, the two BSs are on the same roof, andseparated 20 and 50 meters for 2004 and 2005, respectively.As can be seen, these tables (Tables 8–10) are not symmetric.Thus the correlation of, for example, the shadow fading at BS2005:1 and the angle spread at BS 2005:2 is not the same asthe correlation of the shadow fading of BS 2005:2 and the an-

BS BS BS

MS MS MS

Elevated BS BS at rooftop (2 examples)

Figure 11: Model of correlation between angle spread at base sta-tion and mobile station.

Table 6: Intra-BS correlation of LS parameters for measurementcampaign 2004 site B.

2004:B

SF σAS,MS σAS,BS

SF 1.00 −0.54 −0.69

σAS,MS −0.54 1.00 0.44

σAS,BS −0.69 0.44 1.00

Table 7: Intra-BS correlation of LS parameters for measurementcampaign 2005.

2005

SF σAS,MS σAS,BS

SF 1.00 −0.25 −0.59

σAS,MS −0.25 1.00 0.11

σAS,BS −0.59 0.11 1.00

Table 8: Inter-BS correlation of all studied LS parameters betweensite A and site B from 2004 measurements.

2004:A

SF σAS,MS σAS,BS

2004:BSF −0.14 0.08 −0.06

σAS,MS −0.07 −0.05 0.03

σAS,BS −0.04 −0.09 0.07

Table 9: Inter-BS correlation of all studied LS parameters betweensite B and site C from 2004 measurements.

2004:B

SF σAS,MS σAS,BS

2004:CSF 0.83 −0.23 −0.52

σAS,MS −0.19 0.53 0.18

σAS,BS −0.54 0.22 0.31

gle spread of BS 2005:1 (〈SF2005:1, σ2005:2AS,BS 〉�=〈SF2005:2, σ2005:1

AS,BS 〉),and so on. This is not surprising.

In Figure 12, the correlation coefficient is plotted againstthe angle separating the two base stations with the mobilein the vertex. The large variation of the curve is due to alack of data. This may be surprising in the light of the quite


Table 10: Inter-BS correlation of all studied LS parameters betweensite B1 and B2 from 2005 measurements.

2005:1

SF σAS,MS σAS,BS

2005:2SF 0.85 −0.06 −0.45

σAS,MS −0.05 0.46 0.04

σAS,BS −0.27 0.18 0.33

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Cor

rela

tion

40 60 80 100 120 140 160 180

Angle separating the base stations (deg)

Shadow fadingAngle spread at BSAngle spread at MS

Figure 12: Intersite correlation of the large-scale parameters as afunction of the angle separating the base stations for the 2004 mea-surements.

long measurement routes. However, due to the long decorre-lation distances of the LS parameters (∼100 m), the numberof independent observations is small. The high correlationfor large angles of about 180◦ is mainly due to a very smalldata set available for this separation. Furthermore, this areaof measurements is open with a few large buildings in thevicinity and thus the received power to both BSs is high.

If, on the other hand, the cross-correlation of the large-scale parameters between the two base stations from the 2005measurements is studied, it is found that the correlation issubstantial, see Table 10. Also, note that the correlation in an-gle spread is much smaller than the shadow fading. If the cor-relation is plotted as a function of the angle, separating theBSs as in Figure 13, a slight tendency of a more rapid dropin the correlation of angle spread than that of the shadowfading for increasing angles is seen. The intersite correlationresults shown in Figures 12 and 13 are calculated disregard-ing the relative distance, see Figure 1. However, for the 2005campaign this distance d ≈ 0 is due to the location of thebase stations.

The intersite correlation of the angle spread was cal-culated in the same way as the shadow fading. Only themeasurement locations common to two sectors were usedfor these measurements. The angle spread is shown to have

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

Cor

rela

tion

1 2 3 4 5 6 7 8 9 10

Angle separating the base stations (deg)

Shadow fadingAngle spread at BSAngle spread at MS

Figure 13: Intersite correlation of the large-scale parameters as afunction of the angle separating the base stations for the 2005 mea-surements.

smaller correlation than the shadow fading even for smallangular separations. This indicates that it may be less im-portant to include this correlation in future wireless channelmodels. It should be highlighted that the correlations shownin Table 10 are for angles α < 10◦ and a relative distance|d = log (d1/d2)| ≈ 0.

8. CONCLUSION

We studied the correlation properties of the three large-scaleparameters shadow fading, base station power-weighted an-gle spread, and mobile station power-weighted angle spread.Two limiting cases were considered, namely when the basestations are widely separated, ∼900 m, and when they areclosely positioned, some 20–50 meters apart.

The results in [12] on the distribution and autocorrela-tion of shadow fading and base station angle spread wereconfirmed although the standard deviations of the angularspread and shadow fading were slightly smaller in our mea-surements. The high interbase station shadow fading cor-relation, when base stations are close, as observed in [13],was also confirmed in this analysis. Our results also showthat angular spread correlation exists at both the base stationand the mobile station if the base station separation is small.However, the correlation in angular spread is significantlysmaller than the correlation of the shadow fading. Thus itis less important to model this effect. For widely separatedbase stations, our results show that the base station and mo-bile station angular spreads as well as the shadow fading areuncorrelated.

The angle spread at the mobile was analyzed and a scaledbeta distribution was shown to fit the measurements well.Further, we have also found that the base station and mo-bile station angular spreads are correlated for elevated base


stations but uncorrelated for base station just above rooftop.Correlation can be expected if the scatters are only locatedclose to the mobile station, which is the case for macrocellu-lar environments, as illustrated in Figure 11.

In the future, it will be of interest to assess also the regionin between the two limiting cases studied herein. Note thatthe limiting case of distances of 20–50 meters has a practi-cal interest. For instance, the sectors of three-sector sites aresometimes not colocated but placed on different edges of aroof. The two base stations may also belong to different op-erators and the properties studied here could then be impor-tant when studying adjacent carrier interference.

APPENDICES

A. GENERATION OF ANGLE SPREAD LOOK-UP TABLE

The Laplacian angle of departure distribution is given by

PA(θ) = Ce(−|θ−θ0|)/(σAoD), (A.1)

where θ0 is the nominal direction of the mobile and σAoD isangle-of-departure spread. The variable C is a constant suchthat

∫ π−πPA(θ)dθ = 1. When generating data, the channel

covariance matrix is first estimated as

R =∫ 180◦

θ=−180◦PA(θ)a(θ)a∗(θ)dθ, (A.2)

where a(θ) is the array steering vector which is given by

a(θ) = p(θ)[1, exp

(− j2πdspacingsin(θ))]T

, (A.3)

p(θ) is the (amplitude) antenna element diagrams of the ar-ray, and dspacing is the spacing between the antenna elementsgiven in wavelengths. In our case the element diagrams areapproximated by

p2(θ) = max(101.4cos2(θ), 10−0.2), (A.4)

and the antenna element spacing is 0.56 wavelength. Theprocedure for calculating the look-up table is then (1) fix an-gle spread and nominal angle of arrival, (2) calculate the co-variance matrix R and it is eigendecomposition, (3) generatedata from the model and calculate the envelope correlation.

The choice of Laplacian (power-weighted) AoD distribu-tion over others, such as the Gaussian one, does only affectthe estimation results marginally due to the short antennaspacing distance. This is further explained in [31].

B. EVALUATION OF THE MOBILE STATIONANGLE SPREAD ESTIMATOR

To test the estimator of the (power-weighted) RMS anglespread at the mobile-station side, some propagation chan-nels were generated. Each channel had random number ofclusters which was equally distributed between 1 and 10. TheAoA of each cluster is uniformly distributed between 0◦ and360◦. The powers of the clusters are log-normally distributedwith a standard deviation of 8 dB. Each cluster is modeled

with between 1 and 100 rays (all with equal power) whichare uniformly distributed within the cluster width. The clus-ter widths are uniformly distributed between 0 and 10 de-grees. One-thousand propagation (completely independent)channels are drawn from this model. The powers of the fourantennas are calculated based on the powers of the rays, theirangle of arrival, and the antenna pattern. The true anglespread is first estimated as described in [10, Annex A], andthen the estimation method described in Section 6.3 is ap-plied.

C. LARGE SCALE CORRELATIONS

The correlation coefficient between two variables is definedby the normalized covariance as

ρ = 〈a, b〉 = E[ab]−mamb√(E[a2]−m2

a

)(E[b2]−m2

b

) . (C.5)

At all times when calculating the cross-correlation betweenLS parameters, even for small subsets of data, like when an-alyzing the correlation as a function of angular separationbetween BSs, the mean values are global. Hence the valuesma and mb are calculated using the full data set of each BSssector, respectively. If the mean values would be estimatedlocally, it is equal to assuming that the parameters are locallyzero mean, and this is not what we are studying. What wewant to investigate is if one parameter is large (or small) giventhe other.

ACKNOWLEDGMENTS

This work was sponsored partly within the Antenna Centerof Excellence (FP6-IST 508009), the WINNER project IST-2003-507581, and wireless@KTH.

REFERENCES

[1] G. Foschini and M. J. Gans, “On limits of wireless communica-tions in a fading environmen when using multiple antennas,”Wireless Personal Communications, vol. 6, no. 3, pp. 311–335,1998.

[2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,”European Transactions on Telecommunications, vol. 10, no. 6,pp. 585–595, 1999.

[3] D. S. Baum, H. El-Sallabi, T. Jamsa, et al., “IST-WINNER D5.4,final report on link and system level channel models,” http://www.ist-winner.org/, October 2005.

[4] D. Chizhik, J. Ling, P. Wolniansky, R. Valenzuela, N. Costa,and K. Huber, “Multiple-input-multiple-output measure-ments and modeling in manhattan,” IEEE Journal on SelectedAreas in Communication, vol. 21, no. 3, pp. 321–331, 2003.

[5] V. Eiceg, H. Sampath, and S. Catreux-Erceg, “Dual-polariza-tion versus single-polarization MIMO channel measurementresults and modeling,” IEEE Transactions on Wireless Commu-nications, vol. 5, no. 1, pp. 28–33, 2006.

[6] P. Kyritsi, D. C. Cox, R. A. Valenzuela, and P. W. Wolniansky,“Correlation analysis based on MIMO channel measurementsin an indoor environment,” IEEE Journal on Selected Areas inCommunications, vol. 21, no. 5, pp. 713–720, 2003.


[7] M. Steinbauer, A. F. Molisch, and E. Bonek, “The double-directional radio channel,” IEEE Antennas and PropagationMagazine, vol. 43, no. 4, pp. 51–63, 2001.

[8] R. Stridh, K. Yu, B. Ottersten, and P. Karlsson, “MIMO chan-nel capacity and modeling issues on a measured Indoor radiochannel at 5.8 GHz,” IEEE Transactions on Wireless Communi-cations, vol. 4, no. 3, pp. 895–903, 2005.

[9] J. Wallace and M. Jensen, “Time-varying MIMO channels:measurement, analysis, and modeling,” IEEE Transactions onAntennas and Propagation, vol. 54, no. 11 ,part 1, pp. 3265–3273, 2006.

[10] 3GPP-SCM, “Spatial channel model for multiple inputmultiple output (MIMO) simulations,” TR.25.966 v.6.10,http://www.3gpp.org/, September 2003.

[11] M. Gudmundson, “Correlation model for shadow fading inmobile radio systems,” IEEE Electronics Letters, vol. 27, no. 23,pp. 2145–2146, 1991.

[12] A. Algans, K. I. Pedersen, and E. P. Mogensen, “Experimentalanalysis of the joint statistical properties of azimuth spread,delay spread, and shadow fading,” IEEE Journal on Selected Ar-eas in Communications, vol. 20, no. 3, pp. 523–531, 2002.

[13] V. Graziano, “Propagation correlation at 900MHz,” IEEETransactions on Vehicular Technology, vol. 27, no. 4, pp. 182–189, 1978.

[14] J. Weitzen and T. J. Lowe, “Measurement of angular and dis-tance correlation properties of log-normal shadowing at 1900MHz and its application to design of PCS systems,” IEEETransactions on Vehicular Technology, vol. 51, no. 2, pp. 265–273, 2002.

[15] A. Mawira, “Models for the spatial correlation functions of the(log)-normal component of the variability of VHF/UHF fieldstrength in urban environment,” in Proceedings of the 3rd IEEEInternational Symposium on Personal, Indoor and Mobile Ra-dio Communications (PIMRC ’92), pp. 436–440, Boston, Mass,USA, October 1992.

[16] K. Zayana and B. Guisnet, “Measurements and modelisationof shadowing cross-correlationsbetween two base-stations,” inIEEE International Conference on Universal Personal Commu-nications (ICUPC ’98), vol. 1, pp. 101–105, Florence, Italy, Oc-tober 1998.

[17] E. Perahia, D. C. Cox, and S. Ho, “Shadow fading cross cor-relation between basestations,” in The 53rd IEEE VehicularTechnology Conference (VTC ’01), vol. 1, pp. 313–317, Rhodes,Greece, May 2001.

[18] H. W. Arnold, D. C. Cox, and R. R. Murray, “Macroscopic di-versity performance measured in the 800-MHz portable radiocommunications environment,” IEEE Transactions on Anten-nas and Propagation, vol. 36, no. 2, pp. 277–281, 1988.

[19] T. Klingenbrunn and P. Mogensen, “Modelling cross-correlated shadowing in network simulations,” in The 50thVehicular Technology Conference (VTC ’99), vol. 3, pp. 1407–1411, Amsterdam, The Netherlands, September 1999.

[20] N. Jalden, P. Zetterberg, M. Bengtsson, and B. Ottersten,“Analysis of multi-cell MIMO measurements in an urbanmacrocell environment,” in General Assembly of InternationalUnion of Radio Science (URSI ’05), New Delhi, India, October2005.

[21] L. Garcia, N. Jalden, B. Lindmark, P. Zetterberg, and L. D.Haro, “Measurements of MIMO capacity at 1800MHz with in-and outdoor transmitter locations,” in Proceedings of the Eu-ropean Conference on Antennas and Propagation (EuCAP ’06),Nice, France, November 2006.

[22] L. Garcia, N. Jaldin, B. Lindmark, P. Zetterberg, and L. D.Haro, “Measurements of MIMO indoor channels at 1800MHz

with multiple indoor and outdoor base stations,” EURASIPJournal on Wireless Communication and Networking, vol. 2007,Article ID 28073, 10 pages, 2007.

[23] P. Zetterberg, “WIreless DEvelopment LABoratory (WIDE-LAB) equipment base,” Signal Sensors and Systems (KTH), iR-SB-IR-0316, http://www.ee.kth.se/, August 2003.

[24] http://www.hubersuhner.com/.[25] P. Zetterberg, N. Jalden, K. Yu, and M. Bengtsson, “Analysis

of MIMO multi-cell correlations and other propagation issuesbased on urban measurements,” in Proceedings of the 14th ISTMobile and Wireless Communications Summit, Dresden, Ger-many, June 2005.

[26] T. Rappaport, Wireless Communications: Principles and Prac-tice, Prentice-Hall, Upper Saddle River, NJ, USA, 1996.

[27] T. Trump and B. Ottersten, “Estimation of nominal directionof arrival and angular spread using an array of sensors,” SignalProcessing, vol. 50, no. 1-2, pp. 57–69, 1996.

[28] M. Bengtsson and B. Ottersten, “Low-complexity estimatorsfor distributed sources,” IEEE Transactions on Signal Process-ing, vol. 48, no. 8, pp. 2185–2194, 2000.

[29] M. Tapio, “Direction and spread estimation of spatially dis-tributed signals via the power azimuth spectrum,” in Pro-ceedings of IEEE International Conference on Acoustics, Speech,and Signal Processing (ICASSP ’02), vol. 3, pp. 3005–3008, Or-lando, Fla, USA, May 2002.

[30] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, “A stochas-tic model of the temporal and azimuthal dispersion seen atthe base station in outdoor propagation environments,” IEEETransactions on Vehicular Technology, vol. 49, no. 2, pp. 437–447, 2000.

[31] N. Jalden, “Analysis of radio channel measurements usingmultiple base stations,” Licenciate Thesis, Royal Institute ofTechnology, Stockholm, Sweden, May 2007.


Research ArticleMultiple-Antenna Interference Cancellation for WLAN withMAC Interference Avoidance in Open Access Networks

Alexandr M. Kuzminskiy1 and Hamid Reza Karimi1, 2

1 Alcatel-Lucent, Bell Laboratories, The Quadrant, Stonehill Green, Westlea, Swindon SN5 7DJ, UK2 Ofcom, Riverside House, 2a Southwark Bridge Road, London SE1 9HA, UK

Received 31 October 2006; Accepted 3 September 2007


The potential of multiantenna interference cancellation receiver algorithms for increasing the uplink throughput in WLAN systemssuch as 802.11 is investigated. The medium access control (MAC) in such systems is based on carrier sensing multiple-access withcollision avoidance (CSMA/CA), which itself is a powerful tool for the mitigation of intrasystem interference. However, due tothe spatial dependence of received signal strengths, it is possible for the collision avoidance mechanism to fail, resulting in packetcollisions at the receiver and a reduction in system throughput. The CSMA/CA MAC protocol can be complemented in suchscenarios by interference cancellation (IC) algorithms at the physical (PHY) layer. The corresponding gains in throughput are aresult of the complex interplay between the PHY and MAC layers. It is shown that semiblind interference cancellation techniquesare essential for mitigating the impact of interference bursts, in particular since these are typically asynchronous with respect tothe desired signal burst. Semiblind IC algorithms based on second- and higher-order statistics are compared to the conventionalno-IC and training-based IC techniques in an open access network (OAN) scenario involving home and visiting users. It is foundthat the semiblind IC algorithms significantly outperform the other techniques due to the bursty and asynchronous nature of theinterference caused by the MAC interference avoidance scheme.

Copyright © 2007 A. M. Kuzminskiy and H. R. Karimi. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. INTRODUCTION

Interference at the radio receiver is a key source of degra-dation in quality of service (QoS) as experienced in wirelesscommunication systems. It is for this reason that a great pro-portion of mobile radio engineering is exclusively concernedwith the development of transmitter and receiver technolo-gies, at various levels of the protocol stack, for mitigation ofinterference.

Multiple-antenna interference cancellation (IC) at the re-ceiver has been the subject of a great deal of research in differ-ent application areas including wireless communications [1–3] and others. Despite the considerable interest in this area,IC techniques are typically studied at the physical (PHY)layer and in isolation from the higher layers of the proto-col stack, such as the medium access control (MAC). How-ever, it is clear that any gains at the system level are highlydependent on the nature of cross-layer interactions, partic-ularly if multiple layers are designed to contribute to the in-terference mitigation process. This is indeed the case for theIEEE 802.11 family of wireless local area network (WLAN)systems [4], where the carrier sensing multiple-access with

collision avoidance (CSMA/CA) MAC protocol is itself de-signed to eliminate the possibility of interference at the re-ceiver from other users of the same system.

Although the MAC layer CSMA/CA protocol may be veryeffective for avoidance of intrasystem interference in typicalconditions, certain applications which experience significanthidden terminal problems and/or interference from coexist-ing “impolite” systems may also benefit from PHY layer IC.PHY/MAC cross-layer design is clearly required in such situ-ations.

One important example of the above is an open accessnetwork (OAN) where visiting users (VUs) are allowed toshare the radio resource with home users (HUs) [5]. In manyscenarios, VUs typically experience greater distances from anaccess point (AP) compared to HUs. This means that VUsmay interfere with each other with higher probability com-pared to HUs, leading to throughput reduction for VUs orgaps in coverage. A multiple-antenna AP with IC may be asolution to this problem.

A cross-layer design in such a system is required be-cause the CSMA/CA protocol leads to an asynchronous


“Listen”

DIFS

“Backoff”

Slot time MPDU ACK

t

SIFS

Figure 1: Transmission of MPDU and ACK bursts.

interference structure, where interference bursts appear withrandom delays during the desired signal data burst. Oneway to account for higher-layer effects is to develop inter-ference models that reflect key features of cross-layer inter-action and design PHY-layer algorithms that address these.This is the methodology adopted in [6–11], where semi-blind space-time/frequency adaptive second- and higher-order statistic IC algorithms have been developed in con-junction with an asynchronous (intermittent) interferencemodel. The second-order algorithm is based on the con-ventional least-squares (LS) criterion formulated over thetraining interval, regularized by means of the covariancematrix estimated over the data interval. This simple ana-lytical solution demonstrates performance that is close tothe nonasymptotic maximum likelihood (ML) benchmark[6, 7]. Further analysis is given in [8], which introduces non-stationary interval-based processing and benchmark in theasynchronous interference scenario. The regularized semib-lind algorithms can be applied independently or as an ini-tialization for higher-order algorithms that exploit the finitealphabet (FA) or constant modulus (CM) properties of com-munication signals. The efficiency of these algorithms hasbeen compared to the conventional LS solution [1] by meansof PHY simulations. These involve evaluation of metrics suchas mean square error (MSE), bit-error rate (BER), or packet-error rate (PER), as a function of signal-to-interference ratio(SIR) for given signal-to-noise ratio (SNR), and a number ofindependent asynchronous interferers.

Our goal in this paper is to evaluate cross-layer interfer-ence avoidance/cancellation effects for different algorithmsand estimate the overall system performance in terms ofthroughput and coverage. The combined performance of dif-ferent IC algorithms at the PHY layer and the CSMA/CA pro-tocol at the MAC layer is evaluated in the context of an IEEE802.11a/g-based OAN. This is performed via simulationswhere the links between all radios are modelled at symbollevel based on orthogonal frequency multiplexing (OFDM)as defined in specification [4], subject to path loss, shadow-ing and multipath fading according to the IEEE 802.11 chan-nel models [12, 13]. Conventional and semiblind multiple-antenna algorithms are assumed at the PHY layer in orderto identify possible improvements in system throughput andcoverage for different OAN scenarios with VU and HU ter-minals. Cross-layer effects of continuous and intermittent in-tersystem interference from a coexisting impolite transmitterare also addressed.

The asynchronous interference model is derived in Sec-tion 2 in the context of typical OAN scenarios. The 802.11CSMA/CA protocol is also briefly reviewed in Section 2.Problem formulation is given in Section 3. This is followed in

Section 4 by a description of the conventional and semiblindIC receiver algorithms, along with a demonstration of theirperformance at the PHY layer. Section 5 provides a descrip-tion of the simulation framework and the cross-layer simu-lation results in typical OAN scenarios with intra- and inter-system interference. Conclusions are presented in Section 6.

2. INTERFERENCE SCENARIOS

The MAC mechanism specified in the IEEE 802.11 family ofWLAN standards describes the process by which MAC pro-tocol data units (MPDUs) are transmitted and subsequentlyacknowledged. Specifically, once a receiver detects and suc-cessfully decodes a transmitted MPDU, it responds after ashort interframe space (SIFS) period, with the transmissionof an acknowledgement (ACK) packet. Should an ACK notbe successfully received and decoded after some interval, thetransmitter will attempt to retransmit the MPDU.

Each IEEE 802.11 transmitter contends for access to theradio channel based on the CSMA/CA protocol. This is es-sentially a “listen before talk” mechanism, whereby a radioalways listens to the medium before commencing a trans-mission. If the medium is determined to be already carryinga transmission (i.e., the measured background signal level isabove a specified threshold), the radio will not commencetransmission. Instead, the radio enters a deferral or back-offmode, where it waits until the medium is determined to bequiet over a certain interval before attempting to transmit.This is illustrated in Figure 1.

A “listen before talk” mechanism may fail in the so-called“hidden” terminal scenario. In this case, a transmitter sensesthe medium to be idle, despite the fact that a hidden trans-mitter is causing interference at the receiver, that is, the hid-den terminal is beyond the reception range of the transmitter,but within the reception range of the receiver.

A single-cell uplink scenario is illustrated in Figure 2. AnAP equipped with N antennas is surrounded by K terminals,uniformly distributed up to a maximum distance D. Termi-nals located within distance Dv of the AP are referred to asHUs. Terminals located at a distance greater than Dv are re-ferred to as VUs1 .

One can expect that the extent of possible collisions inthis scenario depends on the distance from the AP. HUs lo-cated near the AP do not interfere with each other becauseof the CSMA/CA protocol. Even if signals from certain VUscollide with the signals from the HUs, the VU signal powerlevels received at the AP are most probably small, and willnot result in erroneous decoding of the HUs’ data. On thecontrary, weaker VU signals are likely to be affected by col-lisions with stronger “hidden” VU and/or HU signals. Thismeans that without IC, the VU throughput may suffer, lead-ing to reduction or gaps in coverage even if the cell radius issufficient for reliable reception from individual users.

1 This distinction is made for illustrative purposes only. In practice, loca-tion bounds of HU and VU may be more complicated than the concentricrings shown in Figure 2.

A. M. Kuzminskiy and H. R. Karimi 3

Dv

D

1

2

1· · ·

N

Accesspoint

Certain terminalsmay not hear each other:

collisionsat the AP are likely

K

3

Home user (HU) terminalVisiting user (VU) terminal

Figure 2: A single-cell OAN scenario with HUs and VUs.

×10−4Real values of the signal received at AP for 4 antennas

1

0.5

0

−0.5

−1

Re[

ampl

itu

de]

0 500 1000 1500 2000 2500 3000 3500 4000

Pilot symbols of the desired signal

Desired signal

CCI 1

Desired signal

CCI 1 CCI 2

0 500 1000 1500 2000 2500 3000 3500 4000

Time, 50 ns samples

×10−5

5

0

−5

Re[

ampl

itu

de]

Figure 3: Typical collision patterns for N = 4.

It is important to emphasize that collisions are typicallyasynchronous with random overlap between the collidingbursts. Typical collision examples are illustrated in Figure 3,which shows the real values of the received signals for N = 4AP antennas, involving the desired signal and one or twocochannel interference (CCI) components. In both cases, thedesired signals correspond to VUs and the interference comesfrom one HU in the first plot, and from two VUs in the sec-ond plot. In both cases, the interference bursts are randomlydelayed with respect to the desired signal because of the ran-dom back-off intervals of the CSMA/CA protocol. The mainconsequence of this asynchronous interference structure forIC is that there is no overlap between the pilot symbols of thedesired signal (located in the preamble) and the interferencebursts.

10 m30 m

VU 2

HU 1 AP

1 N

50 m

VU 3

Walls15 dB

penetration loss Gap in coverageis expected because

of VU 2

Home user (HU) terminalVisiting user (VU) terminal

Figure 4: Residential OAN scenario with home and visiting users.

The single-cell OAN scenario of Figure 2 can be specifiedfor particular home/visitor situations. Figure 4 illustrates aresidential scenario with walls that can be taken into accountby means of a penetration loss. Home user HU 1 would al-ways get a good connection in this scenario. Visiting users2 and 3, however, may not hear each other and their trans-missions may collide in some propagation conditions. Sig-nals received from VU 2 would typically be much strongerthan those from VU 3 due to the shorter distance, resulting inlow throughput for VU 3. Another residential scenario with


1 N

Group 11 2 3

5 m 5 m

70 m

AP

130 m

5 m 5 m

Group 24 5 6

Collisions between users fromgroups 1 and 2 are likely:very low throughput for

group 2 is expected

Gap in coverageis expected becauseof users of group 1

Figure 5: Residential scenario with two groups of visiting users.

two groups of three visiting users each is shown in Figure 5.This scenario illustrates the situation, where gaps in cover-age can be expected for VUs 4–6 without effective IC at thePHY layer because of another group of strong VUs 1–3. Theasynchronous structure of the interference in these scenariosis similar to the one illustrated in Figure 3.


Based on the scenarios in Figures 2, 4, and 5, and other simi-lar OAN scenarios, one may conclude that the MAC layer im-pact on the interference structure can be taken into accountby means of an asynchronous interference model. An exam-ple of such model for three interference components is illus-trated in Figure 6, where random delays and varying burstdurations are assumed. This model can be exploited for de-veloping and comparing different IC algorithms at the PHYlayer. After this cross-layer design, the developed PHY IC al-gorithms can be tested via cross-layer simulations.

The problem formulation, including the main objective,constraints, and system assumptions, as well as the main ef-fects taken into account, is as follows.

Objective

• Increase uplink throughput for VUs in an OAN systembased on OFDM WLAN with CSMA/CA.

Constraints and system assumptions

• Single-antenna user terminals.• Multiple-antenna AP.• CSMA/CA transmission protocol at the AP and termi-

nals.• PHY layer interference cancellation at the AP taking

into account the asynchronous interference model in-duced by the MAC layer.

Preamble(training)

Information Desired signal

Interference 1

Interference 2

Interference 3

TimeData burst

Tt

τ1

τ2

τ3

B2

B1

B3

Figure 6: Asynchronous interference model.

• OAN scenarios with HUs and VUs as well as externalinterference from a coexisting system.

Effects taken into account

• MPDU and ACK structures, interleaving, coding, andmodulation according to the IEEE 802.11a/g PHY.

• Propagation channels: multipath delay spread; pathloss and shadowing; line-of-sight (LoS) and non-LoS(NLoS) conditions; spatial correlation between an-tenna elements at the AP.

4. INTERFERENCE CANCELLATION

Since training symbols are most reliable for estimation ofthe desired signal by means of the conventional LS criterion,the main idea here is to apply regularization of the LS cri-terion by a penalty function associated with the covariance


matrix estimated over the data interval. In the narrow-bandscenario, that is, for each individual OFDM subcarrier, themodified (regularized) LS criterion can be expressed as fol-lows [6, 7]:

w = arg minw

∑

t∈τt

∣

∣s(t)−w∗x(t)∣

∣

2+ ρF(w), (1)

where t is the time index, s(t) is the training sequence for thedesired signal, x(t) is the output N × 1 vector from the re-ceiving antenna array, N is the number of antenna elements,w is the N × 1 weight vector, τt is the interval of Tt knowntraining symbols assuming perfect synchronization for thedesired signal, ρ > 0 is a regularization parameter, F(w) isa regularization function that exploits a priori informationfor specific problem formulations, and (·)∗ is the complexconjugate transpose.

In the considered asynchronous interference scenario,the working interval may be affected by interference com-ponents that are not present during the training interval.Thus, selection of the regularization function such that itcontains information from the data interval increases theability to cancel asynchronous interference. For the second-order statistics class of algorithms, this can be achieved bymeans of the following quadratic function [6, 7]:

F(w) = w∗Rtw − r∗t w −w∗ rt, (2)

leading to the semiblind (SB) solution

wSB =[

(1− δ)Rt + δRb]−1

rt, (3)

where Rt = T−1t

∑

t∈τt x(t)x∗(t) and rt = T−1t

∑

t∈τt x(t)s∗(t)are the covariance matrix and cross-correlation vector esti-mated over the training interval, Rb = T−1

∑ Tt=1x(t)x∗(t) is

the covariance matrix estimated over the whole data burst ofT symbols, and 0 ≤ δ = ρ/(1 + ρ) ≤ 1 is the regularizationcoefficient. Selection of the regularization parameter δ hasbeen studied in [6, 11] and will be discussed below.

One can see that the SB estimator (3) contains the con-ventional LS solution

wLS = R−1t rt, (4)

as a special case for δ = 0.An iterative higher-order statistics estimation algorithm

with projections onto the FA with SB initialization (SBFA)can be described as follows:

wSBFA = w[J],

w[ j] = (XX∗)−1XΘ[

X∗w[ j−1]], j = 1, . . . , J ,

w[0] = wSB,

(5)

w[J] = w[J−1], (6)

where X = [x(1), . . . , x(T)] is the N × T matrix of input sig-nals, w[ j] is the weight vector at the jth iteration, Θ[·] is theprojection onto the FA, and J is the total number of iterationswith stopping rule (6).

LS (δ = 0)

NonasymptoticML benchmark

K = 4, M = 2, SNR = 15 dB, SIR = 0 dB100

10−1

10−2

MSE

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δ

Nt = 8, Nd = 42Nt = 20, Nd = 80Nt = 50, Nd = 450

Figure 7: Typical MSE performance for the SB algorithm for vari-able regularization parameter.

Efficiency of the SB algorithm (3) is studied in [6, 7] bymeans of comparison to the especially developed nonasymp-totic ML benchmark. Typical estimated MSE performancefor different burst structures and variable regularization pa-rameter δ is illustrated in Figure 7 for N = 4, K = 2, SNR =15 dB, SIR = 0 dB, QPSK signals, and independent complexGaussian vectors as propagation channels. The correspond-ing ML benchmark results from [6] are also shown in Fig-ure 7 for comparison. One can see that the SB performanceis very close to the ML benchmark for properly selected reg-ularization parameter. Furthermore, the MSE functions arenot very sharp, which means that some fixed parameter δ canbe used for a wide range of scenarios. Indeed, the results inFigure 7 suggest that δ ≈ 0.1 can be effectively applied forvery different slot structures.

The narrowband versions of the LS, SB, and SBFA algo-rithms can be expanded to the OFDM case. The problemwith this expansion is that the available amount of trainingand data symbols at each subcarrier may not be large enoughto achieve desirable performance. Different approaches canbe applied to overcome this difficulty, such as grouping (clus-tering) or other interpolation techniques [14, 15]. Accordingto the grouping technique, subcarriers of an OFDM systemare divided into groups, and a single set of parameters is es-timated for all subcarriers within a group, using all pilot andinformation symbols from that group.

Next, we compare the LS, SB, and SBFA algorithms at thePHY layer of an OFDM radio link subject to asynchronousinterference. We consider the “D-”channel [13] environmentand apply a group-based technique [14] with Q = 12 groupsof subcarriers. We simulate a single-input multiple-output(SIMO) system (N = 5) for IEEE 802.11g time-frequencybursts of 14 OFDM QPSK modulated symbols and 64 sub-carriers (only 52 are used for data and pilot transmission).


N = 5, K = 3, SNR = 15 dB,“D-” channel, 3/4 code rate, Q = 12, 20000 trials

100

10−1

10−2

10−3

PE

R

−5 0 5 10 15 20

Asynchronous SIR (dB)

LSSB(δ = 0.1)

SB(δ = var)SBFA(δ = 0.1)

Figure 8: Typical PHY-layer OFDM performance for LS, SB, andSBFA.

The transmitted signal is encoded according to the IEEE802.11g standard with a 3/4 code rate [4]. Each packet con-tains 54 information bytes. Each time-frequency burst in-cludes two information packets and two preamble blocks of52 binary pilot symbols. This simulation environment corre-sponds to an over-the-air data rate of 18 Mbit/s.

Figure 8 presents the packet-error rate (PER) curves forLS, SB, and SBFA with a fixed SNR of 15 dB. The SB algo-rithm is presented for fixed (δ = 0.1) regularization as wellas adaptive (δ = var) regularization parameter selected on aburst-by-burst basis based on the CM criterion:2

δ = arg minδ

T∑

t=1

[∣

∣w∗SB(δ)x(t)

∣

∣

2 − 1]2. (7)

In Figure 8, the SIR is varied for two asynchronous inter-ference components, and is fixed at 0 dB for a synchronousinterference component (note that the latter is still asyn-chronous on a symbol basis, but always overlaps with thewhole data burst of the desired signal including the pream-ble).

One can see that the regularized SB solution with thefixed regularization parameter significantly outperforms theconventional LS algorithm for low asynchronous SIR. Partic-ularly, it outperforms LS by 4 dB at 3% PER, and by 7 dB at10% PER. In the high SIR region, the scenario becomes sim-ilar to the synchronous case (asynchronous CCI actually dis-appears), where the LS estimator actually gives the best pos-

2 A simplified switched CM-based selection of the regularization parameteris developed in [11].

sible results [16]. Thus, δ→0 is required for the best SB per-formance in this region. Online adaptive selection of the reg-ularization parameter may be adopted in this case, as illus-trated in Figure 8. However, one can see in Figure 7 that per-formance degradation for fixed δ = 0.1 in the synchronouscase is small and may well be acceptable. The SBFA algorithmbrings additional performance improvement of up to 5 dBfor low SIR at the cost of higher complexity.

OFDM versions of the LS, SB, and SBFA algorithmswith a fixed regularization parameter, together with the con-ventional matched filter (no-IC), will be evaluated next viacross-layer simulations.

5. CROSS-LAYER SIMULATION RESULTS

5.1. Simulation assumptions

We simulate the IEEE 802.11g PHY (OFDM) and CSMA/CAsubject to the following assumptions:

• 2.4 GHz center frequency,• 4-QAM, 1/2 rate convolutional coding,• MPDU burst of 2160 information bits, 50 OFDM sym-

bols, 200 microseconds duration,• ACK burst of 8 OFDM symbols, 32 microseconds slot

duration,• maximum ratio beamforming at the AP for ACK

transmissions,• trial duration of 10 milliseconds,• “E”-channel propagation model [13] (100 nanaosec-

onds delay spread, LOS/NLOS conditions dependingon distance),

• 1-wavelength separation between N = 4 AP antennas,• 20 dBm transmit power for the AP and terminals,• − 92 dBm noise power,• − 82 dBm clear-channel assessment threshold.

A number of simplifying assumptions are made: idealchannel reciprocity (uplink channel estimates are used fordownlink beamforming for ACK transmission); ideal (lin-ear) front-end filters at the AP and terminals; zero fre-quency offset; perfect receiver synchronization at the APand terminals; stationary propagation channels during a 10-millisecond trial. The last assumption is applicable in theconsidered scenario because all channel and weight estimatesare derived on a slot-by-slot basis, and channel variations inWLAN environments are normally negligible over these timescales (200 microseconds).

5.2. Single-cell OAN

Typical histograms for collision statistics in the scenario ofFigure 2 are shown in Figure 9 for D = 150 m. As expected,the average number of colliding MPDUs increases with thetotal number, K , of users contending for the channel.

The VU throughputs are presented in Figure 10 for vari-able visitor radius Dv and total number of users. The conven-tional matched filtering (no-IC), LS, SB, and SBFA (δ = 0.1)algorithms are compared.


K = 100.6

0.5

0.4

0.3

0.2

0.1

0

Sam

ple

prob

abili

ty

0 1 2 3 4 5 6 7 8 9 10

Number of colliding MPDU

(a)

K = 200.6

0.5

0.4

0.3

0.2

0.1

0

Sam

ple

prob

abili

ty

0 1 2 3 4 5 6 7 8 9 10


(b)

K = 300.6

0.5

0.4

0.3

0.2

0.1

0

Sam

ple

prob

abili

ty

0 1 2 3 4 5 6 7 8 9 10


(c)

Figure 9: Collision statistics for D = 150 m with the single-cell scenario of Figure 2.

Dv = 0 m6

5

4

3

2

1

0

Th

rou

ghpu

tfo

r“v

isit

ors”

:d>D

v(M

bits

/s)

10 20 30

Total number of users

No ICLS

SBSBFA

(a)

Dv = 100 m3.5

3

2.5

2

1.5

1

0.5

0

Th

rou

ghpu

tfo

r“v

isit

ors”

:d>D

v(M

bits

/s)

10 20 30


No ICLS

SBSBFA

(b)

Dv = 120 m2.5

2

1.5

1

0.5

0

Th

rou

ghpu

tfo

r“v

isit

ors”

:d>D

v(M

bits

/s)

10 20 30


No ICLS

SBSBFA

(c)

Figure 10: Visiting user throughput for D = 150 and N = 4 for no IC, LS, SB, and SBFA algorithms (left to right for each total number ofusers) with the single-cell scenario of Figure 2.


No IC1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

VU 3

HU 1

VU 2

0 5

Throughput (Mbits/s)

(a)

LS1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

0 5


VU 3

HU 1

VU 2

(b)

SB1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

0 5


VU 3

VU 2

HU 1

(c)

SBFA1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

0 5


VU 3

VU 2

HU 1

(d)

Figure 11: Throughput CDF for the residential scenario of Figure 4.

The VU throughput Uv is calculated as follows:

Uv = 1TSI

I∑

i=1

∑

Dv<dik≤DBik, (8)

where dik is the distance between the AP and the kth terminalat the ith trial, Bik is the total number of bits from the kthterminal successfully received and acknowledged at the AP atthe ith trial, and Ts and I are the duration and number oftrials. The throughput results in Figure 10 are averaged overI = 20 trials of Ts = 10 milliseconds, each with independentuser locations and propagation channel realizations3.

The first plot for Dv = 0 actually shows the total cellthroughput. One can see that all the algorithms show someperformance degradation with growing total number ofusers in the cell. The SB and SBFA algorithms demonstrate asmall improvement over both no-IC and LS for K = [20, 30].The low IC gain is in fact expected in this case since the in-terference avoidance CSMA/CA protocol dominates for userslocated close to the AP, making any IC redundant.

The situation is quite different when we consider thethroughput of the VUs only. One can see in Figure 10, for

3 Cross-layer simulations are very computationally demanding. For eachtrial, we generated (K +1)K/2 independent propagation channels for ran-dom terminal positions (e.g., for K = 30 we generated 465 channels pertrial). Typically, during 10 milliseconds we observed around ten collisionsbetween different users (burst duration is 0.2 milliseconds) leading to ap-proximately 200 collisions for 20 trials. This is why we accepted a lownumber of trials for the single-cell scenario. For particular residential sce-narios with low number of terminals, we simulated around 100–200 trialsto keep a similar level of averaging over different propagation conditions.

Dv = [100, 120] m, that both semiblind solutions signifi-cantly outperform the other two techniques by up to a fac-tor of 4. Furthermore, it appears that the main improvementcomes from the second-order SB solution (3). Iterative pro-jections to the FA in SBFA add up to 25% to the SB gain.

5.3. Residential OAN

Cumulative distribution functions (CDFs) of VU through-put over 200 trials are plotted in Figure 11. These corre-sponds to the residential scenario of Figure 4 with wall pen-etration loss of 15 dB. As expected, home user (HU) 1 is notaffected by VU 2 and 3. On the contrary, visiting user (VU)3 hardly achieves any throughput unless efficient semiblindIC is utilized. Both semiblind estimators demonstrate signifi-cant performance improvement and allow both visiting users(VU) 2 and 3 to share the radio resource almost equally.

The throughput results estimated over 100 trials in thescenario shown in Figure 5 are given in Figure 12. They illus-trate the situation, where gaps in coverage because of strongVUs may be significantly reduced by means of the proposedsemiblind cancellation at the PHY layer.

5.4. Intersystem interference in residential OAN

As mentioned in Section 2, PHY layer IC may also be ef-fective in scenarios where interference from other systemsis not subject to any interference avoidance schemes such asCSMA/CA (i.e., is “impolite”). We illustrate this situation ina residential scenario as presented in Figure 13, which con-sists of one HU, one VU, and a low-power (10 μW) “im-polite” interferer located close to the AP. In this scenario,


Group 2

Group 1SBFA

No IC

SB

LS

K = 6, 100 trials1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

0 0.5 1 1.5 2 2.5

Group throughput (Mbits/s)

Figure 12: Throughput CDF for the residential scenario of Figure 5.

Low-power “impolite”interferer (10 μW)

VU (100 mW)

120 m

5 m

HU (100 mW)

30 m

1 N

AP

5 m

10 m

Home user (HU) terminalVisiting user (VU) terminalInterferer

Figure 13: Residential scenario with intersystem interference.

CSMA/CA for HU and VU is not affected, because the in-terference power at the HU and VU locations is normallybelow the clear-channel assessment threshold. Furthermore,the HU signal received at the AP is much stronger than theinterference. So, the HU is also unaffected at the PHY layer.On the contrary, the VU signal received at the AP may becomparable to the interference level, and so, may be signifi-cantly affected. Again, IC efficiency depends on the temporalinterference structure, as discussed in Section 3.

Figure 14 shows the results for a continuous, white Gaus-sian, 10-μW “impolite” interferer. One can see that the VU

No IC

LS

SB

SBFA

Continuous interference, 200 trials1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

0 1 2 3 4 5 6 7


VUHU

Figure 14: Throughput CDF for residential scenario of Figure 12and a continuous intersystem interferer.

throughput can be significantly improved by means of all theconsidered training-based LS and semiblind SB and SBFA ICalgorithms. This is because the interference always overlapswith the MPDU pilot symbols, resulting in what we classifyin [16] as a synchronous interference.

Again, the situation becomes quite different for intermit-tent intersystem interference. We simulate this as a stream of200 microseconds bursts with duty-cycle of 50%. Typical col-lision patterns between data and interference bursts are plot-ted in Figure 15. Here, the random MPDU back-offs resultin random overlaps between the interference bursts and thetraining symbols. Figure 16 presents the throughput resultsfor intermittent interference. It is not surprising that bothSB and SBFA significantly outperform the conventional no-IC and pilot-based LS IC in this scenario. However, one cansee in Figure 16 that LS demonstrates more significant per-formance improvement over the no-IC solution compared tothe intrasystem interference scenario presented above. This isbecause in the considered intermittent interference scenario,collisions that overlap with the training symbols occur withpractically the same probability as those involving no overlapwith the training symbols. Both types of collisions are illus-trated in Figure 15 in the upper and lower plots, respectively.In the intrasystem interference case, collisions that do notoverlap with the training symbols dominate because of theCSMA/CA protocol, as discussed in Section 2, leading to sig-nificant semiblind gain over the conventional training-basedIC algorithms such as LS.

6. CONCLUSION

The potential gains provided by multiantenna interferencecancellation receiver algorithms, in the context of WLANsystems employing CSMA/CA protocols, were evaluated inthis paper. Cross-layer interactions were captured via joint


×10−5 Real values of the signal received visitingusers for all 4 antennas

2

1

0

−1

−2

Am

plit

ude

0 500 1000 1500 2000 2500 3000 3500 4000

Desired signal

CCI 1

Desired signal

CCI

0 500 1000 1500 2000 2500 3000 3500 4000

Time, 50 ns samples

×10−5

2

1

0

−1

−2

Am

plit

ude

Figure 15: Typical received signal patterns in the intersystem inter-mittent interference scenario.

No IC

LS

SB

SBFA

Intermittent interference, 200 trials1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Pro

b.(t

hro

ugh

put<x-

axis

)

0 1 2 3 4 5 6 7


VUHU

Figure 16: Throughput CDF for residential scenario of Figure 12with an intermittent intersystem interferer (50% duty-cycle).

PHY/MAC simulations involving multiple terminals con-tending for the opportunity to transmit data to the accesspoint. The impact of impolite cochannel interference froma coexisting system was also accounted for. It was shownthat the developed semiblind interference cancellation tech-niques are essential for addressing the asynchronous inter-ference experienced in WLAN. Significant performance gainhas been demonstrated by means of cross-layer simulationsin the OAN scenarios. It has been found that the main ef-fect comes from the regularization in the SB algorithm withcomplexity similar to the conventional LS solution. The morecomplicated SBFA iterations lead to an additional marginalperformance improvement.

ACKNOWLEDGMENTS

The authors would like to thank Professor Y. I. Abramovichfor participating in many fruitful discussions on PHY IC inthe course of this work. Part of this work has been performedwith financial support from the IST FP6 OBAN project andalso part of this work has been presented at ICC ’07 [17].

REFERENCES

[1] A. J. Paulraj and C. B. Papadias, “Space-time processing forwireless communications,” IEEE Signal Processing Magazine,vol. 14, no. 6, pp. 49–83, 1997.

[2] J. G. Andrews, “Interference cancellation for cellular systems:a contemporary overview,” IEEE Wireless Communications,vol. 12, no. 2, pp. 19–29, 2005.

[3] A. M. Kuzminskiy, “Finite amount of data effects in spatio-temporal filtering for equalization and interference rejectionin short burst wireless communications,” Signal Processing,vol. 80, no. 10, pp. 1987–1997, 2000.

[4] IEEE Std 802.11a, “Wireless LAN Medium Access Control(MAC) and Physical Layer (PHY) Specifications,” 1999.

[5] Open BroadbAccess Network (OBAN), IST 6FP Contract no.001889, http://www.ist-oban.org/.

[6] A. M. Kuzminskiy and Y. I. Abramovich, “Second-order asyn-chronous interference cancellation: regularized semi-blindtechnique and non-asymptotic maximum likelihood bench-mark,” Signal Processing, vol. 86, no. 12, pp. 3849–3863, 2006.

[7] A. M. Kuzminskiy and Y. I. Abramovich, “Adaptive second-order asynchronous CCI cancellation: maximum likelihoodbenchmark for regularized semi-blind technique,” in Proceed-ings of IEEE International Conference on Acoustics, Speech, andSignal Processing (ICASSP ’04), vol. 4, pp. 453–456, Montreal,Que, Canada, May 2004.

[8] A. M. Kuzminskiy and Y. I. Abramovich, “Interval-based max-imum likelihood benchmark for adaptive second-order asyn-chronous CCI cancellation,” in Proceedings of IEEE Interna-tional Conference on Acoustics, Speech, and Signal Processing(ICASSP ’07), vol. 2, pp. 865–868, Honolulu, Hawaii, USA,April 2007.

[9] A. M. Kuzminskiy and C. B. Papadias, “Asynchronous inter-ference cancellation with an antenna array,” in Proceedings ofthe 13th IEEE International Symposium on Personal, Indoor andMobile Radio Communications (PIMRC ’02), vol. 1, pp. 260–264, Lisbon, Portugal, September 2002.

[10] A. M. Kuzminskiy and C. B. Papadias, “Re-configurable semi-blind cancellation of asynchronous interference with an an-tenna array,” in Proceedings of IEEE International Conferenceon Acoustics, Speech, and Signal Processing (ICASSP ’03), vol. 4,pp. 696–699, Hong Kong, April 2003.

[11] A. M. Kuzminskiy and Y. I. Abramovich, “Adaptive asyn-chronous CCI cancellation: selection of the regularization pa-rameter for regularized semi-blind technique,” in Proceed-ings of the 7th IEEE Workshop on Signal Processing Advancesin Wireless Communications (SPAWC ’06), pp. 1–5, Cannes,France, July 2006.

[12] V. Erceg, L. Schumacher, P. Kyritsi, et al., “TGn Channel Mod-els (IEEE 802.11-03/940r2),” High Throughput Task Group,IEEE P802.11, March 2004.

[13] L. Schumacher, “WLAN MIMO channel MATLAB program,”http://www.info.fundp.ac.be/∼lsc/Research/IEEE 80211HTSG CMSC/distribution terms.html.


[14] D. Bartolome, X. Mestre, and A. I. Perez-Neira, “Single in-put multiple output techniques for Hiperlan/2,” in Proceed-ings of IST Mobile Communications Summit, Barcelona, Spain,September 2001.

[15] A. M. Kuzminskiy, “Interference cancellation in OFDM withparametric modeling of the antenna array weights,” in Pro-ceedings of the 35th Asilomar Conference on Signals, Systems andComputers, vol. 2, pp. 1611–1615, Pacific Grove, Calif, USA,November 2001.

[16] Y. I. Abramovich and A. M. Kuzminskiy, “On correspondencebetween training-based and semiblind second-order adaptivetechniques for mitigation of synchronous CCI,” IEEE Transac-tions on Signal Processing, vol. 54, no. 6, pp. 2347–2351, 2006.

[17] A. M. Kuzminskiy and H. R. Karimi, “Cross-layer design ofuplink multiple-antenna interference cancellation for WLANwith CSMA/CA in open access networks,” in Proceedings ofIEEE International Conference on Communications (ICC ’07),pp. 2568–2573, Glasgow, Scotland, UK, June 2007.


Research ArticleTransmit Diversity at the Cell Border UsingSmart Base Stations

Simon Plass, Ronald Raulefs, and Armin Dammann

German Aerospace Center (DLR), Institute of Communications and Navigation, Oberpfaffenhofen, 82234 Wessling, Germany

Received 27 October 2006; Revised 1 June 2007; Accepted 22 October 2007


We address the problems at the most critical area in a cellular multicarrier code division multiple access (MC-CDMA) network,namely, the cell border. At a mobile terminal the diversity can be increased by using transmit diversity techniques such as cyclicdelay diversity (CDD) and space-time coding like Alamouti. We transfer these transmit diversity techniques to a cellular environ-ment. Therefore, the performance is enhanced at the cell border, intercellular interference is avoided, and soft handover proceduresare simplified all together. By this, macrodiversity concepts are exchanged by transmit diversity concepts. These concepts also shiftparts of the complexity from the mobile terminal to smart base stations.

Copyright © 2007 Simon Plass et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

The development of future mobile communications systemsfollows the strategies to support a single ubiquitous radio ac-cess system adaptable to a comprehensive range of mobilecommunication scenarios. Within the framework of a globalresearch effort on the design of a next generation mobile sys-tem, the European IST project WINNER—Wireless WorldInitiative New Radio—[1] is also focusing on the identifica-tion, assessment, and comparison of strategies for reducingand handling intercellular interference at the cell border. Forachieving high spectral efficiency the goal of future wirelesscommunications systems is a total frequency reuse in eachcell. This leads to a very critical area around the cell borders.

Since the cell border area is influenced by at least twoneighboring base stations (BSs), the desired mobile termi-nal (MT) in this area has to scope with several signals inparallel. On the one hand, the MT can cancel the interfer-ing signals with a high signal processing effort to recover thedesired signal [2]. On the other hand, the network can man-age the neighboring BSs to avoid or reduce the negative in-fluence of the transmitted signals at the cell border. Due tothe restricted power and processing conditions at the MT, anetwork-based strategy is preferred.

In the region of overlapping cells, handover proceduresexist. Soft handover concepts [3] have shown that the usageof two base stations at the same time increases the robust-ness of the received data and avoids interruption and calling

resources for reinitiating a call. With additional informationabout the rough position of the MT, the network can avoidfast consecutive handovers that consume many resources, forexample, the MT moves in a zigzag manner along the cellborder.

Already in the recent third generation mobile commu-nications system, for example, UMTS, macrodiversity tech-niques with two or more base stations are used to providereliable handover procedures [4]. Future system designs willtake into account the advanced transmit diversity techniquesthat have been developed in the recent years. As the cell sizesdecrease further, for example, due to higher carrier frequen-cies, the cellular context gets more dominant as users switchcells more frequently. The ubiquitous approach of having areliable link everywhere emphasizes the need for a reliableconnection at cell border areas.

A simple transmit diversity technique is to combat flatfading conditions by retransmitting the same signal fromspatially separated antennas with a frequency or time off-set. The frequency or time offset converts the spatial diver-sity into frequency or time diversity. The effective increaseof the number of multipaths is exploited by the forward er-ror correction (FEC) in a multicarrier system. The elemen-tary method, namely, delay diversity (DD), transmits delayedreplicas of a signal from several transmit (TX) antennas [5].The drawback are increased delays of the impinging signals.By using the DD principle in a cyclic prefix-based system, in-tersymbol interference (ISI) can occur due to too large delays.


This can be circumvented by using cyclic delays which resultsin the cyclic delay diversity (CDD) technique [6].

Space-time block codes (STBCs) from orthogonal de-signs [7] improve the performance in a flat and frequencyselective fading channel by coherently adding the signals atthe receiver without the need for multiple receive anten-nas. The number of transmit antennas increases the perfor-mance at the expense of a rate loss. The rate loss could bereduced by applying nearly orthogonal STBCs which on theother hand would require a more complex space-time de-coder. Generally, STBCs of orthogonal or nearly orthogonaldesigns need additional channel estimation, which increasesthe complexity.

The main approach of this paper is the use and inves-tigation of transmit diversity techniques in a cellular envi-ronment to achieve macrodiversity in the critical cell borderarea. Therefore, we introduce cellular CDD (C-CDD) whichapplies the CDD scheme to neighboring BSs. Also the Alam-outi scheme is addressed to two BSs [8] and in the follow-ing this scheme is called cellular Alamouti technique (CAT).The obtained macrodiversity can be utilized for handover de-mands, for example.

Proposals for a next generation mobile communicationssystem design favor a multicarrier transmission, namely,OFDM [9]. It offers simple digital realization due to the fastFourier transformation (FFT) operation and low complexityreceivers. The WINNER project aims at a generalized multi-carrier (GMC) [10] concept which is based on a high flexiblepacket-oriented data transmission. The resource allocationwithin a frame is given by time-frequency units, so calledchunks. The chunks are preassigned to different classes ofdata flows and transmission schemes. They are then used in aflexible way to optimize the transmission performance [11].

One proposed transmission scheme within GMC is themulticarrier code division multiple access (MC-CDMA).MC-CDMA combines the benefits of multicarrier transmis-sion and spread spectrum and was simultaneously proposedin 1993 by Fazel and Papke [12] and Yee et al. [13]. In ad-dition to OFDM, spread spectrum, namely, code divisionmultiple access (CDMA), gives high flexibility due to simul-taneous access of users, robustness, and frequency diversitygains [14].

In this paper, the proposed techniques C-CDD and CATare applied to a cellular environment based on an MC-CDMA transmission scheme. The structure of the paper isas follows. Section 2 describes the used cellular multicarriersystem based on MC-CDMA. Section 3 introduces the cellu-lar transmit diversity technique based on CDD and the ap-plication of the Alamouti scheme to a cellular environment.At the end of this section both techniques are compared andthe differences are highlighted. A more detailed analytical in-vestigation regarding the influence of the MT position for theC-CDD is given in Section 4. Finally, the proposed schemesare evaluated in Section 5.

2. CELLULAR MULTICARRIER SYSTEM

In this section, we first give an outline of the used MC-CDMA downlink system. We then describe the settings of thecellular environment and the used channel model.

2.1. MC-CDMA system

The block diagram of a transmitter using MC-CDMA isshown in Figure 1. The information bit streams of the Nu

active users are convolutionally encoded and interleaved bythe outer interleaver Πout. With respect to the modulationalphabet, the bits are mapped to complex-valued data sym-bols. In the subcarrier allocation block, Nd symbols per userare arranged for each OFDM symbol. The kth data symbolis multiplied by a user-specific orthogonal Walsh-Hadamardspreading code which provides chips. The spreading lengthL corresponds to the maximum number of active users L =Nu,max. The ratio of the number of active users to Nu,max rep-resents the resource load (RL) of an MC-CDMA system.

An inner random subcarrier interleaver Πin allows a bet-ter exploitation of diversity. The input block of the inter-leaver is denoted as one OFDM symbol and Ns OFDM sym-bols describe one OFDM frame. By taking into account awhole OFDM frame, a two-dimensional (2D) interleavingin frequency and time direction is possible. Also an inter-leaving over one dimension (1D), the frequency direction,is practicable by using one by one OFDM symbols. Thesecomplex valued symbols are transformed into time domainby the OFDM entity using an inverse fast Fourier transform(IFFT). This results in NFFT time domain OFDM symbols,represented by the samples

x(n)l = 1

√NFFT

NFFT−1∑

i=0

X (n)i ·e j(2π/NFFT)il, (1)

where l, i denote the discrete time and frequency and n thetransmitting BS out of NBS BSs. A cyclic prefix as a guardinterval (GI) is inserted in order to combat intersymbol in-terference (ISI). We assume quasistatic channel fading pro-cesses, that is, the fading is constant for the duration of oneOFDM symbol. With this quasistatic channel assumption thewell-known description of OFDM in the frequency domainis given by the multiplication of the transmitted data symbol

X (n)l,i and a complex channel transfer function (CTF) value

H(n)l,i . Therefore, on the receiver side the lth received MC-

CDMA symbol at subcarrier i becomes

Yl,i =NBS−1∑

n=0

X (n)l,i H

(n)l,i +Nl,i (2)

with Nl,i as an additive white Gaussian noise (AWGN) pro-cess with zero mean and variance σ2, the transmitter signalprocessing is inverted at the receiver which is illustrated inFigure 2. In MC-CDMA the distortion due to the flat fadingon each subchannel is compensated by equalization. The re-ceived chips are equalized by using a low complex linear min-imum mean square error (MMSE) one-tap equalizer. The re-sulting MMSE equalizer coefficients are

Gl,i =H(n)∗l,i

∣∣H(n)

l,i

∣∣2

+(L/Nu

)σ2

, i = 1, . . . ,Nc. (3)

Furthermore, Nc is the total number of subcarriers. The op-erator (·)∗ denotes the complex conjugate. Further, the sym-bol demapper calculates the log-likelihood ratio for each bit

Simon Plass et al. 3

User 1

User Nu COD

...

COD Πout

...

Πout Map

Map

... MU

X

d(1)1...

d(Nu)1

d(1)Nd...

d(Nu)Nd

CL

...

CL

+

+s1

...

sNd

Πin

X(n)l,1

X(n)l,Nc

OFD

M

D/Ax(n)(t)

Figure 1: MC-CDMA transmitter of the nth base station.

y(t) A/D

IOFD

M ...

Yl,1

Yl,Nc

Π−1in

s1

sNd

...

Eq.

Eq.

...

CHL

CHL

...

DM

UX

Demap....

Demap.

Π−1out

Π−1out

...

DEC

DEC User 1

User Nu

Figure 2: MC-CDMA receiver.

Desired BS

d

d0MT

d1

δ1d0

Interfering BS

Figure 3: Cellular environment.

based on the selected alphabet. The code bits are deinter-leaved and finally decoded using soft-decision Viterbi decod-ing [15].

2.2. Cellular environment

We consider a synchronized cellular system in time and fre-quency with two cells throughout the paper, see Figure 3. Thenth BS has a distance dn to the desired MT. A propagationloss model is assumed to calculate the received signal energy.The signal energy attenuation due to path loss is generallymodeled as the product of the γth power of distance and alog-normal component representing shadowing losses. Thepropagation loss normalized to the cell radius r is defined by

α(dn) =(dnr

)−γ·10η/10 dB, (4)

where the standard deviation of the Gaussian-distributedshadowing factor η is set to 8 dB. The superimposed signalat the MT is given by

Yl,i = X (0)l,i α(d0)H(0)l,i + X (1)

l,i α(d1)H(1)l,i +Nl,i

= S(0)l,i + S(1)

l,i +Nl,i.(5)

Depending on the position of the MT the carrier-to-interference ratio (C/I) varies and is defined by

CI= E{∣∣S(0)

l,i

∣∣2}

E{∣∣S(1)

l,i

∣∣2} . (6)

3. TRANSMIT DIVERSITY TECHNIQUES FORCELLULAR ENVIRONMENT

In a cellular network the MT switches the corresponding BSwhen it is requested by the BS. The switch is defined as thehandover procedure from one BS to another. The handoveris seamless and soft when the MT is connected to both BSs atthe same time. The subcarrier resources in an MC-CDMAsystem within a spreading block are allocated to differentusers. Some users might not need a handover as they are(a) in a stable position or (b) away from the cell border. Inboth cases these users are effected by intercell interferenceas their resource is also allocated in the neighboring cell. Toseparate the different demands of the users, users with sim-ilar demands are combined within time-frequency units, forexample, chunks, in an OFDM frame. The requested param-eters of the users combined in these chunks are similar, like acommon pilot grid. The spectrum for the users could thenbe shared between two cells within a chunk by defining abroadcast region. By this the affected users of the two cellswould reduce their effective spectrum in half. This would bea price to pay avoiding intercellular interference. Intercellu-lar interference could be tackled by intercellular interferencecancellation techniques at complexity costs for all mobileusers. Smart BSs could in addition try to balance the neededtransmit power by risking an increase of intercellular inter-ference also in neighboring cells. The approach presented inthe following avoids intercellular interference by defining theeffected area as a broadcast region and applying transmit di-versity schemes for a cellular system, like cyclic delay diver-sity and STBCs. Part of the ineluctable loss of spectrum ef-ficiency are compensated by exploiting additional diversitygains on the physical layer, avoiding the need of high com-plex intercellular cancellation techniques and decreasing theoverall intercellular interference in the cellular network forthe common good.

In the following, two transmit diversity techniques arein the focus. The first is based on the cyclic delay diversity(CDD) technique which increases the frequency diversity ofthe received signal and requires no change at the receiver to


· · · IFFT 1/√M

Front end of a transmitter

Cyclicprefix

Cyclicprefix

Cyclicprefix

δcyc1

δcycM−1

...

Cyclic delay diversity extension

Figure 4: Principle of cyclic delay diversity.

exploit the diversity. The other technique applies the Alam-outi scheme which flattens the frequency selectivity of the re-ceived signal and requires an additional decoding process atthe mobile.

3.1. Cellular cyclic delay diversity (C-CDD)

The concept of cyclic delay diversity to a multicarrier-basedsystem, that is, MC-CDMA, is briefly introduced in this sec-tion. Later on, the CDD concept will lead to an applicationto a cellular environment, namely, cellular CDD (C-CDD). Adetailed description of CDD can be found in [16]. The ideaof CDD is to increase the frequency selectivity, that is, to de-crease the coherence bandwidth of the system. The additionaldiversity is exploited by the FEC and for MC-CDMA also bythe spreading code. This will lead to a better error perfor-mance in a cyclic prefix-based system. The CDD principle isshown in Figure 4. An OFDM modulated signal is transmit-ted over M antennas, whereas the particular signals only dif-fer in an antenna specific cyclic shift δcyc

m . MC-CDMA modu-lated signals are obtained from a precedent coding, modula-tion, spreading, and framing part; see also Section 2.1. Beforeinserting a cyclic prefix as guard interval, the time domainOFDM symbol (cf. (1)) is shifted cyclically, which results inthe signal

xl−δcycm mod NFFT =

1√NFFT

NFFT−1∑

i=0

e− j(2π/NFFT)iδcycm ·Xi·e j(2π/NFFT)il .

(7)

The antenna specific TX-signal is given by

x(m)l = 1√

M·xl−δcyc

m mod NFFT , (8)

where the signal is normalized by 1/√M to keep the average

transmission power independent of the number of transmitantennas. The time domain signal including the guard inter-val is obtained for l = −NGI, . . . ,NFFT − 1. To avoid ISI, theguard interval length NGI has to be larger than the maximumchannel delay τmax. Since CDD is done before the guard in-terval insertion in the OFDM symbol, CDD does not increasethe τmax in the sense of ISI occurrence. Therefore, the lengthof the guard interval for CDD does not depend on the cyclicdelays δcyc

m , where δcycm is given in samples.

On the receiver side and represented in the frequency do-main (cf. (2)), the cyclic shift can be assigned formally to thechannel transfer function, and therefore, the overall CTF

Hl,i = 1√M

M−1∑

m=0

e− j(2π/NFFT)δcycm ·i·H(m)

l,i (9)

is observed. As long as the effective maximum delay τ′max ofthe resulting channel

τ′max = τmax + maxm

δcycm (10)

does not intensively exceedNGI, there is no configuration andadditional knowledge at the receiver needed. If τ′max � NGI,the pilot grid and also the channel estimation process has tobe modified [17]. For example, this can be circumvented byusing differential modulation [18].

The CDD principle can be applied in a cellular environ-ment by using adjacent BSs. This leads to the cellular cyclicdelay diversity (C-CDD) scheme. C-CDD takes advantageof the aforementioned resulting available resources from theneighboring BSs. The main goal is to increase performanceby avoiding interference and increasing diversity at the mostcritical areas.

For C-CDD the interfering BS also transmits a copy ofthe users’ signal as the desired BS to the designated MT lo-cated in the broadcast area. Additionally, a cyclic shift δcyc

n isinserted to this signal, see Figure 5. Therefore, the overall de-lay in respect to the signal of the desired BS in the cellularsystem can be expressed by

δn = δ(dn)

+ δcycn , (11)

where δ(dn) represents the natural delay of the signal de-pending on distance dn. At the MT the received signal canbe described by

Yl,i=X (0)l,i

(α(d0)H(0)l,i e

− j(2π/NFFT)δ0·i+α(d1)H(1)l,i e

− j(2π/NFFT)δ1·i).

(12)

The transmission from the BSs must ensure that the recep-tion of both signals are within the guard interval. Further-more, at the MT the superimposed statistical independentRayleigh distributed channel coefficients from the differentBSs sum up again in a Rayleigh distributed channel coeffi-cient. The usage of cyclic shifts prevents the occurrence of ad-ditional ISI. For C-CDD no additional configurations at theMT for exploiting the increased transmit diversity are neces-sary.

Finally, the C-CDD technique inherently provides an-other transmit diversity technique. If no cyclic shift δcyc

n is in-troduced, the signals from the different BSs may arrive at thedesired MT with different delays δ(dn). These delays can bealso seen as delay diversity (DD) [5] for the transmitted MC-CDMA signal or as macrodiversity [19] at the MT. Therefore,an inherent transmit diversity, namely, cellular delay diver-sity (C-DD), is introduced if the adjacent BSs just transmitthe same desired signal at the same time to the designatedMT. The C-CDD techniques can be also easily extended tomore than 2 BSs.


Desired cell

· · · IFFT GI d0

2r

d0d1

δ1

GI−1 FFT · · ·

Mobile terminal

GI δcyc1 IFFT · · ·

Interfering cell

Figure 5: Cellular MC-CDMA system with cellular cyclic delay diversity (C-CDD).

3.2. Cellular Alamouti technique (CAT)

In this section, we introduce the concept of transmit diversityby using the space-time block codes (STBCs) from orthogo-nal designs [7], namely, the Alamouti technique. We applythis scheme to the aforementioned cellular scenario. TheseSTBCs are based on the theory of (generalized) orthogonaldesigns for both real- and complex-valued signal constella-tions. The complex-valued STBCs can be described by a ma-trix

B =

←space→⎛

⎜⎜⎝

b0,0 · · · b0,NBS−1...

. . ....

bl−1,0 · · · bl−1,NBS−1

⎞

⎟⎟⎠

↑time↓

, (13)

where l and NBS are the STBC length and the number of BS(we assume a single TX-antenna for each BS), respectively.The simplest case is the Alamouti code [20],

B =(x0 x1

−x∗1 x∗0

)

. (14)

The respective assignment for the Alamouti-STBC to the kthblock of chips containing data from one or more users is ob-tained:

�y(k) =(y(k)

0

y(k)∗1

)

=(h(0,k) h(1,k)

h(1,k)∗ −h(0,k)∗

)

·(x0

x1

)

+

(n(k)

0

n(k)∗1

)

.

(15)

�y(k) is obtained from the received complex values y(k)i or their

conjugate complex y(k)∗i at the receiver. At the receiver, the

vector �y(k) is multiplied from left by the Hermitian of matrixH(k). The fading between the different fading coefficients isassumed to be quasistatic. We obtain the (weighted) STBCinformation symbols

�x = H(k)H·�y(k) = H(k)H·H(k)�x + H(k)H·�n(k)

= H(k)H·�n(k) +�x·1∑

i=0

∣∣h(i,k)

∣∣2

,(16)

corrupted by noise. For STBCs from orthogonal designs,MIMO channel estimation at the receiver is mandatory, thatis, h(n,k), n = 0, . . . ,NBS − 1, k = 0, . . . ,K − 1, must be

MC-CDMAsymbols of BS 0

Time

Oth

erac

tive

use

rsB

S0�D

esir

edch

un

k

......

X0,1

X0,0

−X∗1,1

−X∗1,0

MT 0 MT 1

MC-CDMAsymbols of BS 1

Time

Oth

erac

tive

use

rsB

S1�D

esir

edch

un

k

......

X0,1

X0,0

X∗1,1

X∗1,0

Figure 6: MC-CDMA symbol design for CAT for 2 MTs.

estimated. Disjoint pilot symbol sets for the TX-antennabranches can guarantee a separate channel estimation foreach BS [8]. Since the correlation of the subcarrier fadingcoefficients in time direction is decreasing with increasingDoppler spread—that is, the quasistationarity assumption ofthe fading is incrementally violated—the performance of thisSTBC class will suffer from higher Doppler frequencies. Laterwe will see that this is not necessarily true as the stationarityof the fading could also be detrimental in case of burst errorsin fading channels.

Figure 6 shows two mobile users sojourning at the cellborders. Both users data is spread within one spreading blockand transmitted by the cellular Alamouti technique usingtwo base stations. The base stations exploit information froma feedback link that the two MTs are in a similar location inthe cellular network. By this both MTs are served simultane-ously avoiding any interference between each other and ex-ploiting the additional diversity gain.

3.3. Resume for C-CDD and CAT

Radio resource management works perfectly if all informa-tion about the mobile users, like the channel state informa-tion, is available at the transmitter [21]. This is especially trueif the RRM could be intelligently managed by a single geniemanager. As this will be very unlikely the described schemesC-CDD and CAT offer an improved performance especially


at the critical cell border without the need of any informa-tion about the channel state information on the transmitterside. The main goal is to increase performance by avoidinginterference and increasing diversity at the most critical en-vironment. In this case, the term C/I is misleading (cf. (6)),as there is no I (interference). On the other hand, it describesthe ratio of the power from the desired base station and theother base station. This ratio also indicates where the mo-bile user is in respect to the base stations. For C/I = 0 dBthe MT is directly between the two BSs, for C/I > 0 dB theMT is closer to the desired BS, and for C/I < 0 dB the MT iscloser to the adjacent BS. Since the signals of the neighbor-ing BSs for the desired users are not seen as interference, theMMSE equalizer coefficients of (3) need no modification asin the intercellular interfering case [22]. Therefore, the trans-mit diversity techniques require no knowledge about the in-tercellular interference at the MT. By using C-CDD or CATthe critical cell border area can be also seen as a broadcastscenario with a multiple access channel.

For the cellular transmit diversity concepts C-CDD andCAT, each involved BS has to transmit additionally the sig-nal of the adjacent cell; and therefore, a higher amount ofresources are allocated at each BS. Furthermore, due to thehigher RL in each cell the multiple-access interference (MAI)for an MC-CDMA system is increased. There will be alwaysa tradeoff between the increasing MAI and the increasing di-versity due to C-CDD or CAT.

Since the desired signal is broadcasted by more than oneBS, both schemes can reduce the transmit signal power, andtherefore, the overall intercellular interference. Using MC-CDMA for the cellular diversity techniques the same spread-ing code set has to be applied at the involved BSs for the de-sired signal which allows simple receivers at the MT with-out multiuser detection processes/algorithms. Furthermore,a separation between the inner part of the cells and thebroadcast area can be achieved by an overlaying scramblingcode on the signal which can be also used for synchronizationissues as in UMTS [4].

Additionally, if a single MT or more MTs are aware thatthey are at the cell border, they could already ask for the C-CDD or CAT procedure on the first hand. This would easethe handover procedure and would guarantee a reliable softhandover.

We should point out two main differences between C-CDD and CAT. For C-CDD no changes at the receiver areneeded, there exists no rate loss for higher number of trans-mit antennas, and there are no requirements regarding con-stant channel properties over several subcarriers or sym-bols and transmit antenna numbers. This is an advantageover already established diversity techniques [7] and CAT.The Alamouti scheme-based technique CAT should providea better performance due to the coherent combination of thetwo transmitted signals [23].

4. RESULTING CHANNEL CHARACTERISTICSFOR C-CDD

The geographical influence of the MT for CAT has a symmet-ric behavior. In contrast, C-CDD is influenced by the posi-

tion of the served MT. Due to δcyc0 �=δcyc

1 and the relation in(11), the resulting performance regarding the MT positionof C-CDD should have an asymmetric characteristic. Sincethe influence of C-CDD on the system can be observed atthe receiver as a change of the channel conditions, we willinvestigate in the following this modified channel in termsof its channel transfer functions and fading correlation intime and frequency direction. These correlation characteris-tics also describe the corresponding single transmit antennachannel seen at the MT for C-CDD.

The frequency domain fading processes for differentpropagation paths are uncorrelated in the assumed qua-sistatic channel. Since the number of subcarriers is largerthan the number of propagation paths, there exists correla-tion between the subcarriers in the frequency domain. Thereceived signal at the receiver in C-CDD can be representedby

Yl,i = Xl,i·NBS−1∑

n=0

e− j(2π/NFFT)iδnα(dn)H(n)l,i

︸︷︷︸H′l,i

+Nl,i. (17)

Since the interest is based on the fading and signal character-istics observed at the receiver, the AWGN term Nl,i is skippedfor notational convenience. The expectation

R(l1, l2, i1, i2

) = E{H′l1,i1·H′∗

l2,i2

}(18)

yields the correlation properties of the frequency domainchannel fading. Due to the path propagations α(dn) andthe resulting power variations, we have to normalize the

channel transfer functions H(n)l,i by the multiplication factor

1/√∑ NBS−1

n=0 α2(dn) which is included for Rn(l, i).The fading correlation properties can be divided in three

cases. The first represents the power, the second investigatesthe correlation properties between the OFDM symbols (timedirection), and the third examines the correlation propertiesbetween the subcarriers (frequency direction).

Case 1. Since we assume uncorrelated subcarriers the auto-correlation of the CTF (l1 = l2 = l, i1 = i2 = i) is

R(l, i) =NBS−1∑

n=0

e− j(2π/NFFT)iδn·e+ j(2π/NFFT)iδn︸︷︷︸

=1

α2(dn)

·E{H(n)l,i ·H(n)∗

l,i

}

︸︷︷︸E{|H(n)

l,i |2}=1

=NBS−1∑

n=0

α2(dn),

(19)

and the normalized power is

Rn(l, i) =NBS−1∑

n=0

α2(dn)E

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

∣∣∣∣∣∣∣∣∣∣

H(n)l,i√

NBS−1∑

n=0α2(dn)

∣∣∣∣∣∣∣∣∣∣

2⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

= 1. (20)


60

40

200

Sub-carrier 0200

400600

Distance (m)

0

0.2

0.4

0.6

0.8

1

Cor

rela

tion

fact

orρ

Figure 7: Characteristic of correlation factor ρ over the subcarriersdepending on the distance d0.

Case 2. The correlation in time direction is given byl1�=l2, i1 = i2 = i. Since the channels from the BSs are i.i.d.stochastic processes, E{H(n)

l1,i ·H(n)∗l2,i } = E{Hl1,i·H∗

l2,i} and

R(l1�=l2, i

) = E{Hl1,iH

∗l2,i

}NBS−1∑

n=0

α2(dn),

Rn(l1�=l2, i

) = E

{Hl1,iH

∗l2,i

∑ NBS−1n=0 α2

(dn)

}NBS−1∑

n=0

α2(dn)

= E{Hl1,iH

∗l2,i

}.

(21)

We see that in time direction, the correlation properties ofthe resulting channel are independent of the MT position.

Case 3. In frequency direction (l1 = l2 = l, i1�=i2) the corre-lation properties are given by

R(l, i1�=i2

) = E{Hl,i1H

∗l,i2

}·NBS−1∑

n=0

α2(dn)e− j(2π/NFFT)(i1−i2)δn︸︷︷︸

C-CDD component

.

(22)

For large dn (α(dn) gets small) the influence of the C-CDDcomponent vanishes. And there is no beneficial increase ofthe frequency diversity close to a BS anymore. The normal-ized correlation properties yield

Rn(l, i1�=i2

) = E{Hl,i1H

∗l,i2

}

· 1∑ NBS−1

n=0 α2(dn)·

NBS−1∑

n=0

α2(dn)e− j(2π/NFFT)(i1−i2)δn

︸︷︷︸correlation factor ρ

.

(23)

The correlation factor ρ is directly influenced by the C-CDD component and determines the overall channel corre-lation properties in frequency direction. Figure 7 shows thecharacteristics of ρ for an exemplary system with NFFT = 64,γ = 3.5, NBS = 2, r = 300 m, δ

cyc0 = 0, and δ

cyc1 = 7. One

0 10 20 30 40 50 60

Sub-carrier

0.6

0.7

0.8

0.9

1

Cor

rela

tion

fact

orρ

d0 = 334 md0 = 335 md0 = 336 m

Figure 8: Correlation characteristics over the subcarriers for d0 =[334 m, 335 m, 336 m].

0 10 20 30 40 50 60

Delay

1e − 04

1e − 03

1e − 02

1e − 01

BE

R

0

0.5

1

1.5

2

2.5

SNR

gain

(dB

)

SNR gain at BER = 1e − 03C-CDD, C/I = 0 dB

Figure 9: BER and SNR gains versus the cyclic delay at the cell bor-der (C/I = 0 dB).

sample of the delay represents 320 microseconds or approx-imately 10 m, respectively. In the cell border area (200 m <d0 < 400 m), C-CDD increases the frequency diversity bydecorrelating the subcarriers. As mentioned before, there isless decorrelation the closer the MT is to a BS.

A closer look on the area is given in Figure 8 where the in-herent delay and the added cyclic delay are compensated, thatis, for d0 = 335 m the overall delay is δ1 = δ(265 m) + δ

cyc1 =

−70 m + 70 m = 0 (cf. (11)). The plot represents exemplar-ily three positions of the MT (d0 = [334 m, 335 m, 336 m])and shows explicitly the degradation of the correlation prop-erties over all subcarriers due to the nonexisting delay in thesystem. These analyses verify the asymmetric and δcyc depen-dent characteristics of C-CDD.


Table 1: Parameters of the cellular transmission systems.

Bandwidth B 100.0 MHz

No. of subcarriers Nc 1664

FFT length NFFT 2048

Guard interval length NGI 128

Sample duration Tsamp 10.0 ns

Frame length Nframe 16

No. of active users Nu {1, . . . , 8}Spreading lengh L 8

Modulation — 4-QAM, 16-QAM

Interleaving C-CDD — 2D

Interleaving CAT — 1D, 2D

Channel coding — CC (561, 753)oct

Channel coding rate R 1/2

Channel model — IEEE 802.11n Model C

Velocity — 0 mph, 40 mph

−10 0 10 20 30

C/I (dB)

1e − 04

1e − 03

1e − 02

1e − 01

BE

R

w/o TX diversity, fully loadedw/o TX diversity, half loaded

C-DD, halved TX powerC-CDD, halved TX powerC-DDC-CDD

Figure 10: BER versus C/I for an SNR of 5 dB using no transmitdiversity technique, C-DD, and C-CDD for different scenarios.


The simulation environment is based on the parameter as-sumptions of the IST-project WINNER for next genera-tion mobile communications system [24]. The used chan-nel model is the 14 taps IEEE 802.11n channel model C withγ = 3.5 and τmax = 200 nanoseconds. This model representsa large open space (indoor and outdoor) with non-light-of-sight conditions with a cell radius of r = 300 m. The trans-mission system is based on a carrier frequency of 5 GHz, abandwidth of 100 MHz, and an FFT length of Nc = 2048.One OFDM symbol length (excluding the GI) is 20.48 mi-croseconds and the GI is set to 0.8 microseconds (corre-sponding to 80 samples). The spreading length L is set to

8. The number of active users can be up to 8 depending onthe used RL. 4-QAM is used throughout all simulations andfor throughput performances 16-QAM is additionally inves-tigated. For the simulations, the signal-to-noise ratio (SNR)is set to 5 dB and perfect channel knowledge at the receiveris assumed. Furthermore, a (561, 753)8 convolutional codewith rate R = 1/2 was selected as channel code. Each MTmoves with an average velocity of 40 mph (only for compar-ison to see the effect of natural time diversity) or is static.As described in Section 3, users with similar demands at thecell border are combined within time-frequency units. Weassume i.i.d. channels with equal stochastic properties fromeach BS to the MT. If not stated otherwise, a fully loaded sys-tem is simulated for the transmit diversity techniques, andtherefore, their performances can be seen as upper bounds.All simulation parameters are summarized in Table 1. In thefollowing, we separate the simulation results in three blocks.First, we discuss the performances of CDD; then, the simula-tion results of CAT are debated; and finally, the influence ofthe MAI to both systems and the throughput of both systemsis investigated.

5.1. C-CDD performance

Figure 9 shows the influence of the cyclic delay δcyc1 to the

bit-error rate (BER) and the SNR gain at the cell border(C/I = 0 dB) for C-CDD. At the cell border there is no in-fluence due to C-DD, that is, (δ1 = 0). Two characteristicsof the performance can be highlighted. First, there is no per-formance gain for δ

cyc1 = 0 due to the missing C-CDD. Sec-

ondly, the best performance can be achieved for an existinghigher cyclic shift which reflects the results in [25]. The SNRgain performance for a target BER of 10−3 depicts also theinfluence of the increased cyclic delay. For higher delays theperformance saturates at a gain of about 2 dB.

The performances of the applied C-DD and C-CDDmethods are compared in Figure 10 with the reference sys-tem using no transmit (TX) diversity technique. For thereference system both BSs are transmitting independently


−10 0 10 20 30

C/I (dB)

1e − 05

1e − 04

1e − 03

1e − 02

1e − 01

BE

R

w/o TX diversity, fully loadedw/o TX diversity, half loadedCAT, halved TX power, 0 mphCAT, 0 mph, 2D interleavingCAT, 0 mphCAT, 40 mph

Figure 11: BER versus C/I for an SNR of 5 dB using no transmitdiversity and CAT for different scenarios.

0 0.25 0.5 0.75 1

Resource load

1e− 04

1e − 03

1e − 02

1e − 01

BE

R

C-CDD, C/I = 10 dBCAT, C/I = 10 dB

C-CDD, C/I = 0 dBCAT, C/I = 0 dB

Figure 12: Influence of the MAI to the BER performance for vary-ing resource loads at the cell border and the inner part of the cell.

their separate MC-CDMA signal. From Figure 9, we chooseδ

cyc1 = 30 samples and this cyclic delay is chosen through-

out all following simulations. The reference system is half(RL = 0.5) and fully loaded (RL = 1.0). We observe alarge performance gain in the close-by area of the cell bor-der (C/I = −10 dB, . . . , 10 dB) for the new proposed diversitytechniques C-DD and C-CDD. Furthermore, C-CDD en-ables an additional substantial performances gain at the cellborder. The C-DD performance degrades for C/I = 0 dB be-cause δ = 0 and no transmit diversity is available. The sameeffect can be seen for C-CDD at C/I = −4.6 dB (δ1 = −30,δ

cyc1 = 30 ⇒ δ = 0); see also Section 4. Since both BSs in

C-DD and C-CDD transmit the signal with the same power

−10 0 10 20 30

C/I (dB)

0

20

40

60

80

100

Max

thro

ugh

put

per

use

r(%

)

C-CDD, 4-QAMC-CDD, halved TX power, 4-QAMw/o TX diversity, RL = 0.5, 4-QAMw/o TX diversity, RL = 1, 4-QAMC-CDD, 16-QAMw/o TX diversity, RL = 0.5, 16-QAMw/o TX diversity, RL = 1, 16-QAM

Figure 13: Throughput per user for 4-QAM versus C/I using notransmit diversity or C-CDD with full and halved transmit power.

as the single BS in the reference system, the received signalpower at the MT is doubled. Therefore, the BER performanceof C-DD and C-CDD at δ = 0 is still better than the refer-ence system performance. For higher C/I ratios, that is, in theinner cell, the C-DD and C-CDD transmit techniques lackthe diversity from the other BS and additionally degrade dueto the double load in each cell. Thus, the MT has to copewith the double MAI. The loss due to the MAI can be di-rectly seen by comparing the transmit diversity performancewith the half-loaded reference system. The fully loaded ref-erence system has the same MAI as the C-CDD system, andtherefore, the performances merge for high C/I ratios. To es-tablish a more detailed understanding we analyze the C-CDDwith halved transmit power. For this scenario, the total desig-nated received power at the MT is equal to the conventionalMC-CDMA system. There is still a performance gain due tothe exploited transmit diversity for C/I < 5 dB. The perfor-mance characteristics are the same for halved and full trans-mit power. The benefit of the halved transmit power is a re-duction of the intercellular interference for the neighboringcells. In the case of varying channel models in the adjacentcells, the performance characteristics will be the same but notsymmetric anymore. This is also valid for the following CATperformances.

5.2. CAT performance

Figure 11 shows the performances of the applied CAT in thecellular system for different scenarios. If not stated otherwise,the systems are using a 1D interleaving. In contrast to theconventional system, the BER can be dramatically improvedat the cell border. By using the CAT, the MT exploits the addi-tional transmit diversity where the maximum is given at the


−10 0 10 20 30

C/I (dB)

0

20

40

60

80

100

Max

thro

ugh

put

per

use

r(%

)

CAT, 4-QAMCAT, halved TX power, 4-QAMw/o TX diversity, RL = 0.5, 4-QAMw/o TX diversity, RL = 1, 4-QAMCAT, 16-QAMw/o TX diversity, RL = 0.5, 16-QAMw/o TX diversity, RL = 1, 16-QAM

Figure 14: Throughput per user for 4-QAM and 16-QAM versusC/I using no transmit diversity or CAT with full and halved transmitpower.

cell border. If the MT moves with higher velocity (40 mph),the correlation of the subcarrier fading coefficients in timedirection decreases. This incremental violation of the qua-sistationarity assumption of the fading is profitable compen-sated by the channel code. The total violation of the afore-mentioned constraint of CAT (cf. Section 3.2) is achieved bya fully interleaved (2D) MC-CDMA frame. There is a largeperformance degradation compared to the CAT performancewith a noninterleaved frame. Nevertheless, a residual trans-mit diversity exists, the MT benefits at the cell border, andthe performance is improved. The applied CAT is not onlyrobust for varying MT velocities but also for non-quasistaticchannel characteristics. Similar to C-CDD, there is still a per-formance gain due to the exploited transmit diversity forC/I < 5 dB in the case of halved transmit powers at both BSs.

5.3. MAI and throughput performance ofC-CDD and CAT

The influence of the MAI is shown in Figure 12. The BERperformance versus the resource load of the systems is pre-sented. Two different positions of the MT are chosen: di-rectly at the cell border (C/I = 0 dB) and closer to one BS(C/I = 10 dB). Both transmit diversity schemes suffer fromthe increased MAI for higher resource loads which is in thenature of the used MC-CDMA system. CAT is not influencedby the MAI as much as C-CDD for both scenarios. Both per-formances merge for C/I = 10 dB because the influence ofthe transmit diversity techniques is highly reduced in the in-ner part of the cell.

Since we assume the total number of subcarriers isequally distributed to the maximum number of users per cell,

each user has a maximum throughput of ηmax. The through-put η of the system, by using the probability P(n) of the firstcorrect MC-CDMA frame transmission after n− 1 failed re-transmissions, is given by

η =∞∑

n=0

ηmax

n + 1P(n) ≥ ηmax(1− FER). (24)

A lower bound of the system is given by the right-hand sideof (24) by only considering n = 0 and the frame-error rate(FER). Figures 13 and 14 illustrate this lower bound for dif-ferent modulations in the case of C-CDD and CAT.

C-CDD in Figure 13 outperforms the conventional sys-tem at the cell border for all scenarios. Due to the almost van-ishing performance for 16-QAM with halved transmit powerfor an SNR of 5 dB, we do not display this performance curve.For 4-QAM and C-CDD, a reliable throughput along the cellborder is achieved. Since C-CDD with halved transmit powerstill outperforms the conventional system, it is possible to de-crease the intercellular interference.

The same performance characteristics as in C-CDD re-garding the throughput can be seen in Figure 14 for applyingthe transmit diversity technique CAT. Due to the combina-tion of two signals in the Alamouti scheme, CAT can pro-vide a higher throughput than C-CDD in the cell border area.The CAT can almost achieve the maximum possible through-put in the cell border area. For both transmit diversity tech-niques, power and/or modulation adaptation from the BSsopens the possibility for the MT to request a higher through-put in the critical cell border area. All these characteristicscan be utilized by soft handover concepts.

6. CONCLUSIONS

This paper handles the application of transmit diversity tech-niques to a cellular MC-CDMA-based environment. Ad-dressing transmit diversity by using different base stations forthe desired signal to a mobile terminal enhances the macro-diversity in a cellular system. Analyses and simulation re-sults show that the introduced cellular cyclic delay diversity(C-CDD) and cellular Alamouti technique (CAT) are capa-ble of improving the performance at the severe cell borders.Furthermore, the techniques reduce the overall intercellu-lar interference. Therefore, it is desirable to use C-CDD andCAT in the outer part of the cells, depending on available re-sources in adjacent cells. The introduced transmit diversitytechniques can be utilized for more reliable soft handoverconcepts.

ACKNOWLEDGMENTS

This work has been performed in the framework of the ISTProject IST-4-027756 WINNER, which is partly funded bythe European Union. The authors would like to acknowledgethe contributions of their colleagues. The material in this pa-per was presented in part at the IEEE 64th Vehicular Technol-ogy Conference, Montreal, Canada, September 25–28, 2006.


REFERENCES

[1] IST-2003-507581 WINNER Project, https://www.ist-winner.org.

[2] S. Plass, “On intercell interference and its cancellation in cellu-lar multicarrier CDMA systems,” EURASIP Journal on WirelessCommunications and Networking, vol. 2008, Article ID 173645,11 pages, 2008.

[3] D. Wong and T. J. Lim, “Soft handoffs in CDMA mobile sys-tems,” IEEE Personal Communications, vol. 4, no. 6, pp. 6–17,1997.

[4] M. Schinnenburg, I. Forkel, and B. Haverkamp, “Realizationand optimization of soft and softer handover in UMTS net-works,” in Proceedings of European Personal Mobile Communi-cations Conference (EPMCC ’03), pp. 603–607, Glasgow, UK,April 2003.

[5] A. Wittneben, “A new bandwidth efficient transmit antennamodulation diversity scheme for linear digital modulation,”in Proceedings of IEEE International Conference on Communi-cations (ICC ’93), pp. 1630–1634, Geneva, Switzerland, May1993.

[6] A. Dammann and S. Kaiser, “Performance of low complexantenna diversity techniques for mobile OFDM systems,” inProceedings of International Workshop on Multi-Carrier SpreadSpectrum (MC-SS ’01), pp. 53–64, Oberpfaffenhofen, Ger-many, September 2001.

[7] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-timeblock codes from orthogonal designs,” IEEE Transactions onInformation Theory, vol. 45, no. 5, pp. 1456–1467, 1999.

[8] M. Inoue, T. Fujii, and M. Nakagawa, “Space time transmitsite diversity for OFDM multi base station system,” in Proceed-ings of the 4th EEE International Workshop on Mobile and Wire-less Communication Networks (MWCN ’02), pp. 30–34, Stock-holm, Sweden, September 2002.

[9] S. B. Weinstein and P. M. Ebert, “Data transmission byfrequency-division multiplexing using the discrete Fouriertransform,” IEEE Transactions on Communications, vol. 19,no. 5, pp. 628–634, 1971.

[10] Z. Wang and G. B. Giannakis, “Wireless multicarrier commu-nications: where Fourier meets Shannon,” IEEE Signal Process-ing Magazine, vol. 17, no. 3, pp. 29–48, 2000.

[11] M. Sternad, T. Svensson, and G. Klang, “The WINNERB3G system MAC concept,” in Proceedings of IEEE VehicularTechnology Conference (VTC ’06), pp. 3037–3041, Montreal,Canada, September 2006.

[12] K. Fazel and L. Papke, “On the performance ofconcolutionally-coded CDMA/OFDM for mobile com-munications systems,” in Proceedings of IEEE InternationalSymposium on Personal, Indoor and Mobile Radio Communica-tions (PIMRC ’93), pp. 468–472, Yokohama, Japan, September1993.

[13] N. Yee, J.-P. Linnartz, and G. Fettweis, “Multi-carrier CDMAfor indoor wireless radio networks,” in Proceedings of IEEEInternational Symposium on Personal, Indoor and Mobile Ra-dio Communications (PIMRC ’93), pp. 109–113, Yokohama,Japan, September 1993.

[14] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Sys-tems, John Wiley & Sons, San Francisco, Calif, USA, 2003.

[15] A. Viterbi, “Error bounds for convolutional codes and anasymptotically optimum decoding algorithm,” IEEE Transac-tions on Information Theory, vol. 13, no. 2, pp. 260–269, 1967.

[16] A. Dammann and S. Kaiser, “Transmit/receive-antenna diver-sity techniques for OFDM systems,” European Transactions onTelecommunications, vol. 13, no. 5, pp. 531–538, 2002.

[17] G. Auer, “Channel estimation for OFDM with cyclic de-lay diversity,” in Proceedings of IEEE International Sympo-sium on Personal, Indoor and Mobile Radio Communications(PIMRC ’04), vol. 3, pp. 1792–1796, Barcelona, Spain, Septem-ber 2004.

[18] G. Bauch, “Differential modulation and cyclic delay diversityin orthogonal frequency-division multiplex,” IEEE Transac-tions on Communications, vol. 54, no. 5, pp. 798–801, 2006.

[19] G. L. Stuber, Principles of Mobile Communication, Kluwer Aca-demic Publishers, Norwell, Mass, USA, 2001.


[21] D. Tse and P. Viswanath, Fundamentals of Wireless Communi-cation, Cambridge University Press, New York, NY, USA, 2005.

[22] S. Plass, X. G. Doukopoulos, and R. Legouable, “On MC-CDMA link-level inter-cell interference,” in Proceedings of the65th IEEE Vehicular Technology Conference (VTC ’07), pp.2656–2660, Dublin, Ireland, April 2007.

[23] H. Schulze, “A comparison between Alamouti transmit di-versity and (cyclic) delay diversity for a DRM+ system,” inProceedings of International OFDM Workshop, Hamburg, Ger-many, August 2006.

[24] IST-2003-507581 WINNER, “D2.10: final report on identifiedRI key technologies, system concept, and their assessment,”December 2005.

[25] G. Bauch and J. S. Malik, “Cyclic delay diversity with bit-interleaved coded modulation in orthogonal frequency divi-sion multiple access,” IEEE Transactions on Wireless Commu-nications, vol. 5, no. 8, pp. 2092–2100, 2006.


Research ArticleSmartMIMO: An Energy-Aware Adaptive MIMO-OFDM RadioLink Control for Next-Generation Wireless Local Area Networks

Bruno Bougard,1, 2 Gregory Lenoir,1 Antoine Dejonghe,1 Liesbet Van der Perre,1

Francky Catthoor,1, 2 and Wim Dehaene2

1 IMEC, Department of Nomadic Embedded Systems, Kapeldreef 75, 3001 Leuven, Belgium2 K. U. Leuven, Department of Electrical Engineering, Katholieke Universiteit Leuven, ESAT, 3000 Leuven, Belgium

Received 15 November 2006; Revised 12 June 2007; Accepted 8 October 2007


Multiantenna systems and more particularly those operating on multiple input and multiple output (MIMO) channels are cur-rently a must to improve wireless links spectrum efficiency and/or robustness. There exists a fundamental tradeoff between po-tential spectrum efficiency and robustness increase. However, multiantenna techniques also come with an overhead in siliconimplementation area and power consumption due, at least, to the duplication of part of the transmitter and receiver radio front-ends. Although the area overhead may be acceptable in view of the performance improvement, low power consumption must bepreserved for integration in nomadic devices. In this case, it is the tradeoff between performance (e.g., the net throughput on topof the medium access control layer) and average power consumption that really matters. It has been shown that adaptive schemeswere mandatory to avoid that multiantenna techniques hamper this system tradeoff. In this paper, we derive smartMIMO: anadaptive multiantenna approach which, next to simply adapting the modulation and code rate as traditionally considered, de-cides packet-per-packet, depending on the MIMO channel state, to use either space-division multiplexing (increasing spectrumefficiency), space-time coding (increasing robustness), or to stick to single-antenna transmission. Contrarily to many of such adap-tive schemes, the focus is set on using multiantenna transmission to improve the link energy efficiency in real operation conditions.Based on a model calibrated on an existing reconfigurable multiantenna transceiver setup, the link energy efficiency with the pro-posed scheme is shown to be improved by up to 30% when compared to nonadaptive schemes. The average throughput is, on theother hand, improved by up to 50% when compared to single-antenna transmission.

Copyright © 2007 Bruno Bougard et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

The performance of wireless communication systems candrastically be improved when using multiantenna transmis-sion techniques. Specifically, multiantenna techniques canbe used to increase antenna gain and directionality (beam-forming, [1]), to improve link robustness (space-time cod-ing [2, 3]), or to improve spectrum efficiency (space divi-sion multiplexing [4]). Techniques where multiple anten-nas are considered both at transmit and receive sides cancombine those assets and are referred to as multiple-inputmultiple-output (MIMO). On the other hand, because of itsrobustness in harsh frequency selective channel combinedwith a low implementation cost, orthogonal frequency di-vision multiplexing (OFDM) is now pervasive in broadbandwireless communication. Therefore, MIMO-OFDM schemesturn out to be excellent candidates for next generation broad-band wireless standards.

Traditionally, the benefit of MIMO schemes is character-ized in terms of multiplexing gain (i.e., the increase in spec-trum efficiency) and diversity gain (namely, the increase inimmunity to the channel variation, quantified as the orderof the decay of the bit-error rate as a function of the signal-to-noise ratio). In [5], it is shown that, given a multiple-input multiple-output (MIMO) channel and assuming ahigh signal-to-noise ratio, there exists a fundamental tradeoffbetween how much of these gains a given coding scheme canextract. Since then, the merit of a new multiantenna schemeis mostly evaluated with regard to that tradeoff. However,from the system perspective, one has also to consider theimpact on implementation cost such as silicon area and en-ergy efficiency. When multiantenna techniques are integratedin battery-powered nomadic devices, as it is mostly the casefor wireless systems, it is the tradeoff between the effectivelink performance (namely, the net data rate on top of themedium access control layer) and the link energy efficiency


Transmitter

PDSP+MAC (25%) PTx (8%)

PDAC (6%)

PFE (20%) PPA (41%)

Receiver

PFE (25%) PFEC (35%)

PDSP+MAC (25%) PADC (15%)

Figure 1: Power consumption breakdown of typical single-antennaOFDM transceivers [6, 7]. At the transmit part, the power amplifiercontribution (PTx + PPA), which can scale with the transmit powerand linearity if specific architectures are considered [8], accounts for49%. At the receiver, the digital baseband processing (PDSP), forwarderror correction (PFEC), and medium access control (PMAC) units aredominant and do not scale with the transmit power. The power ofthe analog/digital and digital/analog converters (PADC + PDAC) andthe fixed front-end power (PFE) are not considered because they areconstant.

(total energy spent in the transmission and the reception perbit of data) that really matter. Characterizing how a diversitygain, a multiplexing gain, and/or a coding gain influence thatsystem-level tradeoff remains a research issue.

The transceiver power consumption is generally made oftwo terms. The first corresponds to the power amplifier(s)consumption and is a function of the transmit power, in-ferred from the link budget. The second corresponds to theother electronics consumption and is independent of the linkbudget. We refer, respectively, to dynamic and static powerconsumption. The relative contribution of those terms is il-lustrated in Figure 1 where the typical power consumptionbreakdown of single-antenna OFDM transceivers is depicted.

The impact on power consumption of multiantennatransmission (MIMO), when compared with traditionalsingle-antenna transmission (SISO), is twofold. On the onehand, the general benefit in spectral efficiency versus signal-to-noise ratio can be exploited either to reduce the requiredtransmit power, with impact on the dynamic power con-sumption, or to reduce the transceiver duty cycle with im-pact on both dynamic and static power contributions. Onthe other hand, the presence of multiple antennas requiresduplicating part of the transceiver circuitry, which increasesboth the static and dynamic terms.

The question whether multiantenna transmission tech-niques increase or decrease the transceiver energy efficiencyhas only recently been addressed in the literature [9–11]. In-terestingly, it has been shown that for narrow-band single-carrier transmission, multiantenna techniques basically de-crease the energy efficiency if they are not combined withadaptive modulation [9]. It has also been shown, in the samecontext, that energy efficiency improvement is achievable byadapting the type of multiantenna encoding to the transmis-sion condition [10, 11].

The purpose of this paper is first to extend previouslymentioned system-level energy efficiency studies to the caseof broadband links based on MIMO-OFDM. Therefore, weinvestigate the performance versus energy efficiency tradeoffof two typical multiantenna techniques—space-time blockcode (STBC) [3] and space-division multiplexing (SDM)

[4]—and compare it to the single-antenna case. Both are im-plemented on top of a legacy OFDM transmission chain asused in IEEE 802.11a/g/n and proceed to spatial processingat the receiver only. The IEEE 802.11 MAC has been adaptedto accommodate those transmission modes. For the sake ofclarity, without hampering the generality of the proposed ap-proach, we limit the study to 2 × 2 antennas systems.

Second, we propose smartMIMO, a coarsely adaptiveMIMO-OFDM scheme that, on a packet-per-packet basis,switches between STBC, SDM, and SISO depending on thechannel conditions to simultaneously secure the through-put and/or robustness improvement provided by the mul-tiantenna transmission and guarantees an energy-efficiencyimproved compared with the current standards. Contrarilyin other adaptive scheme [12–16], using SISO still reveal ef-fective in many channel condition because of the saving instatic power consumption.

The remainder of the paper is structured as follows. InSection 2, we present some related work. The MIMO-OFDMphysical (PHY) and medium access control (MAC) layersare described in Section 3. The unified performance and en-ergy models used to investigate the average throughput ver-sus energy-efficiency tradeoff are presented in Section 4. Theimpact of SDM and STBC on the net throughput versusenergy-efficiency tradeoff is discussed in Section 5. Finally,in Section 6, we present the smartMIMO scheme and evalu-ate its benefit on the aforementioned tradeoff.

2. RELATED WORK

The question whether multiantenna techniques increase ordecrease the energy efficiency has only very recently beenaddressed. Based on comprehensive first order energy andperformance models targeted to narrow-band single car-rier transceivers (as usually considered in wireless microsen-sor), Shuguang et al. have evaluated, taking both static anddynamic power consumption into account, the impact onenergy efficiency of single-carrier space-time block coding(STBC) versus traditional single antenna (SISO) transmis-sion [9]. Interestingly, it is shown that in short-/middle-range applications such as sensor networks—and by exten-sion, wireless local area networks (WLANs)—nonadaptiveSTBC actually degrades the system energy efficiency at samedata rate. However, when combined with adaptive mod-ulation in so-called adaptive multiantenna transmission,energy-efficiency can be improved. Liu and Li have extendedthose results by showing that energy-efficiency can further beimproved by adaptively combining multiplexing and diver-sity techniques [10, 11]. Adaptive schemes are hence manda-tory to achieve both high-throughput and energy-efficienttransmissions.

In the context of broadband wireless communication,many adaptive multiantenna schemes have also been pro-posed and are often combined with orthogonal frequency di-vision multiplexing (OFDM). Adaptation is most often car-ried out to minimize the bit-error (BER) probability or max-imize the throughput. In [12], for instance, a scheme is pro-posed to switch between diversity and multiplexing codesbased on limited channel state information (CSI) feedback.

Bruno Bougard et al. 3

In [13], a pragmatic coarse grain adaptation scheme is eval-uated. Modulation, forward error correction (FEC) cod-ing rate and MIMO encoding are adapted according toCSI estimator—specifically, the average signal-to-noise ratio(SNR) and packet error rate (PER)—to maximize the effec-tive throughput. More recently, fine grain adaptive schemeshave been proposed [14, 15]. The modulation and multi-antenna encoding are here adapted on a carrier-per-carrierbasis. The main challenge with such schemes is however toprovide the required CSI to the transmitter with minimaloverhead. This aspect is tackled, for instance in [16].

The approaches mentioned above have been proven tobe effective to improve net throughput and/or bit-error rate(BER). Some are good candidate to be implemented in com-mercial chipset. However, in none of those contributions,the electronics power consumption is considered in the opti-mization. Moreover, most adaptive policies are designed tomaximize gross data-rate and/or minimize (uncoded) bit-error rate without taking into account the coupling betweenphysical layer data rate and bit-error rate incurred in mediumaccess control (MAC) layer [17].

In this paper, adaptive MIMO-OFDM schemes arelooked at with as objective to jointly optimize the average linkthroughput (on top of the medium access control layer) andthe average transceiver energy efficiency. The total transceiverpower consumption is considered, including the terms thatvary with the transmit power and the fixed term due the ra-dio electronics.

To enable low-complexity policy-based adaptation and tolimit the required CSI feedback, coarsely adaptive schemes,as defined in [13], are considered. For this, pragmatic em-pirical performance, energy, and channel state informationmodels are developed based on observation and measure-ments collected on the reconfigurable MIMO-OFDM setuppreviously described in [18].

3. MIMO PHYSICAL AND MAC LAYERS

3.1. MIMO-OFDM physical layer

The multiantenna schemes considered here are orthogonalspace-time block coding (STBC) [2, 3] and space-divisionmultiplexing with linear spatial processing at the receiver(SDM-RX) [4]. Both are combined with OFDM so that mul-tiantenna encoding and/or receive processing is performedon a per-carrier basis. The N OFDM carriers are QAM-modulated with a constellation size set by the link adapta-tion policy presented in [17]. The same constellation is con-sidered for the different carriers of a given symbol, thereforewe refer to “coarse grain” adaptation in opposition to finegrain adaptation where the subcarriers can receive differentconstellation.

Figure 2 illustrates the general setup for MIMO-OFDMon which either SDM or STBC can be implemented.

For SDM processing, a configuration with U transmit an-tenna and A receive antenna is considered. The multiantennapreprocessing reduces to demultiplexing the input stream insub-streams that are transmitted in parallel. Vertical encod-ing is considered: the original bit stream is FEC encoded,

interleaved, and demultiplexed between the OFDM modu-lators. The MIMO processors at the receiver side take careof the spatial interference mitigation on a per-subcarrier ba-sis. We consider a minimum mean-square–error-(MMSE-)based detection algorithm. Although the MMSE algorithm isoutperformed by nonlinear receiving algorithm such as suc-cessive interference cancellation [4], its implementation easekeeps it attractive in low-cost, high-throughput solution suchas wireless local area networks.

In STBC mode, space-time block codes from orthogo-nal designs [2, 3] are considered. Such scheme reduces toan equivalent diagonal system that can be interpreted as aSISO model where the channel is the quadratic average ofthe MIMO sub-channels [3].

Channel encoding and OFDM modulation are done ac-cording to the IEEE 802.11a standard specifications, trans-mission occurs in the 5 GHz ISM band [19].

As mentioned previously, from the transceiver energy-efficiency perspective, it may still be interesting to operatetransmission in single-antenna mode. In that case, a singlefinger of the transmitter and receiver are activated and theMIMO encoder and receive processor are bypassed.

3.2. Medium access control layer

The multiantenna medium access control (MAC) protocolwe consider is a direct extension of the IEEE 802.11 dis-tributed coordination function (DCF) standard [19]. A car-rier sense multiple access/collision avoidance (CSMA/CA)medium access procedure performs automatic medium ac-cess sharing. Collision avoidance is implemented by meanof the exchange of request-to-send (RTS) and clear-to-send(CTS) frames. The data frames are acknowledged (ACK).The IEEE 802.11 MAC can easily be tuned for adaptive mul-tiantenna systems. We assume that the basic behavior ofeach terminal is single-antenna transmission. Consequently,single-antenna exchange establishes the multiantenna fea-tures prior to the MIMO exchange. This is made possible viathe RTS/CTS mechanism and the data header. A signalingrelative to the multiple-antenna mode is added to the physi-cal layer convergence protocol (PLCP) header.

Further, the transactions required for channel estima-tion need to be adapted. In the considered 2 × 2 configura-tion, not only one but four-channel path must be identified.Therefore, the preamble structure is adapted as sketched inFigure 3. The transmitter sends preambles consecutively onantennas 1 and 2. The receiver can then easily identify allchannel paths. The complete protocol transactions to trans-mit a packet of data in the SDM and STBC modes are detailedin Figures 4 and 5, respectively. For SISO transmission, onerelies on the standard 802.11 CSMA/CA transaction and onthe standard preambles.

4. UNIFIED PERFORMANCE AND ENERGY MODEL

The physical (PHY) and medium access control (MAC) lay-ers being known, one wants to compute the net through-put (on top of the MAC) and the energy per bit as func-tions of the transmission parameters, including the type of


MIMOencoder

1

MIMOencoderN

IFFT

IFFT

Par

alle

lto

seri

alP

aral

lel

tose

rial

TX filter

TX filter

RX filter

RX filter

Par

alle

lto

seri

alP

aral

lel

tose

rial

FFT

FFT

MIMO-RXprocessor 1

MIMO-RXprocessorN

Figure 2: Reconfigurable multiantenna transceiver setup supporting SDM, STBC, and SISO transmissions.

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 G2 T1 T2 G2 T1 T2

10× 0.8 = 8μs 2× 0.8 + 2× 3.2 = 8μs 2× 0.8 + 2× 3.2 = 8μs

8 + U × 8μs

2× 0.8 + 2× 3.2 = 8μs 0.8 + 3.2 = 4μs 0.8 + 3.2 = 4μs 0.8 + 3.2 = 4μs

G2 T1 T2 G1 Signal G1 Payload 1 G1 Payload 2

Service + payloadRate lengthChannel U estimation

Signal detectAGCdiversityselection

Coarse frequencyoffset estimationtiming synchronize

Channel and fine frequencyoffset estimation

Channel 2 estimation

Figure 3: Channel estimations from the preamble for a 2 × 2 system.

multiantenna processing, and the channel state. To enablesimple policy-based adaptation scheme and limit the re-quired CSI feedback, a coarse channel state model is neededto capture the CSI in a synthetic way. The proposed modelshould cover the system performance and energy consump-tion of the different multiantenna techniques under consid-eration.

An important aspect is to identify tractable channel stateparameters that dominate the instantaneous packet errorprobability. The average packet errors rate (PER) as tradi-

tionally evaluated misses that instantaneous dimension. Innarrow-band links affected by Rayleigh fading, the signal-to-noise ratio (SNR) suffices to track the channel state. InMIMO-OFDM, however, the impact of the channel is morecomplex. With spatial multiplexing, for instance, the errorprobability for a given modulation and SNR still depends onthe rank of the channel. Moreover, not all the subcarriers ex-perience the same MIMO channel. Finally, a given channelinstance can be good for a specific MIMO mode while beingbad for another one.


(1) TX sends RTS in SISO. the PLCP header contains:

(i) 2 bits specifying the MIMO exchange type (herechannel state extraction at RX)

(ii) 2 bits for the number of TX antennas that will beactivated for the next MIMO exchange (nTX) andthat have to be considered in the channel extraction

(iii) 2 bits for the minimum numbers of RX antennas toactivate for next reception.

(2) RX sends CTS.(3) TX send DATA preamples in time division: 10 short train-

ing sequences for coarse synchronization (t1 · · · t10−8μs)and 2×nTX long training sequences (G2T1T2−8μs, calledC-C sequence). Each TX antenna transmits after eachother its C-C sequence. RX activates its antennas and ex-tracts information about each channel that each antennasees. PLCP header is sent from the last TX antenna andconveys 2 bits for the mode (here SDM-RX) and 2 bitsfor the number of streams. finally, TX sends the DATA inMIMO.

(4) RX sends ACK.

Figure 4: Considered SDM protocol extension.

(1) TX sends RTS in SISO. the PLCP header contains:

(i) 2 bits specifying the MIMO exchange type (herechannel state extraction at RX)

(ii) 2 bits for the number of TX antennas that will beactivated for the next MIMO exchange (nTX) andthat have to be considered in the channel extraction

(iii) 2 bits for the minimum numbers of RX antennas toactivate for next reception.

(2) RX sends CTS.(3) TX send DATA preamples in TDMA: 10 short training se-

quences for coarse synchronization (t1 · · · t10 − 8μs) and2 ×nTX long training sequences (G2T1T2− 8μs, called C-C sequence). Each TX antenna transmits after each otherits C-C sequence. RX activates its antennas and extractsinformation about each channel that each antenna sees.PLCP header is sent from the last TX antenna and con-veys 2 bits for the mode (here STBC) and 4 bits for thecode used. Finally, TX sends the DATA in MI-SO/MO de-pending on the number of recieved antenna (nRX).

(4) RX sends ACK.

Figure 5: Considered STBC protocol extension.

Possible coarse channel state information (CSI) indica-tors for MIMO-OFDM are discussed in [13]. An empiri-cal approach based on multiple statistics of the postprocess-ing SNR (the SNR after MIMO processing) and running-average PER monitoring is proposed. Yet, it is difficult todefine such SNR-based indicators consistently across differ-ent MIMO schemes. Moreover, relying on PER information

results in a tradeoff between accuracy and feedback latency,both with potential impact on stability.

As already proposed in [20], based on the key observationthat energy efficiency and net throughput are actually weakfunctions of the packet error probability [21], we prefer touse the outage probability—that is, the probability that thechannel instantaneous capacity is lower that the link spec-trum efficiency—as indicator of the packet error probability.The instantaneous capacity depends on the average signal-to-noise ratio (SNR), the normalized instantaneous channelresponse H, and the multiantenna encoding. The instanta-neous capacity can be easily derived for the different multi-antenna encoding. Practically, it is convenient to derive thecapacity-over-bandwidth ratio that can be compared to thetransmission spectrum efficiency η instead of absolute rate.

In the remainder of this session, we first derive the instan-taneous capacity expressions for the different transmissionmode considered (Section 4.1). Then, we derive the condi-tion for quasi-error-free packet transmission (Section 4.2).Based on that, we compute the expressions of the netthroughput and the energy per bit (Sections 4.3 and 4.4). Fi-nally, we discuss the derivation of the coarse channel modelrequired to develop policy-based radio link control strategies(Section 4.5).

4.1. Instantaneous capacity

Let H = (hnua) be a normalized MIMO-OFDM channel re-alization. The coefficient hn11 corresponds to the (flat) chan-nel response between the single active transmit antenna andthe single active receive antenna for the subcarrier n (includ-ing transmit and receive filters). The instantaneous capacityof the single-antenna channel is given by (1), where W is thesignal bandwidth and N is the number of subcarriers. SNRis the average link signal-to-noise ratio. If SNR is high com-pared to 1, the capacity relative to the bandwidth can be de-composed in a term proportional to SNR and independentof H and a second term function of H only:

C =W· 1N

N∑

n=1

log 2

(1 + hn2

11SNR), (1)

C

W∼= SNR|dB

10log 102+

1N

N∑

n−1

log 2

(hn2

11

). (2)

In the STBC case, as mentioned in Section 3, the MIMOchannel can be reduced to equivalent SISO channel corre-sponding to the quadratic average of the subchannels be-tween each pairs of transmit and receive antennas [3]. Theinstantaneous capacity can then be computed just as for SISO:

C

W∼= SNR|dB

10 log 2+

1N

N∑

n−1

log 2

(1UA

U∑

u=1

A∑

a=1

hn2ua

). (3)

In the SDM case finally, the compound channel results fromthe concatenation of the transmission channel with the in-terference cancellation filter. The instantaneous capacity canbe computed based on the postprocessing SNR’s (γ)— that is,for each stream, the signal-to-noise-and-interference ratio at


the output of the interference cancellation filter. Let Hn andFn, respectively, denote the MIMO channel realization for thesubcarrier n, and the corresponding MMSE filter (4):

Fn = HnH·(HnHnH + σ2IAxA)−1

. (4)

In the considered 2 × 2 case, let us assume an equal transmitpower at both transmit antennas p1 = p2 = p/2 and let usdenote with fn

1 , fn2 , hn

1 , hn2 , respectively, the first row, second

row, first column, second column of the matrices Hn and Fn.The substream postprocessing SNRs γ1 and γ2 can then becomputed as

γn1=∣∣fn

1 hn1

∣∣2 × P1∣∣fn1 hn

2

∣∣2 × P2 +∣∣fn

1

∣∣2 × σ2=

∣∣fn1 hn

1

∣∣2

∣∣fn1 hn

2

∣∣2+∣∣fn

1

∣∣2 × 2/SNR,

γn2=∣∣fn

2 hn2

∣∣2 × P2∣∣fn2 hn

1

∣∣2 × P1 +∣∣fn

2

∣∣2 × σ2=

∣∣fn1 hn

2

∣∣2

∣∣fn2 hn

1

∣∣2+∣∣fn

2

∣∣2 × 2/SNR.

(5)

SNR is again the average link signal-to-noise ratio. The in-stantaneous capacity can then be computed in analogy with(2) and (3) as follows:

C

W= 1N

N∑

n=1

[log 2

(1 + γn1

)+ log 2

(1 + γn2

)]. (6)

This development can easily be extended to more than 2× 2antenna setups.

4.2. Condition for quasi-error-free packettransmission on a given channel

Because the link throughput and energy efficiency (our ob-jective functions) are weak functions of the packet errorprobability [21], one does not need to estimate the latter ac-curately in order to define adaptation policies that optimizethe formers. It is sufficient to derive a condition under whichthe packet error rate is sufficiently low in order not to sig-nificantly affect the aforementioned objective functions. It iseasy to verify that the accuracy obtained on the throughputon top of the MAC and on the energy efficiency is of the sameorder of magnitude that the packet error rate.

Based on Monte Carlo simulations, we have verified that,for the purpose of computing the average throughput ontop of the MAC and the transceiver energy consumption perbit, the packet error event probability can be approximatedby the outage without significant prejudice to the accuracy(Figure 6). We hence assume that, the channel being known,Pe, equals 1 if the spectrum efficiency η exceeds C/W. To ac-count for the nonoptimality of the coding chain, we apply anempirical margin δ = 0.5 bit/s/Hz, calibrated from simula-tion:

Pe =

⎧⎪⎨⎪⎩

1 ifC

W< η + δ,

∼= 0 otherwise.(7)

10−6

10−5

10−4

10−3

10−2

10−1

100

Pac

ket

erro

rp

roba

bilit

yes

tim

atio

ns

0 1 2 3 4 5 6

Instantaneous capacity estimations (bit/s/Hz)

Approximate outage model

Regression line

δ

Multiple channelrealization

Figure 6: Packet error probability and instantaneous capacity ob-servation on a link with spectrum efficiency 4 for a large set ofchannel realizations. One can observe the strong correlation and thesteep descend of the regression line beyond the point where the in-stantaneous capacity breaks the spectrum efficiency line. The netthroughput and energy per bit are weak function of the packet er-ror probability; these observations motivate considering an outagemodel to derive policy-based adaptive schemes.

4.3. Net throughput

Assuming that the channel capacity criteria is met and,hence, the PER is close to zero, knowing the physical layerthroughput (Rphy) and the details of the protocols, the netthroughput (Rnet) can be computed as follows:

Rnet

∼= LdTDIFS+3·TSIFS + TCW+

((Ld + Lh

)/Rdphy

)+4·Tplcp+Lctrl/R

bphy

.

(8)

To better understand that expression, refer to Figure 7 andnotice that the denominator corresponds to the total time re-quired for the transmission of one packet of data size Ld witha Lh-bit header according to the 802.11 DCF protocol [19].TSIFS is the so-called short interframe time. Rdphy is the phys-ical layer data rate. Lctrl corresponds to the aggregate lengthof all control frames (RTS, CTS, and ACK) transmitted atthe basic rate (Rbphy) and Tplcp is the transmission time of thePLCP header. TDIFS is the minimum carrier sense durationand TCW holds for the average contention time due to theCSMA procedure. The physical layer data rate Rdphy can beexpressed as a function of modulation order (Nmod ) and thecode rate (Rc), considering the number of data carrier perOFDM symbol (N) and the symbol rate (Rs), in addition tothe number of streams U (9). In our study, one has U = 1 forSISO and U = 2 for both SDM and STBC. In the MIMO case(U > 1), one of the four TPLCP’s in (8) must be replaced byTPLCP MIMO given in (10) with TCC Seq being equal 8 μs:

Rdphy = U·N·Nmod ·Rc·Rs, (9)

TPLCP MIMO = TPLCP SISO + (U − 1)× TCC Seq. (10)


4.4. Energy per bit

To compute the energy efficiency, the system power con-sumption needed to sustain the required average SNR mustbe assessed. The latter consists of a fixed term due to the elec-tronics, and a variable term, function of the power consump-tion

Psysten = Pelec +PTx

ε, (11)

where ε denotes the efficiency of the transmitter power am-plifier (PA), that is, the ratio of the output power (PTx) by thepower consumption (PPA). In practical OFDM transmitters,class A amplifiers are typically used. The power consump-tion of the latter component only depends on its maximumoutput power (Pmax ) (12). Next, the transmitter signal-to-distortion ratio (S/DTx) can be derived as a function of thesole backoff (OBO) of the actual PA output power (PTx) toPmax (13)-(14). The latter relation is design dependent andusually not analytical. In this study, we consider an empiricalcurve-fitted model calibrated on the energy-scalable trans-mission chain design presented in [8], which has as key fea-ture to enable both variable output power (PTx) and variablelinearity (S/DTx) with a monotonic impact on the power con-sumption;

PPA = Pmax

2, (12)

OBO = Pmax

PTx, (13)

(S/D)Tx = f (OBO). (14)

The path-loss being known, SNR can then be computed as afunction of OBO and PTx (15). PN is the thermal noise leveldepending of the temperature (T), the receiver bandwidth(W), and noise factor (Nf ), k is the Boltzmann constant:

1SNR

= 1(S/D)Tx

+PN × PLPTx

, (15)

PN = k·T·W·Nf . (16)

The PA power can be expressed as a function of those twoparameters (17):

PPA = 2× (PTX ×OBO). (17)

The achievable PPA versus SNR tradeoff obtained with thedesign as presented in [8] is illustrated for different averagelink path-loss values in Figure 8. Notice that in case the out-put power has to stay constant, the proposed reconfigurablearchitecture still has the possibility to adapt to the linearityrequirements. This has less but still significant impact on thepower consumption.

4.5. Coarse channel model

At this point in the development, we have relations to com-pute the link throughput and the transceiver energy con-sumption per bit for given multiantenna encoding, modu-

lation and code rate, provided that the link SNR is sufficientto satisfy the quasi error free transmission condition.

The latter condition depends also on the actual chan-nel response H (equations (2), (3), (5), (6), and (7)). Exten-sive work has already been done to model broadband chan-nel at that level of abstraction (physical level). In the case ofMIMO-OFDM WLAN as considered here, a reference chan-nel model is standardized by the IEEE [22]. However, to beable to derive simple policy-based adaptation schemes thattake that channel state information into account, one has toderive a model that captures this information in a more com-pact way. A valid approach is to operate an empirical classifi-cation of the channel merit. This can easily be done based onthe instantaneous capacity indicators.

As an example, let us consider the second term of the in-stantaneous SISO capacity. According to [22], the values ofthe carrier fading hnua are Rayleigh distributed. However, dueto the averaging across the carriers, which are only weaklycorrelated, the distribution of the second term of the instan-taneous capacity is almost normal distributed (see Figure 9).Since the first term of the capacity indicator is independentof H and therefore not stochastic, the capacity indicator canthen also be approximated as normal-distributed. One canverify that the same observation holds also for the STBC andSDM instantaneous capacity indicators (see Figure 9).

Let us, respectively, denote the average and variance ofthe instantaneous capacity for a given mode as μmode andσ2

mode. These quantities depend only on SNR. Their evolutionin a function of SNR is plotted in Figure 10 for the differentmultiantenna encoding. It can be observed that for a givenmode and a sufficiently large SNR, μmode grows linearly withSNR in dB while σ2

mode stays sensibly constant. Therefore, alinear regression can be operated. The parameters of the ex-tracted linear model are summarized in Table 1.

Based on the normal distribution of the instantaneouscapacity and the linear models for the parameters of that dis-tribution, a channel merit scale can be defined. A given chan-nel instance receives a merit index for a given multi-antennamode and a given SNR depending on how its actual instanta-neous capacity compared to the capacity distribution for thatmode and that SNR. The empiric scale we consider goes from1 (worst) to 5 (best) with the class boundaries as defined inTable 2 (3 first columns).

For each class index (channel merit), a worst-case error-free transmission condition is defined, comparing the signal-ing spectrum efficiency (η) to the upper bound of the instan-taneous capacity class for this channel merit (Table 2, fourthcolumn).

4.6. Usage of the model

One can now compute, for a given channel merit as definedabove, what will be the link throughput and the transceiverenergy consumption per bit for a given multiantenna mode,a given modulation and a given code rate. The computationoccurs as follows.

Step 1. Knowing the modulation and code rate, hence thesignaling spectrum efficiency, the minimum link SNR to


DIFS

RTS Data

SIFS SIFS SIFS

CTS ACK

Source

Destination

Other

DIFS

NAV (RTS)

NAV (CTS)

CW

Defer Access Backoff

Figure 7: Packet transmission transaction according to the IEEE 802.11 protocol modified to support multiantenna operation.

Table 1: Instantaneous capacity indicator average and standard de-viation as a function of the average SNR.

μ = A× SNR+Bσ

A B

SISO 0.33 −0.84 1.41

SDM 0.6 −2.54 2.41

STBC 0.33 −0.24 0.73

satisfy the worst-case quasi-error-free transmission condi-tion (for the given multi-antenna mode and channel merit)is computed using the appropriate inequality from Table 2,fourth column, and the linear model exposed in Table 1.

Step 2. Knowing the multi-antenna mode, the modulationand the code rate, assuming quasi-packet-error-free trans-mission, the link throughput is computed according to (8).The condition is calibrated for a packet error rate of <1%,yielding an accuracy of 1% of the estimated throughput andenergy efficiency.

Step 3. Based on the power model exposed in Section 4.4, as-suming a given average path loss, the transmitter parameters(output power, backoff) to achieve the link SNR computedin Step 1, and subsequently the transmitter power consump-tion, are computed.

Step 4. From the transmitter power and the net throughput,the energy per bit can be computed.

5. IMPACT OF MIMO ON THE AVERAGE RATEVERSUS AVERAGE POWER TRADEOFF

The proposed performance and energy models enable com-puting the net throughputand energy efficiency as func-tions of the system-level parameters (mode, modulation,code rate, transmit power, and power amplifier backoff). The

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PPA

[W]

−20 −15 −10 −5 0 5 10 15 20 25 30

SNR (dB)

6 9 12 18 24 36 48 54

PL 90 dB PL 80 dB PL 70 dB

PL 60 dB

Required SNRfor rate (Mbps)(SISO)

Figure 8: Power consumption versus link SNR tradeoff achievedwith the energy scalable transmitter for average path-loss, PL =60 dB, 70 dB, 80 dB, 90 dB. The tradeoff curves are compared to theSNR level required for 20 MHz SISO-OFDM transmission with PER< 10% at various rates.

considered settings for those parameters are summarized inTable 3. Capitalizing on those models and using the tech-niques already proposed in [17], one can derive a set of close-to-optimal transmission adaptation policies that optimizethe average energy efficiency for a range of average through-put targets and, then, analyze the resulting tradeoff.

In this section, we derive these tradeoffs separately for theSTBC and SDM modes and compare with SISO. This is donein two steps.

Step 1. For each channel merit, the optimum tradeoff be-tween net throughput and energy per bit (Section 5.1) is de-rived. This results in a Pareto optimal [23] set of workingpoints (settings of the system-level parameters) for each pos-sible channel merit.


Table 2: Definition of the channel merit.

Channel meritChannel instance instantaneous capacity indicator

Maximum spectrum efficiency for quasi error free transmissionMin Max

1 −∞ μmode − 2× σmode —

2 μmode − 2× σmode μmode − σmode η < μmode(SNR)− 2× σmode(SNR)

3 μmode − σmode μmode + σmode η < μmode(SNR)− σmode(SNR)

4 μmode + σmode μmode + 2× σmode η < μmode(SNR) + σmode(SNR)

5 μmode + 2× σmode +∞ η < μmode(SNR) + 2× σmode(SNR)

0

100

200

300

Nu

mbe

rof

chan

nel

inst

ance

s

−5 −4 −3 −2 −1 0 1 2 3 4 5

Capacity indicator (bit/s/Hz)

SISO

(a)

0

100

200

300

Nu

mbe

rof

chan

nel

inst

ance

s

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5


STBC

(b)

0

100

200

300

Nu

mbe

rof

chan

nel

inst

ance

s

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5


SDM

(c)

Figure 9: Distribution of the instantaneous capacity observationfor the different modes (SISO, STBC, SDM) over a large set of chan-nel instances generated with the physical channel model.

Step 2. For a given average throughput target, a policy is de-rived to select which working points from the Pareto optimalset has to be used for each channel merit value in order tominimize the average energy per bit (Section 5.2).

The resulting average throughput versus energy-per-bittradeoffs is finally analyzed in Section 5.3. It should be no-ticed that this approach assume that the channel merit isknown at the transmitter (limited CSI at transmit). That in-formation can be acquired during the reception of the CTS

−10

−5

0

5

10

15

20

25

30

Cap

acit

yin

dic

ator

(bit

/s/H

z)

0 5 10 15 20 25 30 35 40

SNR (dB)

SISO μSISO μ + 3σSISO μ− 3σSDM μSDM μ + 3σ

SDM μ− 3σSTBC μSTBC μ + 3σSTBC μ− 3σ

Figure 10: Capacity indicator mean and standard deviation as afunction of SNR for the various modes.

Table 3: System-level parameters considered.

MIMO mode SISO, SDM2 × 2, STBC2 × 2

Nmod BPSK, QPSK, 16QAM, 64QAM

Rc 1/2, 2/3, 3/4

PTx [dBm] 0, 5, 10, 15, 20, 23

OBO [dB] 6, 8, 10, 12, 14

frame or piggy-backed in the CTS, assuming that the channelis stable during RTS/CTS/packet transaction. The assump-tion is valid in nomadic scenario as considered in case ofWLAN (typical coherence time of 300 milliseconds).

5.1. Net throughput versus energy-per-bit

To derive the optimal net throughput versus energy-per-bit tradeoff for a given mode in a given channel merit, amultiobjective optimization problem has to be solved: from


12

14

16

18

20

22

24

En

ergy

per

bit

(nJ)

26 28 30 32

Goodput (Mbit/s)

60 dB

(a)

20

40

60

80

100

120

En

ergy

per

bit

(nJ)

26 28 30 32

Goodput (Mbit/s)

70 dB

(b)

50

100

150

200

250

En

ergy

per

bit

(nJ)

20 25 30

Goodput (Mbit/s)

80 dB

(c)

0

50

100

150

200

250

300

350E

ner

gyp

erbi

t(n

J)

16 18 20 22 24 26

Goodput (Mbit/s)

90 dB

(d)

Figure 11: Net throughput versus energy-per-bit tradeoff for channel merit 3 and various path-losses. The (·) corresponds to the SISOworking points, the (+) corresponds to the STBC, and the (×) corresponds to the SDMs. In each case, the Pareto optimal set is interpolatedwith a step curve.

all system-level parameter combinations, the ones bound-ing the tradeoff have to be derived. The limited range of thefunctional parameters still allows us to proceed efficiently tothis search with simple heuristics [17]. This optimization canbe proceeded to at design time, which limits to a great extendthe complexity of the adaptation scheme.

The resulting tradeoff points are plotted in Figure 11 fordifferent path losses and an average channel merit (which is3). For each mode, we only keep the nondominated trade-off points, leading to Pareto optimal sets, which are interpo-lated by step curves. We generally observe that SDM enablesreaching higher throughput but that SISO stays more energyefficient for lower rates. STBC becomes attractive in case oflarge path losses. Similar tradeoff shapes can be observed forthe other channel merit values.

5.2. Derivation of the control policies

From the knowledge of the Pareto optimal net throughputversus energy-per-bit tradeoff and the channel merit proba-bilities, which can be obtained from the Monte Carlo anal-ysis of the physical-level channel model, given an averagethroughput constraint, we applied the technique presentedin [17] to derive the adaptation policy that minimizes theenergy per transmitted bit. Such a policy, valid for a givenmultiantenna mode, a given average path loss, and a givenaverage throughput constraints, maps the possible channelmerit to the appropriate setting of the transmission parame-ters.

Let (ri j , ei j) denote the coordinates of the ith Pareto pointin the set corresponding to channel merit j. The average


40

42

44

46

48

50

52

54

56

Ave

rage

ener

gyp

erbi

t(n

J)

0 5 10 15 20 25 30 35

Average rate (Mbs)

Run time controller performance for MATCH 60 dB

(a)

40

42

44

46

48

50

52

54

56

58

60

Ave

rage

ener

gyp

erbi

t(n

J)

0 5 10 15 20 25 30 35

Average rate (Mbit/s)


(b)

40

45

50

55

60

65

70

Ave

rage

ener

gyp

erbi

t(n

J)

0 5 10 15 20 25 30 35



(c)

45

50

55

60

65

70

75

80

85

90A

vera

geen

ergy

per

bit(

nJ)

0 5 10 15 20 25 30



(d)

Figure 12: Average net throughput versus average energy-per-bit tradeoff for SISO (o), STBC (+), and SDM (×) at various path-loss.

power P and rate R corresponding to a given control policy—that is, the selection of one point on each throughput energyefficiency tradeoff—can then be expressed by (18). In theseequations, xi j is l if the corresponding point is selected, 0 oth-erwise, and ψ j is the probability of the channel merit j. Theenergy per bit can be computed as P / R:

P =∑

j

ψ j

∑

i

xi jei j ri j =∑

i

∑

j

xi jψ jei j ri j�=∑

i

∑

j

xi j p′i j ,

R =∑

j

ψ j

∑

i

xi j ri j =∑

i

∑

j

xi jψ jri j�=∑

i

∑

j

xi j r′i j .

(18)

We introduce the notation p′i j and r′i j corresponding, respec-tively, to the power and rate when the channel merit is j andthe ith point is selected on the corresponding curve, both

weighted by the probability to be in that channel state. Onlyone tradeoff point can be selected for a given channel merit,resulting in the following constraints:

∑

i

xi j = 1 ∀ j, xi j ∈ {0, 1}. (19)

For a given average rate constraint R, the optimal control pol-icy is the solution of the following problem:

min∑

i

∑

j

xi j p′i j subject to

∑

i

∑

j

xi j r′i j > R. (20)

This is the classical multiple choice knapsack problem. Weare interested in the family of control policies correspondingto R ranging from 0 to Rmax ,Rmax being the maximum aver-age rate achievable on the link. We call this family the control


1

1.5

2

2.5

3

3.5

4

4.5

5

chan

nel

mer

itfo

rST

BC

1 1.5 2 2.5 3 3.5 4 4.5 5

channel merit for SDM

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 13: Histogram of the channel merit for STBC and SDM.

strategy. Let us denote as kj the index of the point selected onthe jth Pareto curve. Formally, kj = i ⇔ xi j = 1. A controlpolicy can be represented by the vector k = {kj}. The con-trol strategy, denoted {k(n)} corresponds to the set of points

{(R(n),P

(n))} in the average throughput versus average power

plane. A good approximation of the optimal control strategy(i.e., that bounds the tradeoff between R and P) can be de-rived iteratively with the greedy heuristic explained in [17].

5.3. Average throughput versus averageenergy-per-bit

From the knowledge of the Pareto optimal net throughputversus energy-per-bit tradeoff for each channel merit, nextto the channel merit probabilities one can now derive, givenan average rate constraint R, the control policy that mini-mizes the energy per bit. By having the constraint R rangingfrom 0 to its maximum achievable value Rmax , the averagethroughput versus average energy-per-bit tradeoff, when ap-plying the proposed policy-based adaptive transmission, canbe studied. The tradeoff (for each mode separately) is de-picted in Figure 12 for path losses 60, 70, 80, and 90 dB.

The results for SDM and STBC are compared with thetradeoff achieved with a SISO system. One can observe thatfor low path loss (60–70 dB), SISO reveals, on the average,to be the most energy efficient in almost the whole rangeit spans. SDM enables, however, a significant increase ofthe maximum average rate. STBC is irrelevant in this situ-ation. At average path loss (80 dB), a breakpoint rate (around20 Mbps) exists above which both SDM and STBC are moreenergy efficient than SISO, although SDM is still better thanSTBC. At high path loss (90 dB), STBC is the most efficientbetween 20 and 25 Mbps. It is though still beaten by SDM fordata rate beyond 25 Mbps and by SISO for smaller data rate.

6. SMARTMIMO

In the previous section, we have observed that STBC or SDMenable a significant average rate and/or range extension buthardly improve the energy efficiency. This is especially truewhen the average data rate is lower than 50% of the ergodiccapacity of the MIMO channel.

Based on that observation, in this section, we propose toextend the policy-based adaptive scheme not only to adaptthe transmission parameters with a fixed multiantenna en-coding, but also to vary the latter encoding on a packet-per-packet basis. Beside, since it has been observed that SISOtransmission is still most energy-efficient in certain condi-tion, it is also considered as a possible transmission mode inthe adaptive scheme.

Observing the histogram of the channel merits for thereference 802.11n channel model (Figure 13), one can noticethat the merit indexes of a given channel for STBC or SDMare weakly correlated. Since the energy efficiency of a givenmode is obviously better on a channel with a high merit, anaverage energy-efficiency improvement can be expected byletting the adaptation policy select one or the other trans-mission mode depending on the channel state.

6.1. Extended adaptation policy

The approach followed in Section 5 can be generalized tohandle multiantenna mode adaptation, besides the othertransmission parameters. For a given average path-loss and agiven average rate target, the adaptation policy will now mapa compound channel merit (namely, the triplets of channelmerit values for the three possible multiantenna mode) tothe system-level parameter settings, extended with the deci-sion on which multiantenna mode to use.

As previous, the adaptation policies are derived in twosteps.

Step 1. For each possible compound channel merit combi-nation (with our scale, 5 × 5 × 5 = 125 combinations), thePareto optimal tradeoff between throughput and energy perbit is derived. This tradeoff can be derived by combining thesingle-mode Pareto tradeoffs for the corresponding single-mode channel merits. This combined Pareto set correspondsbasically to the subset of nondominated points in the unionof the Pareto sets to be combined.

Step 2. Based on the throughput versus energy-per-bit trade-off for each compound channel merits and the knowledge ofthe compound channel merit probabilities, obtained againby Monte Carlo analysis of the physical-level channel model,one can derive the adaptation policy that minimizes the aver-age energy per bit for a given average throughput target. Thisderivation is identical as in Section 5.

In the remainder, we analyze the average throughput ver-sus energy-per-bit tradeoff achieved by an extended adap-tive transmission scheme and compare to the results fromSection 5.


40

42

44

46

48

50

52

54

56

Ave

rage

ener

gyp

erbi

t(n

J)

0 5 10 15 20 25 30 35



(a)

40

42

44

46

48

50

52

54

56

58

60

Ave

rage

ener

gyp

erbi

t(n

J)

0 5 10 15 20 25 30 35



(b)

40

45

50

55

60

65

70

Ave

rage

ener

gyp

erbi

t(n

J)

0 5 10 15 20 25 30 35



(c)

45

50

55

60

65

70

75

80

85

90A

vera

geen

ergy

per

bit(

nJ)

0 5 10 15 20 25 30



(d)

Figure 14: Average rate versus average energy-per-bit tradeoff for smartMIMO (bold line), superposed to the single-mode results. Theenergy efficiency is improved by up to 30% when compared to the single-mode results.

6.2. Average rate versus average energy-per-bit

By varying the average throughput constraint from 0 to themaximum achievable value (Rmax ), one can derive the setof extended control policies that lead to the Pareto optimaltradeoff between average throughput and average energy perbit. This tradeoff is depicted in Figure 14 for different pathlosses. The results of the multiantenna mode specific trade-off curves are superposed for the sake of comparison.

Globally, it can be observed that the tradeoff achievedwith the extended control policies always dominate thetradeoff achieved with the multiantenna mode-specific poli-cies. An average power reduction up to 30% can be ob-served. The resulting throughput-energy tradeoff even dom-

inates SISO in the whole range, meaning that smartMIMOalways brings a better energy per bit than any single mode.Moreover, this improvement does not affect the maximumthroughput and range extension provided, respectively, bySDM and STBC. The energy benefit comes from a betteradaptation to the channel conditions.

7. CONCLUSIONS

Multiantenna transmission techniques (MIMO) are be-ing adopted in most broadband wireless standard to im-prove wireless links spectrum efficiency and/or robustness.There exists a well-documented tradeoff between potentialspectrum efficiency and robustness increase. However, at


architecture level, multiantenna techniques also come withan overhead in power consumption due, at least, to the du-plication of part of the transmitter and receiver radio frontends. Therefore, from a system perspective, it is the trade-off between performance (e.g., the net throughput on topof the medium access control layer) and the average powerconsumption that really matters. It has been shown, in re-lated works, that, in the case of narrow band single-carriertransceivers, adaptive schemes were mandatory to avoid thatmultiantenna techniques hamper this system-level tradeoff.In the broadband case, orthogonal frequency division mul-tiplexing (OFDM) is usually associated with multiantennaprocessing. Adaptive schemes proposed so far for MIMO-OFDM optimize either the baseline physical layer through-put or the robustness in terms of bit-error rate. Energy ef-ficiency is generally disregarded as well as the effects intro-duced by the medium access control (MAC) layer.

In this paper, the impact of adaptive SDM-OFDM andSTBC-OFDM on the net data rate (on top of the MAC layer)versus energy-per-bit tradeoff has been analyzed and com-pared to adaptive SISO-OFDM. It has been shown that de-pending on the channel conditions, the one or the otherscheme can lead to the best tradeoff. Up to a path loss of80 dB, SISO always leads to the best energy efficiency up toa breakpoint rate (depending on the path loss) from whereSDM is the most energy efficient. STBC improves the energyefficiency in a significant range of data rates only in case oflarge path loss (>90 dB).

Next, we derived and discussed SmartMIMO, an adaptivemultiantenna scheme that controls, packet-per-packet, thebasic OFDM links parameters (carrier modulation, forwarderror correction coding rate) as well as the type of multiple-antenna encoding (SISO, SDM, or STBC) in order to opti-mize the link net data rate (on top of the MAC) versus en-ergy efficiency tradeoff. Based on a model calibrated on anexisting multiantenna transceiver setup, the link energy effi-ciency with the proposed scheme is shown to be improvedby up to 30% when compared to nonadaptive schemes. Theaverage rate is, on the other hand, improved by up to 50%when compared to single-antenna transmission.

ACKNOWLEDGMENTS

This work has been partially published in the Proceedingof the 20th IEEE Workshop on Signal Processing Systems(SiPS06) in October 2006. The project has been partiallysupported by Sony Corporation, the Samsung AdvancedInstitute of Technology (SAIT), and the Flemish Institutefor BroadBand Telecommunication (IBBT, Ghent, Belgium).Bruno Bougard was Research Assistant delegated by the Bel-gian Foundation for Scientific Research (FWO) until October2006.

REFERENCES

[1] A. B. Gershman, “Robust adaptive beamforming: an overviewof recent trends and advances in the field,” in Proceedings ofthe 4th International Conference on Antenna Theory and Tech-niques (ICATT ’03), vol. 1, pp. 30–35, Sevastopol, Ukraine,September 2003.


[3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-timeblock codes from orthogonal designs,” IEEE Transactions onInformation Theory, vol. 45, no. 5, pp. 1456–1467, 1999.

[4] G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. Wolni-ansky, “Simplified processing for high spectral efficiency wire-less communication employing multi-element arrays,” IEEEJournal on Selected Areas in Communications, vol. 17, no. 11,pp. 1841–1852, 1999.

[5] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fun-damental tradeoff in multiple-antenna channels,” IEEE Trans-actions on Information Theory, vol. 49, no. 5, pp. 1073–1096,2003.

[6] M. Zargari, D. K. Su, C. P. Yue, et al., “A 5-GHz CMOStransceiver for IEEE 802.11a wireless LAN systems,” IEEE Jour-nal of Solid-State Circuits, vol. 37, no. 12, pp. 1688–1694, 2002.

[7] A. Behzad, L. Lin, Z. M. Shi, et al., “Direct-conversion CMOStransceiver with automative frequency control for 802.11awireless LANs,” in Proceedings of IEEE International Solid-StateCircuits Conference (ISSCC ’03), vol. 1, pp. 356–357, San Fran-cisco, Calif, USA, February 2003.

[8] B. Debaillie, B. Bougard, G. Lenoir, G. Vandersteen, andF. Catthoor, “Energy-scalable OFDM transmitter design andcontrol,” in Proceedings of the IEEE 43rd Design AutomationConference (DAC ’06), pp. 536–541, San Francisco, Calif, USA,July 2006.

[9] C. Shuguang , A. J. Goldsmith, and A. Bahai, “Energy-constrained modulation optimization for coded systems,” inProceedings of IEEE Global Telecommunications Conference(Globecom ’03), vol. 1, pp. 372–376, San Francisco, Calif, USA,December 2003.

[10] W. Liu, X. Li, and M. Chen, “Energy efficiency of MIMO trans-missions in wireless sensor networks with diversity and multi-plexing gains,” in Proceedings of IEEE International Conferenceon Acoustics, Speech and Signal Processing (ICASSP ’05), vol. 4,pp. 897–900, Philadelphia, Pa, USA, March 2005.

[11] X. Li, M. Chen, and W. Liu, “Application of STBC-encodedcooperative transmissions in wireless sensor networks,” IEEESignal Processing Letters, vol. 12, no. 2, pp. 134–137, 2005.

[12] R. W. Heath Jr. and A. J. Paulraj, “Switching between diver-sity and multiplexing in MIMO systems,” IEEE Transactionson Communications, vol. 53, no. 6, pp. 962–972, 2005.

[13] S. Catreux, V. Erceg, D. Gesbert, and R. W. Heath Jr., “Adaptivemodulation and MIMO coding for broadband wireless datanetworks,” IEEE Communications Magazine, vol. 40, no. 6, pp.108–115, 2002.

[14] M. H. Halmi and D. C. H. Tze, “Adaptive MIMO-OFDM com-bining space-time block codes and spatial multiplexing,” inProceedings of the 8th IEEE International Symposium on SpreadSpectrum Techniques and Applications (ISSSTA ’04), pp. 444–448, Sydney, Australia, August 2004.

[15] M. Codreanu, D. Tujkovic, and M. Latva-aho, “AdaptiveMIMO-OFDM systems with estimated channel state infor-mation at TX side,” in Proceedings of IEEE International Con-ference on Communications (ICC ’05), vol. 4, pp. 2645–2649,Seoul, Korea, May 2005.

[16] P. Xia, S. Zhou, and G. B. Giannakis, “Adaptive MIMO-OFDMbased on partial channel state information,” IEEE Transactionson Signal Processing, vol. 52, no. 1, pp. 202–213, 2004.

[17] B. Bougard, S. Pollin, A. Dejonghe, F. Catthoor, and W. De-haene, “Cross-layer power management in wireless networks


and consequences on system-level architecture,” Signal Pro-cessing, vol. 86, no. 8, pp. 1792–1803, 2006.

[18] M. Wouters, T. Huybrechts, R. Huys, S. De Rore, S. Sanders,and E. Umans, “PICARD: platform concepts for prototyp-ing and demonstration of high speed communication sys-tems,” in Proceedings of the 13th IEEE International Workshopon Rapid System Prototyping (IWRSP ’02), pp. 166–170, Darm-stadt, Germany, July 2002.

[19] IEEE Std 802.11a, “Part 11: wireless LAN medium access con-trol (MAC) and physical layer (PHY) specifications,” IEEE,1999.

[20] S. Valle, A. Poloni, and G. Villa, “802.11 TGn proposal forPHY abstraction in MAC simulators,” IEEE 802.11, 04/0184,February, 2004, ftp://ieee:[email protected]/11/04/11-04-0184-00-000n-proposal-phy-abstraction-in-mac-simulators.doc.

[21] C. Schurgers, V. Raghunathan, and M. B. Srivastava, “Powermanagement for energy-aware communication systems,”ACM Transactions on Embedded Computing Systems, vol. 2,no. 3, pp. 431–447, 2003.

[22] V. Erceg, L. Schumacher, P. Kyritsi, et al., “TGn channel mod-els,” Tech. Rep. IEEE 802.11-03/940r4, Wireless LANs, 2004.

[23] K. M. Miettinen, Non-Linear Multi-Objective Optimization,Kluwer Academic, Boston, Mass, USA, 1999.


Research ArticleCross-Layer Admission Control Policy for CDMABeamforming Systems

Wei Sheng and Steven D. Blostein

Department of Electrical and Computer Engineering, Queen’s University, Walter Light Hall (19 Union Street),Kingston, Ontario, Canada K7L 3N6

Received 31 October 2006; Revised 24 June 2007; Accepted 1 August 2007


A novel admission control (AC) policy is proposed for the uplink of a cellular CDMA beamforming system. An approximatedpower control feasibility condition (PCFC), required by a cross-layer AC policy, is derived. This approximation, however, increasesoutage probability in the physical layer. A truncated automatic retransmission request (ARQ) scheme is then employed to mitigatethe outage problem. In this paper, we investigate the joint design of an AC policy and an ARQ-based outage mitigation algorithmin a cross-layer context. This paper provides a framework for joint AC design among physical, data-link, and network layers.This enables multiple quality-of-service (QoS) requirements to be more flexibly used to optimize system performance. Numericalexamples show that by appropriately choosing ARQ parameters, the proposed AC policy can achieve a significant performancegain in terms of reduced outage probability and increased system throughput, while simultaneously guaranteeing all the QoSrequirements.

Copyright © 2007 W. Sheng and S. D. Blostein. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

In a code division multiple access (CDMA) system, quality-of-service (QoS) requirements rely on interference mitiga-tion schemes and resource management, such as power con-trol, multiuser detection, and admission control (AC) [1–3]. Recently, the problem of ensuring QoS by integratingthe design in the physical layer and the admission control(AC) in the network layer is receiving much attention. In[4, 5], an optimal semi-Markov decision process (SMDP)-based AC policy is presented based on a linear-minimum-mean-square-error (LMMSE) multiuser receiver for constantbit rate traffic and circuit-switched networks. In [6], optimaladmission control schemes are proposed in CDMA networkswith variable bit rate packet multimedia traffic.

The above algorithms [4–6] integrate the optimal ACpolicy with a multiuser receiver, and as a result, are able tooptimize the power control and the AC across the physicaland network layers. However, [4–6] only consider single an-tenna systems, which lack the tremendous performance ben-efits provided by multiple antenna systems [7–17]. Further-more, [4–6] rely on an asymptotic signal-to-interference ra-tio (SIR) expression proposed in [18] which requires a largenumber of users and a large processing gain. This specific

signal model limits the application of the proposed AC poli-cies. Motivated by these facts, in this paper, we investigatecross-layer AC design for an arbitrary-size CDMA systemwith multiple antennas at the base station (BS).

To derive an optimal AC policy, a feasible state space andexact power controllability are required but are hard to eval-uate for the case of multiple antenna systems. This motivatesan approximated power control feasibility condition (PCFC)proposed for admission control of a multiple antenna sys-tem. This approximation, however, introduces outage in thephysical layer, for example, a nonzero probability that a tar-get signal-to-interference ratio (SIR) cannot be satisfied. Toreduce the outage probability in the physical layer, a trun-cated ARQ-based reduced-outage-probability (ROP) algo-rithm can be employed. Truncated ARQ is an error-controlprotocol which retransmits an error packet until correctly re-ceived or a maximum number of retransmissions is reached.It is well known that retransmissions can significantly im-prove transmission reliability, and as a result, can reduce theoutage probability. Although retransmissions increase thetransmission duration of a packet and thus degrade the net-work layer performance, this degradation can be controlledto an arbitrarily small level by appropriately choosing the pa-rameters of a truncated ARQ scheme, such as the maximum


number of allowed retransmissions and target packet-errorrate (PER).

To date, there is no research on cross-layer AC de-sign which considers both link-layer error control schemesand multiple antennas. We remark that this paper differsfrom prior investigations, for example, [4–6], in the fol-lowing aspects: (a) here multiple antenna systems are in-vestigated which provide a large capacity gain, while in[4–6], only single antenna systems are discussed; (b) inthis paper, a cross-layer AC policy is designed by includingerror-control schemes, while in [4–6], no such error con-trol schemes are exploited; (c) prior investigations in [4–6] rely on a large system analysis which requires an infi-nite number of users and infinite length spreading sequences,while here, no such requirements are imposed. In sum-mary, this paper provides a framework for joint optimiza-tion across physical, data-link, and network layers, and as aresult, is capable of providing a flexible way to handle QoSrequirements.

We remark that in the current third generation (3G) sys-tem, the application of more efficient methods for packetdata transmission such as high-speed uplink packet access(HSUPA) has become more important [19]. In HSUPA, athreshold-based call admission control (CAC) policy is em-ployed, which admits a user request if the load reported isbelow the CAC threshold. Although the CAC decision can beimproved upon by taking advantage of resource allocationinformation [19], and it is simple to implement, it is wellknown that the threshold-based CAC policy cannot satisfyQoS requirements in the network layer [5]. Our proposedAC policy provides a solution to guarantee the QoS require-ments in both physical and network layers.

The proposed AC policy can be derived offline and thenstored in a lookup table. Whenever an arrival or departureoccurs, an optimal action can be obtained by table lookup,resulting in low enough complexity for admission controlat the packet level. Similar to call/connection level admis-sion control, in a packet-switched system, a packet admissioncontrol policy decides if an incoming packet can be acceptedor blocked in order to meet quality-of-service (QoS) require-ments. In a packet-switched network, blocking a packet in-stead of blocking the whole user connection can be morespectrally efficient. In this paper, we consider the packet levelAC problem.

The rest of this paper is organized as follows. In Section 2,we present the signal model. In Section 3, an approximatedPCFC and ARQ-based ROP algorithm are discussed. The for-mulation and solution of Markov-decision-process (MDP)-based AC policies are proposed in Section 4. Section 5 sum-marizes the cross-layer design of ARQ parameters. Simula-tion results are then presented in Section 6.

We will use the following notation: ln x is the natu-ral logarithm of x, and ∗ denotes convolution. The super-scripts (·)H and (·)t denote hermitian and transpose, re-spectively; diag(a1, . . . , an) denotes a diagonal matrix withelements a1, . . . , an, and I denotes an identity matrix. Fora random variable X , E[X] is its expectation. The nota-tion and definitions used in this paper are summarized inTable 1.

Table 1: Notation and definitions.

Notation Definition

M Number of antennas at the BS

K Number of users

J Number of classes

Ri Data rate for packet i

pi Transmitted power for packet i

B Bandwidth

Gi Link gain for packet i

ai Array response vector for packet i

λj Arrival rate for class j

μj Departure rate for class j

Ψ j Blocking probability constraint for class j

Dj Connection delay constraint for class j

Lj Maximum number of retransmissions for class j

ρ j Target PER for class j

PERjoverall Achieved overall PER for class j

PERjin Achieved instantaneous PER for class j

γ j Target SIR for class j

Bj Buffer size for class j

wi Beamformer weight for packet i

η0One-sided power spectral density of additive white

Gaussian noise (AWGN)

2. SIGNAL MODEL AND PROBLEM FORMULATION

2.1. Signal model at the physical layer

We consider an uplink CDMA beamforming system, inwhich M antennas are employed at the BS and a single an-tenna is employed for each packet. There are K acceptedpackets in the system, and a channel with slow fading is as-sumed.

To highlight the design across physical and upper layersconsidered in this paper, the effects due to multipath are ne-glected. However, the proposed schemes in this paper canbe extended straightforwardly to the case where multipathexists, provided multipath delay profile information is avail-able.

The received vector at the BS antenna array can be writ-ten as

x(t) =K∑

i=1

√PiGiaisi

(t − τi

)+ n(t), (1)

where Pi and Gi denote the transmitted power and link gainfor packet i, respectively; ai is defined as the array responsevector for packet i, which contains the relative phases of thereceived signals at each array element, and depends on the ar-ray geometry as well as the angle of arrival (AoA); si(t) is thetransmitted signal, given by si(t) =

∑nbi(n)ci(t−nT), where

bi(n) is the information bit stream, and ci(t) is the spreadingsequence; τi is the corresponding time delay and n(t) is thethermal noise vector at the input of antenna array.

W. Sheng and S. D. Blostein 3

It has been shown that the output of a matched filter sam-pled at the symbol interval is a sufficient statistic for the es-timation of the transmitted signal [14]. The matched filterfor a desired packet k is given by cHk (−t). The output of thematched filter is sampled at t = nT , where T denotes sym-bol interval. Hence, the received signal at the output of thematched filter is given by [14]

xk(n) = x(t)∗cHk (−t)|t=nT

=K∑

i=1

√PiGiai

∫ nT+τk

(n−1)T+τk

∑

m

bi(m)ci(t −mT − τi

)

× ck(t − nT − τk

)dt + nk(n),

(2)

where nk(n) = n(t)∗cHk (−t)|t=nT .In order to reduce the interference, we employ a beam-

forming weighting vector wk for a desired packet k. We canwrite the output of the beamformer as

yk(n)

= wHk x(n)

=K∑

i=1

√PiGiwH

k ai

∫ nT+τk

(n−1)T+τk

∑

m

bi(m)ci(t −mT − τi

)

×ck(t − nT − τk

)dt + wH

k nk(n).(3)

We assume the signature sequences of the interferingusers appear as mutually uncorrelated noise. As shown in[14], the received signal-to-interference ratio (SIR) for a de-sired packet k can be written as

SIRk = B

Ri

pkφ2kk∑

l�=i plφ2il + η0B

, (4)

where B and Ri denote the bandwidth and data rate forpacket i, respectively, and the ratio B/Ri represents the pro-cessing gain; pi = PiG

2i denotes the received power for packet

i, and η0 denotes the one-sided power spectral density ofbackground additive white Gaussian noise (AWGN); the pa-rameters φ2

ii and φ2ik are defined as

φ2ik =

∣∣wHk ai

∣∣2(5)

which capture the effects of beamforming. In the following,we consider a spatially matched filter receiver, for example,wk = ak.

QoS requirements in the physical layer

In a wireless communication network, we must allow foroutage, defined as the probability that a target SIR, or equiv-alently, a target packet-error rate (PER), cannot be satisfied.The QoS requirement in the physical layer can be representedby a target outage probability.

In this paper, we rely on a relationship between a targetSIR and a target PER. Although an exact relationship maynot be available, we can obtain the target SIR according to

an approximate expression of PER. As discussed in [20], ina system with packet length Np (bits), the target SIR for adesired packet i, denoted by γi, can be approximated by

γi =1g

[ln a− ln ρi

](6)

for γi ≥ γ0 dB, where ρi denotes the overall target PER; a,g, and γ0 are constants depending on the chosen modulationand coding scheme. In the above expression, the interferenceis assumed to be additive white Gaussian noise, which is rea-sonable in a system with enough interferers.

2.2. Signal model in data-link and network layers

We consider a single-cell CDMA system which supports Jclasses of packets, characterized by different target PERs ρj ,different blocking probability requirements Ψ j , and differentconnection delay requirements Dj , where j = 1, . . . , J . Re-quests for packet connections of class j are assumed to bePoisson distributed, with arrival rates λj , j = 1, . . . , J .

The admission control (AC) is performed at the BS. AnAC policy is derived offline, and stored in a lookup table.When a packet is generated at the mobile station (MS), theMS sends an access request to the BS. In this request, theclass of this packet is indicated. After receiving the request,the BS makes a decision, which is then sent back to the MS,on whether the incoming packet should be either accepted,queued in the buffer, or blocked. Similarly, whenever a packetdeparts, the BS decides whether the packet in the queue canbe served (transmitted).

Once a packet is accepted, its first transmission roundwill be performed, and then the receiver will send back anacknowledgement (ACK) signal to the transmitter. A posi-tive ACK indicates that the packet is correctly received whilea negative ACK indicates an incorrect transmission.

If a positive ACK is received or the maximum number ofretransmissions, denoted by L, is reached, the packet releasesthe server and departs. Otherwise, the packet will be retrans-mitted. Therefore, the service time of a packet can compriseat most L+ 1 transmission rounds. Each transmission roundincludes the actual transmission time of the packet and thewaiting time of an ACK signal (positive or negative). The du-ration of a transmission round for a packet in class j is as-sumed to have an exponential distribution with mean dura-tion 1/μj , j = 1, . . . , J . However, in this paper, a sub-optimalsolution is also provided for a generally distributed duration.

If the packet is not accepted by the AC policy, it will bestored in a queue buffer provided that the queue buffer isnot full. Otherwise, the packet will be blocked. Each class ofpackets shares a common queue buffer, and Bj denotes thequeue buffer size of class j.

The QoS requirements in the network layer can be rep-resented by the target blocking probability and connectiondelay, denoted by Ψ j and Dj for class j, respectively. For eachclass j, where j = 1, . . . , J , there are Kj packets physicallypresent in the system, which have the same target packet-error-PER, blocking probability, and connection delay con-straints.


We note that there are two types of buffers in the system:queue buffers and server buffers. The queue buffer accom-modates queued incoming packets, while the server bufferaccommodates transmitted packets in the server in case anypacket in the server requires retransmission. For simplicity,we assume that the size of the server buffer is large enoughsuch that all the packets in the server can be stored. In the fol-lowing, the generic term “buffer” refers to the queue buffer.

2.3. Problem formulation

The AC policy considered in this paper is for the uplinkonly. However, with an appropriate physical layer model forpower allocation, the methodology can be extended straight-forwardly to the downlink AC problem. The uplink AC isperformed at the BS, and the following information is nec-essary to derive an admission control policy: traffic model inthe system, such as arrival and departure rate, and QoS re-quirements in both physical and network layers.

The overall system throughput is defined as the numberof correctly received packets per second, given by

Throughput =J∑

j=1

(1− P j

b

)(1− ρj

)(1− P j

out)λj , (7)

where Pjb, ρj and P

jout denote the blocking probability, target

PER, and outage probability for class j packets, respectively.In this paper, we aim to derive an optimal AC policy

which incorporates the benefits provided by multiple an-tennas and ARQ schemes. The objective is to maximize theoverall system throughput given in (7), while simultaneouslyguaranteeing QoS requirements in terms of outage probabil-ity, blocking probability, and connection delay.

The above optimization problem can be formulated as aMarkov decision process (MDP). With a required power con-trol feasibility condition (PCFC), combined with an ARQ-based reduced-outage-probability (ROP) algorithm, a targetoutage probability constraint can be satisfied. Blocking prob-ability and connection delay requirements can be guaranteedby the constraints of this MDP.

In the following, we first derive an approximate PCFCcombined with an ARQ-based reduced-outage-probability(ROP) algorithm that can guarantee the outage probabilityconstraint. Based on these results, we then formulate the ACproblem as a Markov decision process. Afterward, we discusshow to design ARQ parameters optimally in order to achievea maximum system throughput.

3. PHYSICAL LAYER INVESTIGATION: PCFCDERIVATION AND OUTAGE REDUCTION

To investigate the physical layer performance, we must de-rive an approximate PCFC, which ensures a positive powersolution to achieve target SIRs. Due to the approximation ofthe derived PCFC, we then propose an ARQ-based ROP al-gorithm to reduce the resulting outage probability.

3.1. PCFC

In the physical layer, the SIR requirements of packet i can bewritten as

SIRi ≥ γi (8)

for i = 1, . . . ,K , where SIRi is given in (4).Inserting the SIR expression in (4) into (8), and letting

SIRi achieve its target value, γi, we have the matrix form [15]

[I−QF]p = Qu, (9)

where I is the identity matrix, p = [p1, . . . , pK ]t, u =η0B[1, . . . , 1]t,

Q = diag{

γ1R1/B

1 + γ1R1/B, . . . ,

γKRK/B

1 + γKRK/B

},

F =

⎡⎢⎢⎢⎣

F1,1 F1,2 · · · F1,K

F2,1 F2,2 · · · F2,K

· · · · · · · · · · · ·FK ,1 FK ,2 · · · FK ,K

⎤⎥⎥⎥⎦

(10)

in which Fi j = φ2i j /φ

2ii.

To ensure a positive solution for power vector p, we re-quire the following power control feasibility condition [15],

ρ(QF) < 1, (11)

where ρ(·) denotes the maximum eigenvalue.The outage probability can be obtained as the probabil-

ity that the above condition is violated. Although the statespace, required by an optimal AC policy, can be formulatedby evaluating the above outage probability, this evaluationrelies on the number of packets as well as the distribution ofAoAs for all the packets in the system, and thus results in avery high computation complexity. An approach to evaluatethe above outage probability with reasonably low complexityis currently under investigation.

In this paper, we propose an alternative solution, whichemploys an approximated PCFC, and as a result can dramat-ically simplify the formulation of the state space.

Without loss of generality, we consider an arbitrarypacket i in class 1, where i = 1, . . . ,K1. By considering spe-cific traffic classes and letting SIR achieve its target value, theexpression in (4) can be written as

γi =piφ

2ii

(B/R1

)

∑ K1l=1,l�=i plφ

2il +

∑ K2l=1plφ

2il + · · ·∑ KJ

l=1plφ2il + σ2

,

(12)

where σ2 � η0B denotes noise variance, and pi representsreceived power for packet i.

It is not difficult to show that packets in the sameclass have the same received power. By denoting the re-ceived power in class j as pj , where j = 1, . . . , J , the above


expression can be written as

γi =p1φ

2ii

(B/R1

)

∑ K1l=1,l�=i p1φ

2il + · · · +

∑ KJl=1pJφ

2il + σ2

= p1φ2ii

(B/R1

)

p1(K1 − 1

)β1 +

∑ Jj=2pjKjβ j + σ2

,

(13)

where β1 = (1/(K1 − 1))∑ K1

l=1,l�=iφ2il and βj = (1/Kj)

∑ Kj

l=1φ2il,

in which j = 2, . . . , J .By exchanging the numerator and denominator, (13) is

equivalent to

p1(K1 − 1

)β1 +

∑ Jj=2pjKjβ j + σ2

p1(B/γ1R1

) = φ2ii, (14)

where i = 1, . . . ,K1.Summing the above K1 equations, and calculating the

sample average, we obtain

p1(K1 − 1

)α1 +

∑ Jj=2Kj pjαj + σ2

p1(B/γ1R1

) = 1K1

K1∑

i=1

φ2ii, (15)

where α1 = (1/K1)∑ K1

i=1β1 and αj = (1/K1)∑ K1

i=1βj .When the number of packets is large enough, by the weak

law of large numbers, the above α1, . . . ,αJ can be approxi-mated by their mean values, and (15) can be further simpli-fied as

p1(K1 − 1

)E11

[φint

]+

∑ Jj=2Kj pjE1 j

[φint

]+ σ2

p1(B/γ1R1

) = E1[φdes

]

(16)

in which Emn[φint] is the expected fraction of an interfererpacket in class n passed by a beamforming weight vectorfor a desired packet in class m, where m,n = 1, . . . , J , whileEj[φdes] is the expected fraction of a desired packet in class jpassed by its beamforming weight vector, where j = 1, . . . , J .

The AoAs of active packets in the system are assumed tobe independent and identically distributed, that are indepen-dent of a packet’s specific class. Therefore, it is reasonable toassume that Emn[φint] is also independent of specific classesm and n, which can be denoted by E[φint]. Similarly, Ej[φdes]is independent of class j, and can be denoted by E[φdes].E[φdes] and E[φint] represent the expected fractions of thedesired packet’s power and interference, respectively.

From the above discussion, (16) can be written as

p1(K1 − 1

)E[φint

]+

∑ Jj=2Kj pjE

[φint

]+ σ2

p1(B/γ1R1

) = E[φdes

].

(17)

By exchanging the numerator and denominator of theabove equation, we have

p1B

γ1R1

/(p1

(K1 − 1

) E[φint

]

E[φdes

]

+J∑

j=2

Kj pjE[φint

]

E[φdes

] +σ2

E[φdes

])= 1.

(18)

The QoS requirement for class 1 in (18) can be extendedto any class j,

pjB

γ jRj

/(pj

(Kj − 1

) E[φint

]

E[φdes

]

+J∑

m=1,m�= jKmpm

E[φint

]

E[φdes

] +σ2

E[φdes

])= 1,

(19)

where j = 1, . . . , J .The power allocation solution can be obtained by solving

the above J equations [21]

pj = σ2

E[φint

]/((

1 +B

γ jRj(E[φint

]/E

[φdes

]))

×[

1−J∑

j=1

Kj

1 +(B/γ jRj

(E[φint

]/E

[φdes

]))])

,

(20)

where j = 1, . . . , J .Positivity of the power solution implies the following

power control feasibility condition:

J∑

j=1

Kj

1 +(B/γ jRj

(E[φint

]/E

[φdes

])) < 1. (21)

As shown in [22], E[φint] and E[φdes] can be determinednumerically from (5) for a beamforming system.

We note that the above approximated power control fea-sibility condition is independent of the angle of arrivals, andthus can provide a less-complicated offline AC policy, whichdoes not require estimation of the current AoA realizationsof each packet. However, due to the randomness of the ac-tual SIR, this deterministic power control feasibility condi-tion introduces outage. In the next section, we discuss howto mitigate the outage.

3.2. ARQ-based ROP

We first define two types of PERs. The overall achieved PER,

denoted by PERjoverall, is defined as the probability that a class

j packet is incorrectly received after its maximum number ofARQ retransmissions is reached, for example, an error occursin each of the Lj + 1 transmission rounds, where Lj denotesthe maximum number of retransmissions. The achieved in-stantaneous PER, denoted as PER

jin(l), is defined as the prob-

ability that an error occurs in a single transmission round lfor a class j packet.

Under the assumption that each retransmission round isindependent from the others, by using an ARQ scheme witha maximum of Lj retransmissions for class j, the achievedoverall PER is constrained by [20]

PERjoverall =

Lj+1∏

l=1

PERjin(l),

≤ ρj ,

(22)

where ρj denotes the target overall PER for class j.


The achieved outage probability for class j, denoted by

Pjout, can be written as

Pjout = Prob

{PERoverall

j > ρj

}

= Prob

{Lj+1∏

l=1

PERjin(l) > ρj

},

(23)

where Prob{A} denotes the probability of event A. By main-taining PCFC, PERin

j (l) remains unchanged. Therefore, byincreasing Lj , the outage probability in the above equationcan be reduced.

4. AC PROBLEM FORMULATION BY INCLUDING ARQ

In the previous section, we have derived an approximatedPCFC combined with an ARQ-based ROP algorithm in thephysical layer. In the following, we discuss how to derive anAC policy in the network layer.

An optimal semi-Markov decision process (SMDP)-based AC policy as well as a low-complexity generalized-Markov decision process (GMDP)-based AC policy is dis-cussed.

4.1. SMDP-based AC policy

Traditionally, the decision epoches are chosen as the time in-stances that a packet arrives or departs. In the system underconsideration, the duration of each packet may include sev-eral transmission rounds due to ARQ retransmissions, and asa result, the time duration until next system state may not beexponentially distributed. Therefore, the SMDP formulationapproach discussed in [4–6], which assumes an exponentiallydistributed duration, cannot be applied here.

In the following, we propose a novel formulation inwhich the decision epoch is chosen as the arrival and de-parture of each transmission round. Based on these decisionepoches, the time duration until the next state remains ex-ponentially distributed. The components of a Markov deci-sion process, such as state space, action space, and dynamicstatistics, are modified accordingly to represent the charac-teristics of different transmission rounds. The formulationof this SMDP as well as its LP solution are now described.

State space and action space

Class j packets are divided into Lj +1 subclasses, in which thestate of the ith subclass can be represented by the number ofpackets which are under the ith round transmission, that is,the (i− 1)th retransmission, where i = 1, . . . ,Lj + 1.

In admission problems, the discrete-value (finite) state attime t, s(t), can be written as

s(t) =[n1q(t), k1,1(t), . . . , k1,L1+1(t)

︸︷︷︸, . . . ,

nJq(t), kJ ,1(t), . . . , kJ ,LJ+1(t)︸︷︷︸

]T,

(24)

where k j,i(t) represents the number of active packets in class

j and subclass i served in the system, and njq(t) denotes the

number of packets in the queue buffer of class j. Since thearrival and departure of packets are random, {s(t), t > 0}represents a finite state stochastic process [4]. From here on,we will drop the time index.

The state space S is comprised of any state vector s, inwhich SIR requirements can be satisfied or, equivalently, thepower control feasibility condition (PCFC) holds,

S ={s : n

jq ≤ Bj , j = 1, . . . , J ;

J∑

j=1

(∑ Lj+1

l=1k j,l

)

1 +(B/γ jRj

(E[φint

]/E

[φdes

])) < 1

},

(25)

where Bj denotes the buffer size of class j. We have men-tioned that the PCFC for the case of no ARQ is used in ourAC problem, no matter how many retransmissions are al-lowed.

At each state s, an action is chosen that determines howthe admission control will perform at the next decision mo-ment [4]. In general, an action, denoted as a, can be definedas a vector of dimension

∑ Jj=1Lj + 2J ,

a =[a1,d1

1, . . . ,dL1+11︸︷︷︸

. . . , aJ ,d1J , . . . ,d

LJ+1J︸︷︷︸

]T, (26)

where aj denotes the action for class j if an arrival occurs,j = 1, . . . , J . If aj = 0, the new arrival is placed in the bufferprovided that the buffer is not full or is blocked if the bufferis full; if aj = 1, the arrival is admitted as an active packet,and the number of servers of class j is incremented by one.

The quantity dij , where 1 ≤ i ≤ Lj , denotes the actionfor class j packet if the ith transmission round is finished,and is received correctly. If dij = 0, where 1 ≤ i ≤ Lj , k j,i isdecremented by one, and no packets that are queued in thebuffer are made active; if dij = 1, the number of servers ismaintained by admitting a packet at the buffer as an activepacket.

The quantity dLj+1j denotes the action for class j packet if

a connection has finished its (Lj+1)th transmission round. If

dLj+1j = 0, no packets that are queued in the buffer are made

active, and k j,Lj+1 is decremented by one; if dLj+1j = 1, the

number of servers is maintained by admitting a packet at thebuffer as an active packet.

The admissible action space for state s, denoted byAs, canbe defined as the set of all feasible actions. A feasible actionensures that after taking this action, the next transition stateis still in space S [4].

State dynamics psy(a) and τs(a)

The state dynamics of an SMDP are completely specified bystating the transition probabilities of the embedded chainpsy(a) and the expected holding time τs(a) : psy(a) is definedas the probability that the state at the next decision epoch is


Table 2: Expression of transition probability psy .

y psy(a)

y = s + q j λjajτs(a)

y = s + bj λj(1− aj)δ(Bj − njq)τs(a)

y = s + cji

(1− ρj)[μjkj,i(1− dij)τs(a)]

+(1− ρj)[μjkj,idij(1− δ(n

jq))τs(a)]

y = s + r j∑ Lj+1

i=1 (1− ρj)μjk j,idijτs(a)δ(njq)

y = s + eji ρ jμjk

j,iτs(a)

y = s + f j μjkj,Lj+1d

Lj+1j δ(n

jq)τs(a)

y = s + g jμjk

j,Lj+1(1− dLj+1j )τs(a)

+μjkj,Lj+1d

Lj+1j (1− δ(n

jq))τs(a)

Otherwise 0

y if action a is selected at the current state s, while τs(a) is theexpected time until the next decision epoch after action a ischosen in the present state s [4].

Derivations of τs(a) and psy(a) rely on the statisticalproperties of arrival and departure processes [4]. Since thearrival and departure processes are both Poisson distributedand mutually independent, it follows that the cumulativeprocess is also Poisson, and the cumulative event rate is thesum of the rates for all constituent processes [4]. Therefore,the expected sojourn time, τs(a), can be obtained as the in-verse of the event rate,

τs(a)−1 = λ1a1 + λ1(1− a1

)δ(B1 − n1

q

)

+L1+1∑

i=1

μ1

(k1,i) + · · · + λJaJ + λJ

(1− aJ

)δ(BJ − nJq

)

+LJ+1∑

i=1

μJ(kJ ,i

),

(27)

where

δ(z) ={

1 if z > 0,

0 if z = 0.(28)

To derive the transition probabilities, we employ the de-composition property of a Poisson process, which states thatan event of a certain type occurs with a probability equal tothe ratio between the rate of that particular type of event andthe total cumulative event rate 1/(τs(a)) [4]. Transition prob-ability psy(a) is shown in Table 2, where ρj denotes the tar-get packet-error rate for class j packets. The set of vectors

{qj , bj , c ji , r j , eji , f

j , gj} represents the possible state transi-tions from current state s. Each vector in this set has a dimen-sion of

∑ Jj=1Lj + 2J , and contains only zeros except for one

or two positions. The nonzero positions of this set of vectors,as well as the possible state transitions represented by thesevectors, are specified in Tables 3 and 4, respectively.

Policy and cost criterion

For any given state s ∈ S, an action a, which decides if thenew packet at the next decision epoch will be blocked or ac-

Table 3: Definition of vectors in Table 2: each vector defined in thistable has a dimension of

∑ Jj=1Lj + 2J , which contains only zeros

except for the specified positions.

Vector Nonzero positions

q j Position 2( j − 1) +∑ j−1

t=1Lt + 2 contains a 1

bj Position 2( j − 1) +∑ j−1

t=1Lt + 1 contains a 1

cji Position 2( j − 1) +

∑ j−1t=1Lt + i + 1 contains a −1

r j Position 2( j − 1) +∑ j−1

t=1Lt + 1 contains a −1

eji

Position 2( j − 1) +∑ j−1

t=1Lt + i + 1 contains a −1

and position 2( j − 1) +∑ j−1

t=1Lt + i + 2 contains a 1

f j Position 2( j − 1) +∑ j−1

t=1Lt + 1 contains a −1

g j Position 2( j − 1) +∑ j−1

t=1Lt + Lj + 2 contains a −1

Table 4: Representation of vectors in Table 2: each defined vectorrepresents a possible state transition from current state s.

Notation State transition

s + q j An increase in subclass 1 of class j by 1

s + bj An increase in queue j by 1

s + cji A decrease in subclass i of class j by 1

s + r j A decrease in queue j by 1

s + eji

An increase of subclass i + 1 by 1,

and a decrease in subclass i of class j by 1

s + f j A decrease in queue j by 1

s + g j A decrease in subclass Lj + 1 of class j by 1

cepted, is selected according to a specified policy R. A station-ary policy R is a function that maps the state space into theadmissible action space.

We consider average cost criterion [4]. The cost criterionfor a given policy R and initial state s0, which includes block-ing probability as a special case, is given as follows:

JR(s0

) = limt→∞

1TE

{∫ T

0c(s(t), a(t)

)dt

}, (29)

where c(s(t), a(t)) can be interpreted as the expected cost un-til the next decision epoch and is selected to meet the net-work layer performance criteria [4].

In the system under investigation, we are interested inblocking probability and connection delay constraints. If thecost criterion JR(s0) represents blocking probability, we have

c(s, a) = (1 − aj)(1 − δ(Bj − njq)), and if the cost criterion

JR(s0) represents connection delay, we have c(s, a) = njq.

An optimal policy R∗ that minimizes an average cost cri-terion JR(s0) for any initial state s0 exists,

JR∗(s0) = minR∈R

JR(s0), ∀s0 ∈ S (30)

under the weak unichain assumption [23], where R is theclass of admissible AC policies.

Solving the AC policy by linear programming (LP)

The optimal AC policy, which can minimize the blockingprobability, can be obtained by using the decision variableszsa, s ∈ S, a ∈ As.


The optimal AC policy R∗ in (30) can be obtained bysolving the following linear programming (LP):

minzsa≥0,s,a

∑

s∈S

∑

a∈As

J∑

j=1

ηj(1− aj

)(1− δ(Bj − njq

))τs(a)zsa

(31)

subject to

∑

a∈Ay

zya −∑

s∈S

∑

a∈Aspsy(a)zsa = 0, y ∈ S,

∑

s∈S

∑

a∈Asτs(a)zsa = 1,

∑

s∈S

∑

a∈As

(1− aj

)(1− δ(Bj − njq

))τs(a)zsa ≤ Ψ j ,

∑

s∈S

∑

a∈Asnjqτs(a)zsa ≤ Dj ,

(32)

where Dj and Ψ j denote the connection delay and blockingprobability constraints, respectively, and ηj is the coefficientrepresenting the weighting of the cost function for a particu-lar class j, where j = 1, . . . , J .

The optimal policy will be a randomized policy: the op-timal action a∗ ∈ As for state s, where As is the admissi-ble action space, is chosen probabilistically according to theprobabilities zsa/

∑a∈Aszsa.

We remark that the above randomized AC policy can op-timize the long-run performance. The decision variables, zsa,where s ∈ S and a ∈ Ax, act as the long-run fraction of de-cision epoches at which the system is in state s and actiona. At each state s, there exists a set of feasible actions, andeach action induces a different cost c(s, a). The long-run per-formance can be optimized by appropriately allocating thesetime fractions, and the allocation leads to a randomized ACpolicy. When a deterministic policy is desired, a constraintregarding the decision variables zsa should be imposed intothe above optimization problem, in order to ensure that ateach state s, there is one and only one nonzero decision vari-able. It is obvious that the more constraints we impose, theworse the achieved performance becomes. We choose a ran-domized AC policy in order to achieve long-run optimal per-formance.

4.2. GMDP-based AC policy

In the above, we provide an optimal SMDP formulation. Thestate space has dimension of 2J +

∑ Jj=1Lj for J classes of traf-

fic. For large J and retransmission number, this leads to acomputation problem of excessive size.

In order to reduce complexity, we consider the decisionepoch as the time instances that a packet arrives or departs.As we discussed in the previous section, based on these de-cision epoches, the time interval until the next state is notexponentially distributed. Therefore, we have a generalizedMarkov decision process (GMDP). While an optimal solu-tion for this GMDP problem is hard to obtain, a linear pro-gramming approach provides a suboptimal solution [5].

We remark that the formulation of a GMDP is very simi-lar to the AC problem formulation employed in [4–6], exceptthat the state space has been modified to include beamform-ing and the mean duration of a packet is modified to considerthe impact of ARQ schemes.

In the formulated GMDP, decision epoches are chosen asthe time instances that a packet arrives or departs. The arrivalprocess for class j is assumed to have a Poisson distributionwith arrival rate λj . The duration of the class j packets mayhave a general distribution, with mean (1/μj)(1 + ρj + · · · +

ρLjj ), where μj denotes the departure rate for each transmis-

sion round for the class j packets.The state space S is comprised of any state vector s, which

satisfies SIR requirements,

S ={s = [

n1q, k1, . . . ,nJq, kJ

]T: n

jq ≤ Bj ,

j = 1, . . . , J ;J∑

j=1

k j

1 +(B/γ jRj

(E[φint

]/E

[φdes

])) < 1

},

(33)

where k j denotes the number of active packets for class j.At each decision epoch, an action is chosen as a =

[a1,d1 . . . , aJ ,dJ]T , where aj denotes the action for class j if

an arrival occurs, j = 1, . . . , J and dj denotes the action forclass j packet if a packet in this class departs. The admissibleaction space for state s, denoted by As, can be defined as theset of all feasible actions.

The state dynamics of a SMDP are completely specifiedby stating the transition probabilities of the embedded chainpsy(a) and the expected holding time τs(a), which are givenin [4, 5].

After formulating the AC problem as a GMDP, the ACpolicy, which minimizes the blocking probability, can be ob-tained by using the decision variables zsa, s ∈ S, a ∈ As fromlinear programming which is presented in (31).

In a low instantaneous PER region, the suboptimal solu-tion proposed in the above is very close to the SMDP-basedAC policy. Intuitively, when the PER is very low, retransmis-sion occurs only occasionally, and the duration of a packetwould be very close to an exponential distribution. In thiscase, the LP approach would provide an optimal solution tothe above GMDP.

We remark that unlike the SMDP-based AC policy inwhich the transmission round is assumed to have an expo-nential distribution, the GMDP-based AC policy discussedin the subsection can be applied to a system with a generallydistributed transmission round.

4.3. Complexity

SMDP or GMDP-based AC policies are always calculated of-fline and stored in a lookup table. Whenever an arrival ordeparture occurs, an optimal action can be obtained by ta-ble lookup using the current system state. This facilitates theimplementation of packet-level admission control.


Initial ARQ parameters[L1, . . . ,LJ ] = [0, . . . , 0]

[ρ1 , . . . , ρJ ] = [ρ01 , . . . , ρ0

J ]

j = 0

AC policy

Lj = Lj + 1 j = j + 1Evaluate Pjout

If Pjout ≤ target value

Loptj = Lj

If j = J

yes

yes

No

No

Stop

Figure 1: Search procedure of the optimal number of retransmissions.

Once system parameters change, an updated policy is re-quired. However, in the system we investigate, the policy onlydepends on buffer sizes, long-term traffic model, and QoS re-quirements. These parameters are generally constant for theprovision of a given profile of offered services. Therefore, anSMDP or GMDP-based policy has a very reasonable compu-tation complexity.

5. CROSS-LAYER DESIGN OF ARQ PARAMETERS

In the previous sections, we discuss how to derive the PCFCin the physical layer and how to derive admission control inthe network layer. These derivations assume that ARQ pa-rameters such as Lj and ρj , where j = 1, . . . , J , are alreadyknown. In this section, we discuss how to choose these pa-rameters in order to guarantee outage probability constraintsand optimize overall system throughput.

The search procedures for optimal ARQ parameters, de-noted as vectors Lopt = [L

opt1 , . . . ,L

optJ ] and ρopt = [ρ

opt1 , . . . ,

ρoptJ ], are demonstrated in Figures 1 and 2, respectively. The

initial parameters are set to [L1, . . . ,LJ] = [0, . . . , 0] and[ρ1, . . . , ρJ] = [ρ0

1, . . . , ρ0J ], where ρ0

j represents the upper

bound target PER for class j, which can be specified for thesystem. In Figure 2, Δ j represents the adjustment step size.

From the search procedures presented in Figures 1 and2, it is observed that the number of allowed retransmissionsL

optj , which can achieve a target outage probability, is mini-

mized; and as a result, the network layer performance degra-dation can be minimized. Thus, network layer QoS require-ments in terms of blocking probability and connection de-lay can be guaranteed by formulating the AC problem as anSMDP or GMDP.

Summing above, by choosing ARQ parameters in a cross-layer context, QoS requirements in the physical and networklayers can be guaranteed, and the overall system throughputcan be maximized.


We consider a 3-element circular antenna array, for example,M = 3, with a uniformly distributed angle of arrival (AoA)over [0, 2π) [22]. Numerical values of parameters E[φdes] andE[φint] in (21), derived in [22], are shown in Table 5. We re-mark that the proposed AC policies can be applied to anyother array geometry and AoA distribution. Without loss of


Initial ARQ parameters

[L1, . . . ,LJ ] = [Lopt1 , . . . ,L

optJ ]

[ρ1, . . . , ρJ ] = [ρ01, . . . , ρ0

J ]

Thr past = 0

Derive PCFC

SMDP-AC policy

ρj = ρj − Δ j j = j + 1Evaluate throughputstore in Thr current

If Thr current <Thr past

ARQ parameter

ρj = ρoptj

Let Thr past = Thr current

ρoptj = ρj

j = J ?

yes

yes

No

No

Stop

Figure 2: Search procedure of the optimal target PER.

Table 5: Numerical values of E[φdes] and E[φint] in (20) and (21).

M 1 2 3 4 5 6

E[φdes] 1.0 1.0 1.0 1.0 1.0 1.0

E[φint] 1.0 0.5463 0.3950 0.3241 0.2460 0.2058

generality, we consider a single-path channel and a two-classsystem with a QPSK and convolutionally coded modulationscheme with rate 1/2 and a packet length Np = 1080. Underthis scheme, the parameters of a, g, and γ0 in (6) can be ob-tained from [20]. For simplicity, no buffer is employed in thesimulation. Simulation parameters are presented in Table 6.

6.1. Performance of SMDP-based AC policies

Here, we investigate how the ARQ scheme can reduce outageprobability while only slightly degrading the network layerperformance.

We examine the case in which only the class 2 packets canbe retransmitted once, for example, L1 = 0 and L2 = 1, andan optimal SMDP-based AC policy is employed. The targetPER for the class 1 packets is set to 10−4, while different target

Table 6: Simulation parameters.

B 3.84 MHz a 90.2514

g 3.4998 γ0 1.0942 dB

R1 144 kbps R2 384 kbps

λ1 1 λ2 0.5

μ1 0.25 μ2 0.1375

Ψ1 0.1 Ψ2 0.2

D1 2.25 D2 0.5360

M 3 η0 10−6

η1 0.5 η2 0.5

PERs for class 2 are evaluated. We focus on the performancefor the class 2 packets since only these packets are allowed re-transmission. Figure 3 presents the analytical and simulatedblocking probabilities as a function of ρ2. It is observed thatthe simulation results are very close to the analytical results.Figure 4 presents the outage probability and throughput forthe class 2 packets. It is observed that at a reasonably lowPER, the outage probability can be reduced dramatically, andoverall system throughput can be significantly improved byallowing only one retransmission. Figure 5, which presents


10−5 10−4 10−3 10−2 10−1 100

Target PER

00.010.020.030.040.050.060.07

P1 b

Analytical resultSimulation result

(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

0

0.05

0.1

0.15

0.2

P2 b


(b)

Figure 3: SMDP-based AC policy: analytical and simulated block-ing probabilities as a function of ρ2 in which L1 = 0, L2 = 1, andρ1= 10−4.

the network layer performance degradation by employingARQ, shows that the degradation can be ignored in a lowPER region.

6.2. Performance of GMDP-based AC policies

In the above, we discussed the performance of SMDP-basedAC policies, which require high computation. To reducecomplexity, a GMDP-based AC policy can be employed. Thetarget PER for class 1 is set to 10−4, while different target PERrequirements for class 2 are considered.

Figure 6 shows the analytical and simulated blockingprobabilities as a function of target PER for the class 2 pack-ets. The gap between the simulated and analytical results isdue to the non-exponential distribution of the packet dura-tion.

Figure 7 demonstrates that for a small number of re-transmissions, SMDP and GMDP-based AC policies havesimilar performance. Although performance comparison forlarge Lj is not presented here since an SMDP-based AC pol-icy would involve excessive computation, it is expected thatfor low PER, these two AC policies would still have similarperformance. For a high PER, however, the packet durationis far from exponentially distributed, and thus linear pro-gramming cannot provide an optimal solution to a GMDPand its performance would be inferior to that of SMDP. Insummary, GMDP-based AC policy provides a simplified ap-proach which is capable of achieving a near-optimal system

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−3

10−2

10−1

100

Ou

tage

pro

babi

lity

Without ARQWith ARQ

(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

0

0.1

0.2

0.3

0.4

0.5

Th

rou

ghpu

t(p

acke

ts/s

econ

d)Without ARQWith ARQ

(b)

Figure 4: SMDP-based AC policy: outage probability and through-put for class 2 packets as a function of ρ2 in which L1 = 0, L2 = 1,and ρ1= 10−4.

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−4

10−3

10−2

10−1

P1 b

Without ARQWith ARQ

(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−4

10−3

10−2

10−1

100

P2 b

Without ARQWith ARQ

(b)

Figure 5: SMDP-based AC policy: blocking probability degradationas a function of ρ2 in which L1 = 0, L2 = 1, and ρ1= 10−4.


10−5 10−4 10−3 10−2 10−1 100

Target PER

00.010.020.030.040.050.060.07

P1 b


(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

0

0.05

0.1

0.15

0.2

P2 b


(b)

Figure 6: GMDP-based AC policy: analytical and simulated block-ing probabilities as a function of ρ2 in which L1 = 0, L2 = 1, andρ1= 10−4.

performance for a system with low PER or a small number ofretransmissions.

Figures 8–10 compare the performance among differentnumbers of retransmissions in which ρ1 = ρ2, and L1 = L2.From here on, Lj is denoted by L in the figures. We investi-gate the performance for Lj = 0, 1, and 2, respectively. Theresults for large Lj can be extended straightforwardly. It is ob-served that in a low PER region, for example, ρj ≤ 0.01, withan increased Lj , outage can dramatically be reduced, whilethe blocking probability is only slightly degraded. With onlyone retransmission allowed, the throughput can be improvedby 100%. However, when Lj is increased beyond a certainlevel, for example, Lj = 2 in the system under consideration,the outage reduction and throughput improvement are notsignificant. Beyond this threshold, further increasing Lj mayeven lead to a performance degradation due to a degradednetwork layer performance. From Figures 8–10, we also con-clude that at high PER, the proposed ARQ-based ROP algo-rithm is not as efficient as in low PER.

6.3. Performance of a complete-sharing-basedadmission control policy

For a complete-sharing (CS)-based policy, whenever a packetarrives, the power control feasibility condition in (21) is eval-uated by incorporating information of this newly arrivedpacket. If this condition is satisfied, the incoming packet canbe accepted, otherwise, the packet is stored in a buffer orblocked if the buffer is full. CS-based AC policy provides a

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−4

10−2

100

Blo

ckin

gp

roba

bilit

y

SMDPGMDP

(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−4

10−2

100

Ou

tage

pro

babi

lity

SMDPGMDP

(b)

10−5 10−4 10−3 10−2 10−1 100

Target PER

00.2

0.40.60.8

Th

rou

ghpu

t(p

acke

ts/s

econ

d)

SMDPGMDP

(c)

Figure 7: Performance comparison between SMDP and GMDP-based AC policies as a function of ρ2 in which L1 = 0, L2 = 1,and ρ1= 10−4.

simple admission control algorithm but ignores the QoS re-quirements in the network layer.

We now provide a simple example for complete-sharing(CS)-based AC policy. For comparison purposes, the simula-tion results for a GMDP-based AC policy is also presented. Inthis example, both classes of packets are allowed to retrans-mit twice, for example, L1 = L2 = 2.

We note that in a system with relaxed blocking proba-bility constraints, even a CS-based AC policy can satisfy allthe QoS requirements. To illustrate the shortcoming of a CS-based AC policy, we now restrict the blocking probabilityconstraint for class 2 to 0.05 without loss of generality, andall the other parameters in Table 6 remain unchanged.

The results for a GMDP-based AC policy and a CS-based

AC policy are shown in Table 7, in which Pjb denotes the

blocking probability for class j packets, where j = 1, 2 andPb denotes the overall blocking probability. It is observedthat for a CS-based AC policy, the blocking probability con-straint cannot be guaranteed. For example, when the buffersize is [0, 3], the blocking probability for class 1 packets is0.1185, which exceeds its constraint 0.1. When the buffer size


10−5 10−4 10−3 10−2 10−1 100

Target PER

10−6

10−4

10−2

100

P1 b

L = 0L = 1L = 2

(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−6

10−4

10−2

100

P2 b

L = 0L = 1

L = 2

(b)

Figure 8: GMDP-based AC polices: blocking probability as a func-tion of target PER in which ρ1 = ρ2 and L1 = L2.

Table 7: Comparison between CS-based and GMDP-based ACpolicies, in which L1 = L2 = 2, ρ1 = ρ2= 10−4. The blocking prob-ability constraint is set to [0.1, 0.05], and the connection delay con-straint is [2.25, 0.5360].

[B1,B2]GMDP: P1b GMDP: P2

b GMDP: Pb CS: P1b CS: P2

b CS: Pb[0, 2] 0.0714 0.0359 0.0537 0.0978 0.0390 0.0684

[0, 3] 0.0764 0.0280 0.0522 0.1185 0.0171 0.0678

[1, 2] 0.0434 0.0412 0.0423 0.0505 0.040 0.0452

[3, 2] 0.0179 0.0379 0.0279 0.0210 0.0569 0.0389

is [3, 2], the blocking probability for class 2 packets is 0.0569,which exceeds its blocking probability constraint 0.05. How-ever, for the same buffer sizes, GMDP-based AC policy canalways guarantee blocking probability constraints for bothclasses.

6.4. Choosing ARQ parameters

As discussed in Section 5, ARQ parameters, such as Lj andρj , should be chosen appropriately in order to achieve max-imum throughput while simultaneously satisfying the QoSrequirements in the physical and network layers.

We now provide a simple example to illustrate how toobtain optimal ARQ parameters by using the algorithm pro-posed in Section 5. The initial target PERs ρ0

j = 0.05, where

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−4

10−3

10−2

10−1

100

P1 ou

t

L = 0L = 1

L = 2

(a)

10−5 10−4 10−3 10−2 10−1 100

Target PER

10−5

100

P2 ou

t

L = 0L = 1L = 2

(b)

Figure 9: GMDP-based AC polices: outage probability as a functionof target PER in which ρ1 = ρ2 and L1 = L2.

j = 1, 2, are given by the system which represents the upperbound of the target PER.

Using the algorithm presented in Section 5, the optimalARQ parameters are derived as L

opt1 = 1, L

opt2 = 1, ρ

opt1 =

0.005, and ρopt2 = 0.005, respectively, for outage probability

constraint [0.01, 0.01] and blocking probability constraints[0.1, 0.2]. If the blocking probability constraint remains un-changed, and the outage probability constraint is reduced to[10−3, 10−3], the optimal ARQ parameters can be derived asL

opt1 = 2, L

opt2 = 2, ρ

opt1 = 0.01, and ρ

opt2 = 0.01, respectively.

6.5. Sensitivity of the proposed algorithm totraffic load

In this subsection, we study the sensitivity of the pro-posed AC policy to different traffic loads. Traffic load canbe represented by the packet occupancy ratio, defined as[λ1/μ1, λ2/μ2]. The following traffic loads are investigated:[(1, 1/2); (2, 1(1/2)); (3, 2(1/2)); (4, 3(1/2)); (5, 4(1/2))].

Let λ and μ denote the overall arrival rate and the averagedeparture rate, respectively, which can be expressed as λ =λ1 +λ2 and μ = (λ1/(λ1 +λ2))μ1 +(λ2/(λ1 +λ2))μ2. The overalltraffic load is represented by λ/μ. In the following examples,the target PER is assumed to be 10−3 for both classes, and aGMDP-based AC policy is employed, which would achievea very similar performance to an optimal SMDP-based ACpolicy due to the low target PER under investigation.


10−5 10−4 10−3 10−2 10−1 100

Target PER

0

0.5

1

1.5

Th

rou

ghp

ut

(pac

kets

/sec

ond

)

L = 0L = 1L = 2

Figure 10: GMDP-based AC polices: throughput as a function oftarget PER in which ρ1 = ρ2 and L1 = L2.

Figure 11 presents the average blocking probability, out-age probability and throughput as a function of overall trafficload. With an increased traffic load, there will be an increasedinterfering power and thus the performance is degraded. Weremark that for all the traffic loads investigated, the proposedARQ-based ROP algorithm is able to reduce the outage prob-ability significantly at the cost of a slightly degraded networklayer performance. Therefore, the proposed ARQ-based ROPalgorithm can be applied to a wide variety of traffic condi-tions.

7. CONCLUSIONS

This paper provides a novel framework which exchanges in-formation among physical, data-link, and network layers,and as a result provides a flexible way to handle the QoSrequirements as well as the overall system throughput. Inthis paper, we propose a cross-layer AC policy combinedwith an ARQ-based ROP algorithm for a CDMA beamform-ing system. Both optimal and suboptimal admission controlpolicies are investigated. We conclude that in a low PER re-gion, for example, less than 10−2, the proposed AC policiesare capable of achieving significant performance gain whilesimultaneously satisfying all QoS requirements. Numericalexamples show that the throughput can be improved by100% by employing only one retransmission. Although ARQschemes may degrade network layer performance, this degra-dation can be adequately controlled by appropriately choos-ing ARQ parameters. Furthermore, the proposed AC policyand ARQ-based ROP algorithm can be applied to any trafficload.

1 2 3 4 5 6 7 8 9

Traffic load (λ/μ)

00.05

0.10.15

0.2

Blo

ckin

gp

roba

bilit

y

L = 0L = 1L = 2

(a)

1 2 3 4 5 6 7 8 9


10−5

100

Ou

tage

pro

babi

lity

L = 0L = 1L = 2

(b)

1 2 3 4 5 6 7 8 9


0.5

1

1.5

Th

rou

ghp

ut

(pac

kets

/sec

ond

)

L = 0L = 1L = 2

(c)

Figure 11: Blocking probability, outage probability, and through-put as a function of overall traffic load in which ρ1 = ρ2= 10−3.

ACKNOWLEDGMENT

The support of the Natural Sciences and Engineering Re-search Council of Canada, discovery Grant 41731, is grate-fully acknowledged.

REFERENCES

[1] M. Andersin, Z. Rosberg, and J. Zander, “Soft and safe admis-sion control in power-controlled mobile systems,” IEEE/ACMTransactions on Networking, vol. 5, no. 2, pp. 255–265, 1997.

[2] Y. Bao and A. S. Sethi, “Performance-driven adaptive admis-sion control for multimedia applications,” in Proceedings ofIEEE International Conference on Communications (ICC ’99),vol. 1, pp. 199–203, Vancouver, BC, Canada, June 1999.

[3] T.-K. Liu and J. Silvester, “Joint admission/congestion controlfor wireless CDMA systems supporting integrated services,”IEEE Journal on Selected Areas in Communications, vol. 16,no. 6, pp. 845–857, 1998.

[4] C. Comaniciu and H. V. Poor, “Jointly optimal power and ad-mission control for delay sensitive traffic in CDMA networks


with LMMSE receivers,” IEEE Transactions on Signal Process-ing, vol. 51, no. 8, pp. 2031–2042, 2003.

[5] S. Singh, V. Krishnamurthy, and H. V. Poor, “Integratedvoice/data call admission control for wireless DS-CDMA sys-tems,” IEEE Transactions on Signal Processing, vol. 50, no. 6, pp.1483–1495, 2002.

[6] F. Yu, V. Krishnamurthy, and V. C. M. Leung, “Cross-layer op-timal connection admission control for variable bit rate multi-media traffic in packet wireless CDMA networks,” IEEE Trans-actions on Signal Processing, vol. 54, no. 2, pp. 542–555, 2006.

[7] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,”European Transactions on Telecommunications, vol. 10, no. 6,pp. 585–595, 1999.

[8] A. Yener, R. D. Yates, and S. Ulukus, “Combined multiuserdetection and beamforming for CDMA systems: filter struc-tures,” IEEE Transactions on Vehicular Technology, vol. 51,no. 5, pp. 1087–1095, 2002.


[10] B. Hassibi and B. M. Hochwald, “High-rate codes that are lin-ear in space and time,” IEEE Transactions on Information The-ory, vol. 48, no. 7, pp. 1804–1824, 2002.

[11] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-timecodes for high data rate wireless communication: performancecriterion and code construction,” IEEE Transactions on Infor-mation Theory, vol. 44, no. 2, pp. 744–765, 1998.

[12] G. J. Foschini, “Layered space-time architecture for wirelesscommunication in a fading environment when using multi-element antennas,” Bell Labs Technical Journal, vol. 1, no. 2,pp. 41–59, 1996.

[13] S. D. Blostein and H. Leib, “Multiple antenna systems: theirrole and impact in future wireless access,” IEEE Communica-tions Magazine, vol. 41, no. 7, pp. 94–101, 2003.

[14] F. Rashid-Farrokhi, L. Tassiulas, and K. J. R. Liu, “Joint op-timal power control and beamforming in wireless network-susing antenna arrays,” IEEE Transactions on Communications,vol. 46, no. 10, pp. 1313–1324, 1998.

[15] G. Song and K. Gong, “Performance comparison of optimumbeamforming and spatially matched filter in power-controlledCDMA systems,” in Proceedings of IEEE International Confer-ence on Communications (ICC ’02), vol. 1, pp. 455–459, NewYork, NY, USA, April-May 2002.

[16] A. M. Wyglinski and S. D. Blostein, “On uplink CDMA cell ca-pacity: mutual coupling and scattering effects on beamform-ing,” IEEE Transactions on Vehicular Technology, vol. 52, no. 2,pp. 289–304, 2003.

[17] K. I. Pedersen and P. E. Mogensen, “Directional power-basedadmission control for WCDMA systems using beamformingantenna array systems,” IEEE Transactions on Vehicular Tech-nology, vol. 51, no. 6, pp. 1294–1303, 2002.

[18] J. Evans and D. N. C. Tse, “Large system performance of linearmultiuser receivers in multipath fading channels,” IEEE Trans-actions on Information Theory, vol. 46, no. 6, pp. 2059–2078,2000.

[19] S. Brueck, E. Jugl, H.-J. Kettschau, M. Link, J. Mueckenheim,and A. Zaporozhets, “Radio resource management in HSDPAand HSUPA,” Bell Labs Technical Journal, vol. 11, no. 4, pp.151–167, 2007.

[20] Q. Liu, S. Zhou, and G. B. Giannakis, “Cross-layer combiningof adaptive modulation and coding with truncated ARQ overwireless links,” IEEE Transactions on Wireless Communications,vol. 3, no. 5, pp. 1746–1755, 2004.

[21] A. Sampath, P. S. Kumar, and J. M. Holtzman, “Power controland resource management for a multimedia CDMA wirelesssystem,” in Proceedings of the 6th IEEE International Sympo-sium on Personal, Indoor and Mobile Radio Communications(PIMRC ’95), vol. 1, pp. 21–25, Toronto, Canada, September1995.

[22] A. Wyglinski, “Performance of CDMA systems using digitalbeamforming with mutual coupling and scattering effects,”M.S. thesis, Queen’s University, Kingston, Ontario, Canada,2000.

[23] H. C. Tijms, Stochastic Modelling and Analysis: A Computa-tional Approach, John Wiley & Sons, Chichester, UK, 1986.