1

A comparative analysis of spatial multiplexingtechniques for outdoor MIMO-OFDM systems with

limited feedback constraintStefano Savazzi, Monica Nicoli and Mikael Sternad

Abstract— In this paper we analyze spatial multiplexing tech-niques for the downlink of a multiple-input-multiple-outputorthogonal-frequency-division-multiplexing system. Our study isfocused on outdoor environments characterized by moderateangular spread. We consider two techniques that are able toseparate the downlink data streams associated to different usersand to guarantee a fixed error probability by exploiting alimited feedback from each user. The proposed Adaptive Gridof Beams (AGoB) and Grid of Beams (GoB) differ in the waythe precoders are designed (by adaptive or fixed processing) andin the scheduling policy. The new AGoB is able to harness apartial knowledge of the downlink channel spatial structure tobetter select the users and adjusts their precoders for downlinktransmission. The performances of GoB and AGoB are comparedin this paper in terms of throughput and cell coverage capability.The radio interface is adapted to fit the requirements for theAMC-AAS (adaptive modulation and coding with advancedantenna system) mode of the IEEE 802.16-2005 standard [9].Numerical results show that, as far as the channel exhibits alimited angular spread at the base station, the AGoB technique isable to provide significant throughput gains compared to the fixedGoB approach. On the other hand, large angular spreads areproved to have a substantial impact on the system performanceas the benefits of the adaptation are significantly reduced.

I. INTRODUCTION

Next generation mobile communication systems are ex-pected to place stringent demands for high quality and highdata rates over mobile radio channels. Recent studies onmultiple-input-multiple-output (MIMO) antenna systems withorthogonal-frequency-division-multiplexing (OFDM) signal-ing have shown that spatial-frequency parallel processing atthe transmitter and the receiver, combined with fast adaptivetransmission, is able to provide significant increases in spectralefficiency [1].

Motivated by capacity analysis of MIMO transmission insingle-user scenarios [2], the study of multi-user MIMOsystems has recently become crucial for applications wheremultiple users have to be served simultaneously by a singlebase station (BS), thus generating multi-user interference(MUI) [3].

Manuscript received May 2, 2007, revised Dec. 21, 2007. This workis partially supported by MIUR-FIRB Integrated System for Emergency(InSyEme) project under the grant RBIP063BPH. The authors are with theDipartimento di Elettronica e Informazione, Politecnico di Milano, I-20133Milano, Italy and Signals and Systems, Dept. of Engineering Sciences, Upp-sala University, SE-751 21, Uppsala, Sweden. (e-mail: savazzi@elet.polimi.it;nicoli@elet.polimi.it, ms@signal.uu.se). The material in this paper was pre-sented in part at the IEEE International Conference on Communications, 11-15June, 2006.

In multi-user systems, a traditional time division multipleaccess (TDMA) scheme may be used by the BS to serveone user at a time selected as the one with the best channelcondition. Still, this strategy is known to be suboptimal as themultiplexing gain that can be achieved by simultaneous trans-mission to multiple users is not exploited [4]. The capacityfor the multi-user MIMO channel has been analyzed from aninformation theoretic viewpoint in [5] by assuming that thetransmitter perfectly knows the interfering signals.

Within the class of linear processing techniques for multi-user transmission, space-division multiple access (SDMA)consists in serving multiple users simultaneously by separatespatial precoding (e.g., beamforming) of each stream. Intra-cell bandwidth reuse is thus accomplished by creating spatiallymultiplexed channels [6] that can be associated to differentusers competing for the same network resource (i.e., for thesame time slots and subcarriers, in the OFDM case). Comparedto the optimal solution [5], spatial multiplexing is known tobe a simpler method to exploit the high spectral efficiencyoffered by MIMO systems: it provides higher multiplexinggains with respect to traditional TDMA schemes (e.g., space-time coding) but it is very sensitive to the choice of the codingstrategy (i.e., the beamforming weights) and to the amountof the available channel state information (CSI) of the linkstoward the receivers. To provide the necessary robustness tochannel fading and MUI, linear precoding should be thereforebased on the knowledge of the CSI that can be provided byeach receiver using a feedback channel. Optimal solution to theprecoding problem with perfect CSI at the transmitter has beenproposed in [7], where it is shown how to set up interference-free spatially multiplexed channels by proper design of theprecoding matrices or vectors.

This paper focuses on SDMA techniques for outdoor envi-ronments where the BS is located higher than the surroundingscattering (as for cellular systems). Low complexity precodingtechniques are designed so as to cope with a feedback channelof limited bandwidth that constrains partial (and quantized)CSI at the transmitter [8]. In particular, the downlink of amulti-user frequency division duplex (FDD) MIMO-OFDMsystem (or MIMO broadcast channel) conforming to the IEEE802.16-2005 standard [9] is investigated, with adaptive mod-ulation and coding (AMC) transmission. Such a scheme usesadjacent subcarriers to form subchannels allocated to differentusers. In addition, the advanced antenna system (AAS) optionof the IEEE 802.16-2005 standard is adopted to enable the useof multiple antenna at both the transmitter and at the receiver

2

for SDMA, at the price of an increased number of pilots.

A. Related worksIn FDD systems, CSI needs to be delivered to the BS

through a feedback channel of limited bandwidth. Performanceanalyses of spatial multiplexing in realistic conditions shouldtherefore include the effects of partial or quantized CSI. Whenthe feedback is severely limited, important issues are howto quantize the information (the channel gains) needed atthe transmitter and how many bits of feedback are requiredto achieve the desired performances in terms of spectralefficiency or error rate [10]-[11].

Linear beamforming techniques based on partial CSI inMIMO broadcast channels are commonly divided into twoclasses. The fixed beamforming based transmission consistsin designing pre-determined and fixed beamforming vectors(random [12] or optimized [13]-[14]) that are associated toeach user based on feedback information. On the other hand,the adaptive beamforming approach consists in adaptivelyadjusting the beamforming weights so as to minimize the MUIexperienced by each user. Precoding vectors are thus designedbased on information of the interference configuration that isconveyed through the feedback channel. The most commonlyused linear adaptive techniques are zero-forcing beamform-ing (ZF-BF) and minimum mean squared error beamforming(MMSE-BF) [7], [15]-[16], where the first technique is knownto be simpler as it does not need the estimation of the noisepower and it is appropriate when inter-user interference isdominant [18]-[19].

By reducing frequency-selective channels into an equiv-alent set of frequency-flat subchannels, OFDM signallinghas emerged as an effective solution to combat inter-symbolinterference (ISI) caused by wireless multi-path fading channel[17]. OFD Multiple Access (OFDMA) systems add moredegrees of freedom in beamforming design as they allowthe allocation of different sets of subchannels (or subcarri-ers) to different sets of terminals, while users allocated forthe same frequencies can be separated by means of spatialprocessing. Typical settings [9] prescribe a specific time-frequency resource unit (that consists of a number of adjacentOFDM subcarriers and time slots) that is allocated to themost favorable subset of users based on the available CSIso as to minimize MUI and maximize the overall throughput[20]-[21]. The joint exploitation of beamforming and OFDMprocessing, together with availability of a large number ofusers that are competing for the same communication resource,give increasing chances to schedule spatially separated usersin parallel. As a consequence, designing scheduling algorithmsto optimally select the users to be allocated for the sametransmission resource is of fundamental importance to fullybenefit from multiuser diversity [12]-[19].

In outdoor propagation environments the BS is typicallypositioned higher than the surrounding scatterers so that thetransmitted signals propagate through a multipath channel witha moderate angular spread [22]-[23]. In this case the fadingthat impairs the channels to different users is often spatiallycorrelated so that scheduling algorithms can be designed to

efficiently arrange the users into groups (or clusters). Thesesubsets are used to separate the users that are allowed toshare the same transmission resources from those that mustbe served in different time or frequency units. Recent studies[24] focused on the uplink case and suggested a clustering ofusers based on their mutual spatial correlation, or, alternatively,on their main direction of arrival (DOA) in case of moderateangular spread at the BS. In [25], a tree-based schedulingalgorithm is proposed so as to make use of perfect CSI atthe transmitter and to organize an arbitrary number of usersinto groups based on their spatial channel properties.

B. Paper organization and main contributions

In this paper two SDMA linear precoding strategies areinvestigated and compared for downlink FDD transmission.

The first one, referred to as grid of beams (GoB) (seeD2.7 from [26]), employs a set of fixed (pre-determined)beamforming precoders (beams) that are switched in time:each mobile station (MS) chooses the best precoder as theone that maximizes the signal-to-noise-plus-interference ratio(SINR) (computed at the output of the beamformer at thereceiving antenna array), and conveys this information to theBS by using a limited feedback rate channel. A simple usersubset selection algorithm is then employed to find the usergroup with the highest sum-throughput (the sum of achievablespectral efficiencies for each spatially multiplexed channel).

Second, an alternative, more flexible, approach referred toas adaptive GoB (AGoB) is proposed. Differently from GoB,adaptation is now introduced in both user subset selection andprecoder computation. User subset selection results as a trade-off between maximizing the sum-throughput and minimizingthe mutual spatial correlation among the users’ channels; thiscorrelation is evaluated from feedback quantized measuresof the main directions of departure (DODs) of the downlinksignals. ZFBF is used for precoder computation based on thefeedback spatial channel features.

The relative benefits of adaptive versus fixed beamformingare herein investigated in outdoor environments where themultipath channel experiences angular spread [22] that mightrange from small to larger values1. A performance comparisonbetween fixed (GoB) and adaptive (AGoB) spatial multi-plexing techniques is performed under varying propagationconditions. For both the approaches, the optimum minimumvariance distortionless response (MVDR) linear filter [28] isadopted at the receiving side, while the modulation and codingparameters are selected based on the SINR at the output ofthe MVDR filter so as to satisfy a given bit error rate (BER)requirement (as prescribed in the IEEE 802.16-2005 standard[9]). We show that as far as the channel exhibits a limitedangular spread at the base station (as for outdoor environments[23]), adaptation of user subset and precoders is able to providesignificant throughput gains compared to the fixed strategy. Onthe other hand, simulation results prove that the benefits of theadaptation vanish for large angular spreads.

1Angular spreads in outdoor environments typically do not exceed 40−50deg with common values ranging between 10 deg and 30 deg [23].

3

11cP

MM cP

1a

L active users

Feedback

1y

1s 1w TN

Ly

1H

LH

1y

Ly

Scheduler

Ls Mw

LaTN

RN

RN

Feedback

Fig. 1. Downlink multiuser MIMO system.

The structure of the paper is as follows. After the definitionof the system and the channel model in Section II and III,the GoB spatial multiplexing solution is recalled in SectionIV. Section V deals with the proposed structure for the BSand the MS processing according to the AGoB solution.A performance analysis is carried out in Section VI forboth the AGoB and the GoB methods, for varying spatialchannel conditions. The performance of opportunistic randombeamforming [12] is introduced as a reference. The impactof the considered scheduling solutions on the cell coverage isalso investigated.

II. SYSTEM MODEL

We consider the downlink channel of a multi-user MIMO-OFDMA wireless system, where L active MSs share the samecell. The BS is equipped with a uniform antenna array withNT ∆T-spaced elements, while each MS has NR ∆R-spacedreceiving antennas (NR ≤ NT). FDD is used to separateuplink and downlink communications.

In the system under study multiple access is handled bya combination of time, frequency and space division. Asshown in Fig. 2, the OFDM channel is indeed organizedin time-frequency resource units, each consisting of a frameof W subsequent slots (or bursts) of D OFDM symbolsand a frequency bin (or subchannel) of B adjacent ∆f -spaced subcarriers. The same time-frequency unit is allocatedto a subset S of M ≤ NT users separated by means ofspatial multiplexing. The setting under consideration can beequivalently adapted to fit the radio interface requirements forIEEE 802.16-2005 standard [9], where the AMC mode assignsa modulation and coding scheme per time-frequency unit andit can be also used in conjunction with multiple antenna at thetransmitter and at the receiver (AAS option).

We assume that channel estimation (or prediction [34] [35])can be performed by each user from pilot subcarriers (orpreambles) included in each slot, as indicated in Fig. 2. Noticethat this feature is available for the AMC mode (with AASoption) of the IEEE 802.16-2005 standard [9].

As shown in Fig. 1, a scheduler at the BS chooses the bestsubset S of M users and assigns to them M spatially separatedchannels by means of M precoding vectors wmMm=1. Userselection is performed in each time-frequency unit based

on some channel measurements drawn by the MS during aprevious training phase and transmitted to the BS through afeedback channel.

In any given time-frequency unit, the NR×1 signal receivedby the kth user (k = 1, . . . ,M ), on a single subcarrier andwithin a single OFDM symbol, can be modelled as

yk =pPkHkwk| z

hkk

ck+

+PM

m=1,m6=kpPmHkwm| z

hkm

cm + nk, (1)

where, for the kth user, Hk denotes the NR × NT chan-nel matrix, wk is the NT × 1 precoding vector, Pk is thetransmitted power and ck ∈ C(n) is the transmitted symbolwith E[|ck|2] = 1. Adaptive transmission is used and it isadjusted to attain a fixed BER target (BER=10−6 from TableI). The complex symbol ck can belong to any of the Navailable modulation sets C(n)Nn=1 (e.g., according to theIEEE 802.16-2005 standard [9]). Furthermore, the additivenoise nk is assumed to be zero-mean white complex Gaussianwith E[nknHk ] = σ2INR . In (1), we defined as hkm = Hkwm

the NR × 1 equivalent single input multiple output (SIMO)channel between the BS and the kth MS when the transmittedsignal is precoded by wm. The channel response is hereinconsidered as frequency-flat within the frequency bin, thusthe bin size (the number of adjacent subcarriers) needs tobe adequately designed according to the specific propagationenvironment [30].

As shown in Fig. 1, MVDR spatial filtering yk = aHk ykis performed at the receiver side on the signal (1) in order tomaximize the SINR [28]:

ak =1√Pk× Q−1M hkk

hHkkQ−1M hkk

. (2)

Here QM =PM

m=1,m6=k PmhkmhHkm + σ2INR denotes the

noise-plus-interference spatial covariance at the kth MS. Thisbeamforming technique is known to minimize the interferenceunder the constraint of a distortionless response to the desiredchannel (i.e., aHk hkk = 1). CSI is required at the MS for theevaluation of the MVDR filter but also at the BS (in the formof partial or quantized information) for scheduling purposes.For all the considered schemes, specific training phases arecarried out so as to estimate the required parameters.

As regards the CSI at the BS, optimum selection of theuser set S and, if necessary, of the corresponding precodingset wmMm=1 (in case AGoB is employed) would requirethe knowledge of all the MIMO channels HmMm=1 anda consequent intensive feedback transmission. To limit thefeedback rate, in this paper the channel quality informationto be evaluated at each MS is reduced to a SINR value ρk,

ρk =Pk¯aHk hkk

¯2PMm=1,m6=k Pm

¯aHk hkm

¯2+ σ2aHk ak

, (3)

that has to be fed back for each time-frequency resource. Sincean MVDR beamformer is employed at each MS, the SINR (3)

4

Feedback data

SINR+

Beam Selection

from codebook 1

SINR SINR

SINR+

Beam Selection

from codebook 2

1 2 W

Downlinkfrequency bin

D OFDM symbols

Bsu

bcar

riers

Frequency-time resource unit

: GoB/AGoB: for estimation of SIMO channels (for the current codebook)

: GoB: for estimation of SIMO channels (for the next codebook)AGoB: for estimation of MIMO channel

1P

2PPilot

symbols

Downlinkfrequency bin

Feedback data

D OFDM symbols

Bsu

bcar

riers

Worst-case SINR

+Spatial freq.

update

SINR SINR

Worst-case SINR

+Spatial freq.

update

λth downlink frame(λ-1)th downlink frame

Frequency-time resource unit

mk ,h

kH

a) GoB

b) AGoB

Feedback data

SINR+

Beam Selection

from codebook 1

SINR SINR

SINR+

Beam Selection

from codebook 2

SINR+

Beam Selection

from codebook 1

SINR SINR

SINR+

Beam Selection

from codebook 2

1 2 W

Downlinkfrequency bin

D OFDM symbols

Bsu

bcar

riers

Frequency-time resource unit

: GoB/AGoB: for estimation of SIMO channels (for the current codebook)

: GoB: for estimation of SIMO channels (for the next codebook)AGoB: for estimation of MIMO channel

1P

2PPilot

symbols

Downlinkfrequency bin

Feedback data

D OFDM symbols

Bsu

bcar

riers

Worst-case SINR

+Spatial freq.

update

SINR SINR

Worst-case SINR

+Spatial freq.

update

Worst-case SINR

+Spatial freq.

update

SINR SINR

Worst-case SINR

+Spatial freq.

update

λth downlink frame(λ-1)th downlink frame

Frequency-time resource unit

mk ,h

kH

a) GoB

b) AGoB

Fig. 2. Overview of the uplink fedback quantities and the downlink framestructure for the GoB scheme (on top) and for the AGoB strategy (on thebottom).

reduces to the following expression:

ρk = PkhHkkQ

−1M hkk. (4)

Notice that each SINR value needs to be quantized beforetransmission. The number of quantization levels shall be atleast equal to the number of allowed transmission modes (i.e.,the modulation-coding sets C(n)).

In conjunction with the SINR value, additional feedbackis required so as to help the BS to decide the best subsetof users to be allocated. The required quantity depends onthe specific multiplexing technique (see Section IV for thefixed technique, GoB, and Section V for the adaptive one,AGoB). As a brief outline, for GoB, at least dlog2Me bitsare required for each user to identify the best precoder amongthe set wmMm=1; for AGoB, since the precoder needs to beadapted to the user channel, a quantized information relatedto the main DOD has to be fed back to the BS. As regards thepower allocation, in this paper we restrict our attention to asimplified scheduling system with uniform power assignment,i.e. P1 = P2 = . . . = PM , for both the training and the datatransmission phases.

III. CHANNEL STRUCTURE

Assume the kth MS, for k = 1, ..., L, to be sufficiently faraway from the BS so that the Kronecker model holds [31],the NR ×NT MIMO channel matrix Hk can be described as

Hk =PPn=1

ck,naR(ϕk,n)aTT(αk,n), (5)

kα

kΦ

TN

krkd

Scatterers antennas

RN antennas

BSTSk

Fig. 3. Spatial structure of the downlink channel.

that is the superposition of P independent paths (gener-ated from P scatterers) having DOD αk,n and DOA ϕk,n,for n = 1, . . . , P . The NT × 1 vector aT(αk,n) de-notes the response of the BS antenna array to the DODαk,n; its ith element, i = 1, . . . , NT, is [aT(αk,n)]i =exp

£−j 2πλ (i− 1)∆T cos(αk,n)

¤, ∆T being the antenna spac-

ing and λ the carrier wavelength. Similarly, the NR ×1 vector aR(ϕk,n) represents the response of the MSantenna array to the DOA ϕk,n, with

£aR(ϕk,n)

¤i=

exp£−j 2πλ (i− 1)∆R cos(ϕk,n)

¤, and ∆R denoting the an-

tenna spacing at the MS. The complex fading term ck,n ∼CN(0, σ2k,n) accounts for the amplitude and the phase shiftof the wave scattered by the nth scatterer towards the kth MS.According to the wide sense stationary uncorrelated scattering(WSSUS) model for propagation, the fading terms are assumedto be uncorrelated to each other, i.e. E[ck,dc∗k,n] = 0 ∀d 6= n.

Based on the assumptions above, the spatial correlationmatrices for the signals at the BS (RT,k) and at the MS (RR,k)arrays are, respectively,

RT,k =1

NRE[H

HkHk] =

PPn=1

σ2k,n a∗T(αk,n)a

TT(αk,n), (6)

and

RR,k =1

NTE[HkH

Hk ] =

PPn=1

σ2k,n aR(ϕk,n)aHR(ϕk,n). (7)

Notice that (6) depends only on the DODs from the localscatterers around the terminal, while (7) relies on the DOAs.

As a model for the channel spatial correlation matrices,we adopt the propagation layout introduced in [23], as brieflyrecalled in the following. As depicted in Fig. 3, the MS antennaarray is assumed to be surrounded by a ring of uniformlydistributed scatterers of radius rk, with equal mean powersσ2k,n = 1/P . On the contrary, the BS, at distance dk À rkfrom the MS, is not surrounded by any local scatterer (thisassumption is generally valid in cellular systems, as the BSposition is high compared to the average ground level). Itfollows that the signal to the kth MS is characterized byangular spread Φk ' 2rk/dk around a main DOD that is heredenoted by αk. The DOD spread Φk will in general be smallerthan the beamwidth of the BS array. Under such conditions,for P → ∞ the MIMO channel correlation matrices (6)-(7)assume a simplified expression that has been derived in [23].In particular, the correlation of the signals from the pth and theqth transmitting antennas (p, q = 1, ..., NT) is therein shown

5

to be:

limP→∞

[RT,k]pq = exp

∙j2π

λ(p− q)∆T cos(αk)

¸×

× I0

µ2π

λ∆T(p− q)Φk sin(αk)

¶, (8)

where I0(·) is the zeroth-order modified Bessel function of thefirst kind. By appropriately adjusting the angular spread Φk in(8), a propagation environment with either moderate (smallΦk) or rich (large Φk) scattering can be modelled.

IV. OVERVIEW OF FIXED GRID OF BEAMS (GOB)

Fixed beams for SDMA can be seen as an extension of thecellular structure with reuse of frequencies [26]: in order toincrease the throughput, a spatial reuse of beams is employedusing a beam selection information fed back by each usertogether with a SINR value calculated according to (4). TheM -precoder set (or codebook) wmMm=1 is chosen amonga group of K predetermined fixed codebooks. To providefull coverage (for uniform distribution of users over the cell),codebook selection is done on a frame-by-frame basis sothat all the K available codebooks are selected during Ksubsequent frames.

Fig. 4 shows the design of precoders (or beams) proposedin [26] for a scheme with K = 2 codebooks: herein thebeampattern is plotted versus the main DOD αk of the signalfrom the BS, for a transmitting array of NT = 4 antennas.A simple heuristic rule can be given to minimize the MUI:only non-neighboring beams form a codebook (e.g., in Fig. 4,beams 1, 3, 5 form the codebook 1, while beams 2, 4, 6form the codebook 2); users allocated to the M beams of agiven codebook can share the same time-frequency resource(M = 3 in the example of Fig. 4). Assuming an antennaarray aperture of 120 deg at the BS, the K = 2 codebooksspan the whole BS aperture and they can be switched fromframe to frame to cover the whole cell. Chebychev tapering isproposed in [26] to optimize the shape of the beams; a side-lobe suppression of 21 dB is adopted in this paper accordingto the scheme therein proposed2.

At the receiving front end, each user has the knowledge ofthe transmitted codebook, so that optimal beam selection canbe carried out at each MS by maximizing the SINR (4) at theoutput of the MVDR filter. As indicated in Fig. 2, to allowfor the SINR evaluation, a specific training phase (indicatedby P1) needs to be carried out in each slot by activatingseparately at the BS the M precoders of the current code-book, wmMm=1, and estimating at each MS the whole sethkmMm=1 of M precoded channels. Using the M estimatedchannels, the kth MS can then evaluate the M SINR valuesthat correspond to the M possible selections of the precoderwk: each SINR value is derived from the expression (4) byusing one of the estimated beams as hkk and the remaining

2The use of Chebychev tapered beamforming weights is motivated primarilyby practical constraints. Optimal fixed precoders shall depend in general onthe particular environment (see for example the Grassmannian beamformingsolution [13] for i.i.d. Rayleigh fading and [27] for the correlated Rayleighfading case).

-60 -40 -20 0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

DOD αk [deg ]

1 2 3 4 5 6

Bea

mpa

ttern

Fig. 4. Beampattern from 6 equispaced Chebychev tapered beams

M − 1 beams as the interfering set hkmm6=k. The optimalprecoder for the kth MS is then selected as the one thatyields the maximum SINR. Such a beam selection informationhas to be fed back to the BS (e.g., using dlog2(M)e bits)together with the corresponding maximum SINR value, beforebeginning the downlink frame. A second kind of training phase(indicated by P2 in Fig. 2) is carried out in the last burst ofeach frame to allow for a new user subset selection that ismaintained during the next frame. Similarly as before, this isdone by activating at the BS the M precoders of the codebookthat is chosen for the upcoming frame (that is different fromthe current one) and by estimating at each MS the new set ofM precoded channels and thus the SINRs to perform a newbeam selection.

The scheduling procedure based on these feedback mea-surements consists of two distinct phases:

1) Beam allocation. The user subset S (recall that precodersare pre-determined and maintained during the whole frame) isselected by the scheduler for a specific time-frequency unitaccording to the measurements that are fed back by eachuser, i.e. the beam selection information and the correspondingSINR value. The optimal subset of M users is then foundby grouping users with different precoders and by choosingthe group that maximizes the throughput. Other optimizationmetrics that cope with fairness aspects might be taken intoaccount as well.

2) Transmission mode selection. The SINR informationis updated over each slot by the MS (so as to account forpossible channel variations) and it is used by the BS so as todesign and update the transmission modes to be associated toeach selected terminal.

Notice that both GoB and AGoB require each beamformingweight be computed in hardware that is limited by finiteresolution and calibration errors [29].

V. ADAPTIVE GRID OF BEAMS (AGOB)Here, a novel adaptive version of the fixed GoB approach

[26] is proposed. As compared to GoB, by allowing a limited

6

increase in the feedback rate and in the number of pilotsubcarriers for each slot (see Fig. 2), the method is able toadapt the design of the beamforming vectors and of the usersubset to the spatial structure of the users’ channels based onquantized CSI.

The training phase P1 remains unaltered compared to GoB,while P2 (that in GoB was used to estimate the SIMO channelsfor the next codebook) is now exploited to draw informationabout the DODs of the downlink signals so as to adaptscheduling and precoding to the spatial channels’ structure.Differently from GoB, the training procedure P2 is carriedout without the use of precoding, to allow the estimation ofthe MIMO channel matrix Hk at the kth MS. The numberof pilot symbols for the estimation of Hk is typically largerthan that required to estimate the SIMO channels hkmMm=1in GoB. Beside that, if the number of antennas is moderate,this difference can be regarded as negligible.

The MIMO channel is used by the kth MS to evaluate thespatial correlation matrix RT,k

3 and to estimate the spatialfrequency fk = cos(αk) associated to the main DOD αk (seeSection V-A). Additional information to be evaluated at theMS is a lower bound ρk for the SINR (4) based on a worst-case choice of the interference configuration4 (see Section V-C). Both the measurements, fk and ρk, are quantized andtransmitted on the feedback channel. The feedback rate clearlydepends on the velocity of the variations of the second-orderchannel statistics (i.e., the matrix RT,k) and thus on the thedegree of MS mobility.

The scheduling procedure based on the above measurementsconsists of two phases:

1) Spatial channel allocation. The user subset S, and nowalso the precoders wkMk=1 to be maintained during the wholeframe, are selected by the scheduler according to the feedbackmeasurements ρk and fk. Based on the SINRs ρk a suboptimalchoice of the transmission modes is also carried out by usingthe mapping scheme in Table I. ZF based beamformers aredesigned at the BS based on fk; the pilot symbols P1 are thentransmitted to the chosen terminals, using these precoders (seeFig. 2).

2) Transmission mode selection. Once the precoded chan-nels hkmMm=1 have been estimated by the selected terminalsfrom the pilot symbols P1, the real SINR measurementsρkMk=1 can be evaluated from (4) and be fed back (afterquantization) for each slot. Transmission mode decisions arethen updated by the scheduler according to the new SINRmeasurements and are valid for the remaining W − 1 bursts.This aspect will be further analyzed in Section VI-A.

A. Spatial frequency estimation

From (8), under the condition of a limited angular spread(Φk ' 0) such that I0

¡2πλ ∆T(p− q)Φk sin(αk)

¢≈ 1, the

3The matrix RT,k can be estimated or predicted by averaging overa number of channel realizations from different time-frequency units (seeAppendix VIII-A for further details).

4Being the precoders adaptively designed by the BS, the worst case SINRsρk can be only a measure of the maximum expected level of interferenceexperienced by each user.

elements of the spatial correlation matrix RT,k reduce to

[RT,k]pq ≈ exp∙j2π

λ(p− q)∆T cos(αk)

¸for p, q = 1, ..., NT and k = 1, ..., L. The spatial correlationstructure at the transmitting array can be thus approximatedby the rank-one matrix (see Appendix VIII-A):

RT,k ≈ a∗T(αk)aTT(αk) = b(fk)bH(fk),

where the change to the variable fk has been introduced tosimplify the analysis in the remainder of the paper, being fk ∈[−1,+1] the spatial frequency corresponding to the main DODαk, and b(fk) = a∗T(αk) the steering vector expressed as afunction of the spatial frequency fk. The main DOD αk, orequivalently the corresponding spatial frequency fk, can beestimated at the MS using the spatial correlation matrix RT,k

drawn from the pilots P2. The problem reduces to solve arank-1 constrained least-squares optimization:

fk=argminf||RT,k − b(f)bH(f)||2=

=argmaxfbH(f)RT,kb(f), (9)

where k·k denotes the Frobenius norm operator.Once estimated at the MS, the spatial frequency fk needs

to be transmitted to the BS for scheduling purposes. Since inpractical systems DODs can be considered as slowly varyingamong each time-frequency unit, only a limited feedback rateis actually required for the transmission of fk. In particular,since the frame duration is designed according to the fast-fading channel parameters, the spatial frequency has to beestimated during the mobile switching-on phase, then it maybe simply updated every W slots (thus on every new frame)using a lower number of bits. Appendix VIII-B gives furtherdetails on how to perform both the estimate and the update inthe most efficient and simplest way.

B. Scheduling and precoder computationHerein the focus is on the processing to be carried out at

the BS. In particular, both the problem of finding the best usersubset S and how to set up the precoders for a specific set ofM fed back spatial frequencies are considered. To minimizethe complexity, a suboptimal solution to the scheduling prob-lem is proposed. The transmission modes C(n) are selectedbased on the SINR measurements (4) according to the targetBER, as exemplified in Table I (from the transmission modesin the IEEE 802.16-2005 standard [9]). We also assume thatF scheduler modules are operating independently on the Favailable frequency bins (see Table II).

1) User subset selection: With L users competing for aspecific time-frequency resource, the scheduler exploits thefeedback SINR lower bounds ρkLk=1 and the spatial frequen-cies fkLk=1 in order to: 1) find all sets of M users fulfilling aspatial separation constraint (to avoid critical MUI situations);2) among the selected sets, find the set that maximizes thewhole throughput.

Let R0 = SiNSi=1 indicate the collection of all the

NS = L!/(M !(L − M)!) subsets of M users that can be

7

obtained from the whole set of L active MSs. For the i-th subset, Si = ki,1, . . . , ki,M, the corresponding SINRlower bounds and spatial frequency values will be indicatedby ρki,1 , . . . , ρki,M and fki,1 , . . . , , fki,M, respectively.For any subset Si we define the minimum spatial separationbetween its users as

L(Si) = minki, ,ki,m∈Si 6=m

¯fki, − fki,m

¯. (10)

We also define the sum-throughput as

γ(Si) =PM

m=1 γ(ρki,m) (11)

where γ(·) [bit/payload symbol/user] denotes the maximumspectral efficiency that can be achieved given the SINR valueargument, according to Table I. Herein a maximum sum-throughput based scheduling is employed, nevertheless, theproposed approach could be easily extended in case otherthroughput measurements or scheduling schemes (such asproportional fair scheduling [32]) are adopted.

From the whole collection of possible subsets R0, the setR1 ⊆ R0 is generated so that:

R1 = Si ∈ R0|L(Si) > β . (12)

This new set R1 contains all the subsets that fulfill a minimumspatial separation constraint. To avoid critical MUI situations,each selection belonging to R1 has a spatial separation atleast greater than β (that needs to be computed numericallyas shown in Appendix VIII-C and Section VI). The next stepis to generate a second subset R2, with R2 ⊆ R1, as

R2 = Si ∈ arg maxSi∈R1

γ(Si), (13)

this is thus obtained by picking all the selections Si ∈ R1 thathave the highest throughput metric γ(Si) in (11).

If the subset R2 contains more than one candidate subset5,the best user subset S is finally selected as the user group thatmaximizes the minimum spatial separation metric L(·):

S = arg maxSi∈R2

L (Si) . (14)

This is done to find a subset that provides the maximumvalue for the sum-throughput γ(Si) (as Si ∈ R2) and tojointly maximize the angular separation between users (tominimize the mutual interference). As shown in Section VI, adrawback of this approach is that the selected users tend to beclustered with high probability to specific and fixed regionsof the angular cell sector. This aspect might be critical forfixed applications where uniform cell coverage is required toguarantee fairness.

2) Precoder computation: Once the set S is chosen, ap-propriate precoding vectors have to be set up for the Mselected users according to the feedback spatial frequenciesf = [f1 · · · fM ]. Defining as B(f) = [b(f1) · · ·b(fM )] theNT × M matrix gathering the BS antenna array responses

5Notice that solution to (13) can be clearly not unique even for smallL as the number of allowed transmission modes (with associated spectralefficiencies in Table I) is finite.

TABLE IIEEE 802.16-2005 TRASMISSION MODES AND REQUIRED SNR RANGES

(OVER AWGN CHANNELS).

Modulation Coding SNR ρ Efficiency γrate [dB] [bit/carrier]

QPSK 1/2 9.4-11.2 1QPSK 3/4 11.2-16.4 1.5

16-QAM 1/2 16.4-18.2 216-QAM 3/4 18.2-22.7 364-QAM 1/2 18.2-22.7 364-QAM 2/3 22.7-24.4 464-QAM 3/4 >24.4 4.5

to the M spatial frequencies, the precoding matrix W(f) =[w1· · ·wM ] is calculated as

W(f) = B(f)£BH(f)B(f)

¤−1. (15)

A ZFBF is thus accomplished as in [18]. Notice that thischoice makes the mth precoder, wm, orthogonal to all othervectors b(fk) within S for k 6= m. Therefore, in the ideal caseof null angular spread at the BS and for perfect knowledge ofeach user DOD (fk = fk), this orthogonal precoding leadsto a null MUI in the MS signal (1); on the other hand,in more realistic propagation conditions that exhibit largerangular spreads, perfect interference rejection is no longerguaranteed.

C. Lower-bound SINR computation at MSDuring the spatial channel allocation phase the L active

MSs are in charge of sending to the BS a measure oftheir SINR value. However, having no knowledge of the Mprecoders wmMm=1 that influence the inter-cell interferenceconfiguration, the MS cannot evaluate the exact SINR valueρk from (4). Hence, we propose to feedback from the kthMS a lower bound ρk corresponding to the worst case ofinterference configuration. From constraint (12), the worstconfiguration of interferers for the k-th user is associated tothe set of β-spaced spatial frequencies f (k)m Mm=1 distributedwithin the cell sector around the kth value f

(k)k = fk

6. Thecorresponding precoding vectors wmMm=1 can be computedas in (15) from the selected frequencies f (k)m Mm=1. Theprecoders are then plugged in (4) together with the powervalues PkMk=1 = P/M , yielding the SINR lower bound ρk.

VI. PERFORMANCE COMPARISON AND NUMERICALRESULTS

In this section, a MIMO downlink radio interface is simu-lated in accordance with the IEEE 802.16-2005 standard [9].The main system parameters are summarized in Table II. Thenumber of antennas at the BS and at the MS is NT = 4and NR = 2, respectively. The minimum angular separationamong users is selected as β = sin(20 deg) according to [33]

6As an example, for M = 3 and f1 = 0, the two interferers for user 1 areplaced at the two spatial frequencies having spatial separation from f1 equalto β, i.e. f(1)2 = β and f

(1)3 = −β.

8

10 12 14 16 18 20 22 24 26 28 302

4

6

8

10

12

14

[deg] )(asin β

Bursts 2,...,W

Burst 1

Sum

Spe

ctra

l Eff

icie

ncy

[bit/

payl

oad

sym

bols

]

L=12

L=6

W=4

Fig. 5. Average sum spectral efficiency versus β for L = 6 (no marker)and L = 12 (circle marker) achievable by scheduled users. The sum spectralefficiency is evaluated in the first burst (dashed lines), during bursts 2÷W(dotted lines) or averaged over the whole frame of W = 4 bursts (solid lines).Statistics for angular spreads are the same as for figure 6.

(see also Appendix VIII-C). The basic allocation unit [9] isherein defined through a bin, which is a set of nine contiguoussubcarriers observed over a single OFDMA symbol. In thiswork, according to the second type of AMC subchannel, aslot is defined as a unit of two bins (i.e., B = 18 adjacentsubcarriers) by D = 3 OFDMA symbols. Since in the AMCpermutation subcarriers assigned to a particular subchannel donot overlap with those of any other subchannel, for a physicallevel simulation, performance in noise and interference limitedsystems can be completely assessed by restricting the attentionto a single time-frequency unit (and subcarrier). Each slotcarries D · B = 54 symbols; of these, as shown in Fig.2, at least M should be allocated for the estimation of theprecoded channels hikMi=1 for the current frame (i.e., thepilots P1). Other pilots should be available at the beginningof each frame (i.e., the pilots P2) for the estimation of theMIMO channel matrix Hk (for AGoB) or for estimation of theprecoded channels for the next frame (for GoB). A number ofsymbols is also reserved for downlink control messaging andsynchronization. At the BS, spectral efficiencies are computedfrom SINR measurements as reported in Table I. The numberof bits used for spatial frequency feedback is b = 6, alower number of bits is required during the tracking mode(see Appendix VIII-B)7. For performance assessment we willassume perfect knowledge of hkmMm=1 and RT,k at the kthMS (as for a large number of pilot symbols).

The performances of AGoB and fixed GoB are hereinevaluated for varying channel conditions and compared tothose of the (opportunistic) orthonormal random beamformingapproach (ORBF [12]). This is a non-adaptive scheme asthe GoB one, but is based on randomly chosen orthonormal

7Here the 26 available spatial frequencies are chosen so as to spanthe 120deg cell sector uniformly in the angular domain (other solutionsmight be employed as well). The resulting maximum quantization error isapproximately 1deg.

TABLE IISYSTEM PARAMETERS AND DOWNLINK RADIO INTERFACE (SEE AMC

MODE FOR IEEE 802.16-2005 [9])

Number of antennas NT = 4, NR = 2Antenna spacings ∆ = 0.5λMax. number of treams M = 3Cell sectors 120 degBackground SNR SNR= 18dBMinimum spatial separation β = sin(20 deg)Maximum terminal velocity v = 50km/hCarrier frequency fc = 3.5 GHzAvailable bandwidth Bw = 10 MHzSubcarrier spacing ∆f = 10.9 kHzOFDM symbol period 102.8 μsBurst (slot [9]) D = 3 (333μs)Frequency bin length B = 18 (181kHz)(9x2 subcarriers)Frequency bins F ≤ 48Frame length W = 4 (1.2ms)

precoders wmMm=1. In other words, the same signal fromeach antenna is modulated by an independent gain whosephase and magnitude is random and uniformly generated sothat each beamformer is orthogonal to each other. As for theGoB scheme, the beam selection information together withthe SINR measurement are tracked by each user and fedback to the BS to form a basis for scheduling. Compared tobeamforming solutions such as GoB and AGoB, this schemedoes not need any antenna array calibration.

A. Performance comparison for GoB, AGoB and ORBFIn this section, a first performance comparison is carried

out for AGoB, GoB and ORBF. The propagation channel issimulated according to the multipath structure in Section III.More specifically, the fading coefficients in the matrix Hk areassumed to be Rayleigh distributed, constant within a frameinterval but varying from frame to frame. Temporal variationsare simulated according to the Clarke model and approximatedby a second-order autoregressive (AR-2) random process [35].As regards the spatial channel structure, users are placed at afixed distance from the BS with main DOD αk uniformlydistributed within the cell sector. Each DOD varies slowlyover the frames according to the AR-1 model [36]: αk( ) =αk( − 1) + ξk, where ξk ∼ N (0, σ2α) and σα = 0.5 deg8.Since multipath channels in urban, suburban, and rural macro-cells are usually characterized by angular spread ranging from10 deg to 30 deg [22]-[23], herein we choose a one-sidedGaussian random variable, Φk ∼ N (15 deg, σ2Φ), for theangular spread at the BS, with mean 15 deg and standarddeviation σΦ = 5deg (Φk ≥ 0 ∀k)9. Spatial correlation of the

8In general, the statistics of the DOD variations shall be related to thepropagation environment and to the kinematics of the mobile terminals (e.g.,the Doppler frequency). The simplified model used in this section is adoptedonly to ease the numerical analysis, without affecting the main reasoning ofthe paper.

9For this propagation scenario we observed that the spatial frequencies canbe tracked by sending just 1 bit for the frequency update (tracking mode),with negligible perfomance degradation. A periodical estimate refresh thatuses 6 bits is also needed. Refresh period can last several frames, dependingon the spatial frequency variability.

9

1

2

3

4

1

2

3

4

3 4 5 6 7 8 9 10 11 121

2

3

4

Spec

tral e

ffic

ienc

y [b

it/ca

rrie

r/use

r]

Number of users L

AGoB - bursts 2 ÷WAGoB - burst 1Perfect CSIGoB

Fig. 6. Average spectral efficiency on each scheduled channel vs. the numberL of active users for AGoB, GoB and the interference free case. For AGoB,the spectral efficiency is evaluated in either the first or the remaining W − 1bursts in the frame.

9

17

25

3 4 5 6 7 8 9 10 11 129

17

25

9

17

25

AGoBPerfect CSIGoB

Ave

rage

SIN

R [d

B]

Number of users L

Fig. 7. Average SINR at the MS after MVDR on each scheduled channelvs. the number L of active users for AGoB, GoB and the interference freecase.

channel gains at the transmitter and the receiver is modelledaccording to [23].

Fig. 6 shows the spectral efficiency for each scheduled chan-nel (ordered by SINR value ρk) averaged over user positions,fading channels (frames) and data. Accordingly, Fig. 7 givesthe average SINR at the MS after MVDR filtering. Both plotsare drawn versus the number L of active users. Since in mostpractical cases the number of active users competing for aspecific resource unit has a wide and unpredictable range, weconsidered L values ranging from L = 3 to L = 12 to modelthe cases of either small or large cell loading, respectively.For the evaluation of AGoB spectral efficiency and SINR weconsidered a single OFDM payload symbol within each time-frequency unit, taken either from the first slot (AGoB - burst1) or from any of the remaining W − 1 slots (AGoB - burst2÷W ) of the downlink frame.

As shown in Fig. 6, when considering the first slot, wefound that, for the considered number of active users L, anaverage spectral efficiency gain of at least 0.5 bit/carrier per

10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

Number of users L

Sum

spec

trale

ffic

ienc

y[b

it/pa

yloa

dsy

mbo

l]

AGoB (burst 2÷W)

ORBFGoB

AGoB (burst 1)

Fig. 8. Average sum-spectral efficiency vs. the number L of active users forAGoB, GoB and ORBF (dashed line). For AGoB, the spectral efficiency isevaluated in either the first or the remaining W − 1 bursts in the frame.

user (or, equivalently, per channel) can be achieved comparedto the GoB solution. The gain is doubled (at least 1 bit/carrierper user) in case of payload symbol placed in any of theremaining W − 1 bursts. From Fig. 7, we can also concludethat a SINR gain of at least 4dB can be obtained for eachuser in any burst of the frame. It is also worth noticing thatthe performance gains are uniformly distributed among thescheduled channels. Benefits in spectral efficiency and SINRvalues are more significant when few users are competing forthe same resource unit (i.e., for low L).

The MUI reduction capability of the proposed AGoB systemcan be inferred from both Fig. 6 and 7 through a comparisonwith the spectral efficiency (Fig. 6) and the SINR (Fig. 7)that can be achieved when full CSI of each user is available(dashed lines). For this ideal case, the user subset selectionis carried out as in [25] where precoders are computed froma zero-forcing technique that exploits perfect knowledge ofthe users’ channel matrices. For a fair comparison with ourscheme, one single stream is associated to each user. Noticethat, since full CSI is available at the BS, each column ofthe matrix B in (15) contains the leading right eigenvectorfor each channel matrix Hk, k = 1, ...,M . Fig. 7 shows thatfor realistic channel conditions the SINR loss experienced byAGoB with respect to this ideal case (perfect CSI) is less than2.5dB, for the considered number of users L.

Under the same spatial channel conditions, Fig. 8 shows theaverage sum-spectral efficiency versus the number L of activeusers. Here the AGoB method is also compared with the ORBFcase (dashed line). Although the problem of antenna arraycalibration is avoided by ORBF, a significant performancedegradation is observed in case of random beamforming dueto the low degree of multi-user diversity offered by theconsidered multi-user scenario. For the considered setting, anhigher number of users (at least L > 40 users competingfor the same time-frequency unit) is indeed required by therandom beamforming scheme to reach the same performancesas for GoB and AGoB.

From these results we can conclude that, even with partial

10

5 9 13.5 18 22.5 27 31.5 36 40.5 45 49.50

5

10

15

20

25

30

L=9

L=20

AGoBORBFGoB

SIN

R (3

rd c

hann

el) [

dB]

Angular spread Φk [deg]

Fig. 9. Average lowest SINR at the output of the scheduler for varyingchannel conditions and number of active users L = 9 or L = 20. Angularspread is Φk ∼ N (E[Φk], σ2Φ), Φk ≥ 0, σΦ = 5deg; the average spread,E[Φk], ranges from 9 deg to 48 deg.

CSI at the transmitter, the proposed AGoB scheduling andprecoding approach at the BS, jointly with MVDR filtering atthe MS, is effectively able to reduce MUI, thus increasing theoverall spectral efficiency.

B. Impact of the propagation environment on the performanceIn this section, the impact of the propagation environment

on the system performance is investigated by analyzing theeffect of the channel angular spread on the average throughput.The scattering scenario ranges from line of sight or LOS (forlow angular spread Φk), to rich scattering (for large angularspread Φk). The corresponding performances are shown inFig. 9 in terms of the (average) lowest SINR at the outputof the scheduler for a number of active users L = 9 orL = 10. We show that a critical threshold angular spreadexists above which the performances of the fixed GoB aresimilar to those of AGoB. This means that, for outdoorenvironments with large angular spread, fixed beamformingsolution (GoB) overcomes precoder adaptation (AGoB). Inparticular, for the considered simulation setting, the thresholdangular spread is found to have marginal dependence on thenumber of users (see also Fig. 10) and it is approximatelyequal to 25-27 deg. The performances of opportunistic randombeamforming (dashed lines in Fig. 9) scale with the angularspread, as expected. For the considered setting, an angularspread of 35 deg makes L = 20 users be sufficient to havecomparable performances to those of GoB and AGoB.

In Fig. 10 the (average) lowest SINR at the output of thescheduler is plotted versus both the number L of active usersand the angular spread Φk. As expected, since in practice thenumber of users competing for the same time-frequency unit islimited, the adaptive strategy is shown to be a reasonable can-didate for outdoor environments where multipath propagationis characterized by a moderate angular spread.

C. Cell CoverageIn this section, a cell coverage analysis is dealt with under

the simplifying assumption that the spatial frequency fk can

5

10

15

20 0 9 18 27 36 45 54

0

5

10

15

20

25

Angular spread

Number of usersL

SIN

R (3

rd c

hann

el) [

dB]

2

4

6

8

10

12

14

16

18

20

22

AGoB

ORBF

GoB

SINR [dB]

[deg] kΦ

5

10

15

20 0 9 18 27 36 45 54

0

5

10

15

20

25

Angular spread

Number of usersL

SIN

R (3

rd c

hann

el) [

dB]

2

4

6

8

10

12

14

16

18

20

22

AGoB

ORBF

GoB

SINR [dB]

[deg] kΦ

Fig. 10. Average lowest SINR at the output of the scheduler for varyingchannel conditions (angular spread), and number of active users L rangingfrom 5 to 20.

be assumed as a reliable measure of the angular positionof the kth MS within its cell10. A performance comparisonis firstly carried out by analyzing the probability pM thatall the M channels can be reliably set up: this probabilityis pM = Pr[γ(min

k∈S(ρk)) ≥ 1] for the AGoB, and pM =

Pr[γ(mink∈S

(ρk)) ≥ 1] for the GoB. It is evaluated in Fig. 11for both methods, for M = 3 and for varying number L ofactive users. As before, the angular spread Φk at the BS is aone-sided Gaussian random variable, Φk ∼ N (15 deg, σ2Φ),Φk ≥ 0, with mean 15 deg and standard deviation σΦ = 5deg.Opportunistic random beamforming is not considered in thisanalysis as its performance was proved to be unsatisfactoryin moderate angular spread environments (see the results inSection VI-B). On the other hand, the comparison betweenAGoB and GoB in Fig. 11 shows that, in order to scheduleM users for the same time-frequency unit, the first methodrequires a total number of MSs that is lower compared tothe second approach: for L = 7 AGoB guarantees a thirdchannel to be set up with probability larger than 90%, whileGoB needs an higher L value to achieve the same reliability.We may thus conclude that, whenever a moderate number ofactive users is considered, the AGoB approach can lead to asignificant increase in the average number of scheduled users.

To further comment on this aspect, let us consider thefollowing metric

ξ(αk) , Pr(αk, |S| =M | k ∈ S) == Pr(αk|k ∈ S, |S| =M) · pM (16)

that represents the probability density function of the angularposition αk for any user belonging to the selected subset Swith probability pM (thus cardinality |S| = M ). The metric(16) is chosen so as to compare the angular distributions of thescheduled users that result from the two spatial multiplexingtechniques, under the constraint that the BS is simultaneouslyserving M = 3 users (full loaded BS). The function ξ(αk) is

10Notice that this is not true for NLOS channels, while it holds true whenthe spatial scattering is symmetrically distributed around the LOS direction.

11

30.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of users L4 5 6 7 8 9 10 11 12

AGoBGoB

3rd

setu

p ch

anne

lpro

babi

lity

p M

Fig. 11. 3rd set up channel probability pM vs. the number of users L forAGoB and GoB (M = 3).

-54 -36 -18 0 18 36 540.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

DOD αk [deg ]

AGoBGoB

1 2 3

ξ (α

k)

Fig. 12. Probability density function for the angular position αk of any userk that belongs to the selected user subset S, for AGoB and GoB (L = 7).The user subset S is constrained to have cardinality M .

used so as to analyze the coverage capabilities of the proposedtechniques. Notice that the distance of the MSs towards thedestination is fixed. This is done to highlight the advantagesof the adaptive and fixed scheduling schemes rather than theimpact of the path-loss or/and shadowing on the downlinkperformances. In Fig. 12, the probability function ξ(αk) isplotted versus the angular position αk of the allocated usersfor both the AGoB and the GoB schemes for L = 7. Althoughfor AGoB the probability pM of serving M users is higher withrespect to the GoB, the adaptive technique induces a clusteringof the selected users into M groups to minimize the MUI(those groups are indicated as 1, 2, 3 in Fig. 12 for M = 3). Onthe other hand, the switching phase of the fixed GoB is hereinshown to provide a more uniform cell coverage compared withAGoB. While for mobile applications as in IEEE 802.16-2005standard [9] this aspect might be less relevant, this is not thecase for fixed applications (see IEEE 802.16-2004 standard[9]), where fixed or nomadic terminals are randomly placedwithin the cell and are competing for the same time-frequencyresource unit.

VII. CONCLUDING REMARKS

In this paper spatial multiplexing techniques for a MIMOOFDM FDD system have been compared based on varyingchannel conditions ranging from pure LOS (with negligibleangular spread) to rich scattering (i.e. for large angular spread).An adaptive technique (AGoB) that enhances the existingGoB scheme has been proposed, based on the simultaneoustransmission of fixed weighted beams that are adapted to thechannel and MUI conditions.

For outdoor environments featuring a moderate angularspread, although users’ subset at the BS is suboptimallydesigned according to partial CSI in the form of spatial fre-quencies and SINR lower bounds, AGoB is shown to providesignificant benefits compared to the fixed GoB approach. Thesame time-frequency unit can be therefore scheduled to severalusers in the same coverage area leading to a more efficient useof channel resources.

The main conclusions that can be derived from the nu-merical analysis can be summarized as follows: i) precoderadaptation in the form of AGoB reveals as a better choicein practice with respect to fixed beams solution (GoB) whenoutdoor environments exhibit small angular spread (e.g., Φ ≤25); ii) the additional bandwidth cost (in terms of feedbackrate and pilot subcarriers) is negligible as far as second orderchannel statistics (spatial correlations) are slowly varying (aswith the spatial frequency).

When environments with richer scattering (larger Φ) areconsidered, the benefits of precoder adaptation vanish: being,in that case, the interference from other in-cell users spreadover the spatial domain, the effectiveness of zero-forcing basedadaptation is significantly reduced, while the use of simplerfixed beams for side-lobe suppression still gives valuableimprovements.

Finally, the switching phase of the fixed GoB is shown toprovide a more uniform cell coverage with respect to AGoB.This aspect is mostly relevant in fixed or nomadic applications(see IEEE 802.16-2004 standard [9]) where each user hasto be served with a guaranteed quality and regardless of itsposition in the cell. On the other hand, in mobile applicationswhere users are expected to rapidly cover the whole cell sector,precoder adaptation (as in AGoB) is more effective in that itmaximizes the number of users that are served simultaneously.

VIII. APPENDIX

A. Spatial correlationUnder small angular spread assumption, the spatial correla-

tion can be approximated as a function of the main DOD, sothat αn,k ' αk, ∀n, therefore

RT,k =PP

n=1 σ2k,na

∗T(αn,k)a

TT(αn,k) '

' a∗T(αk)aTT(αk)PP

n=1 σ2k,n = a

∗T(αk)a

TT(αk). (17)

An estimate of αk, herein referred to as αk, can be easilyobtained by calculating, at first, the spatial correlation. GivenF channel matrix realizations Hk,1, ...,Hk,F , the estimationis obtained as

RT,k = F−1PF

q=1HHk,qHk,q (18)

12

by averaging over a number of fading realizations. If uncor-related fading over each frequency bin can be assumed (samespatial structure needs to be observed), then summation in (18)should be performed over the F bins in the frequency domain.For applications where the channel is frequency flat (e.g., fixedor nomadic applications) averaging should instead be obtainedover time (e.g., during a specific preamble). By applying (17)the estimation of αk reduces to the situation described in (9).

B. Spatial frequency estimation

The estimation for the spatial frequency fk in (9) has to becarried out with no prior information whenever the terminal isturned on. Notice that, under the assumption of quantizationof fk over b bits, the maximization in (9) requires O(N2

T2b)

flops that can be easily reduced to O(NT − 1)2b) taking intoaccount the symmetric structure of RT,k.

The frequency estimate can be updated frame by frame dur-ing each spatial channel allocation phase. A low-complexityalgorithm is proposed for the update. The estimate fk, forthe spatial frequency in the -th frame is calculated fromthe previous frame estimate fk, −1 and the current channelcorrelation RT,k according to:

fk, = fk, −1 + μ∂

∂f

¯bH(f)RT,kb(f)

¯f=fk, −1

. (19)

The gradient in (19) is herein approximated by exploiting theHermitian symmetry of RT,k. It can be shown indeed, bytedious but straightforward algebraic computations, that thefollowing approximation holds:

bH(f)RT,kb(f) ' Γ0[RT,k] + Re[Ω(RT,k,f)] (20)

where the term Γn[RT,k] = 1NT−i

PNT−ii=1 [RT,k]i,i+n

represents the average of the n-th diagonal elementsof matrix RT,k, while Re[Ω(RT,k,f)] is thereal part of the complex quantity Ω(RT,k,f) =2NT

PNT−1n=1 Γn [RT,k] exp

¡−j2πfn∆T

λ

¢. The gradient

of (20) can be easily written as

∂

∂fbH(f)RT,kb(f) '

∂

∂fRe[Ω(RT,k,f)] =

=2π

λ∆T Im [Ω(RT,k,f)] . (21)

Including the term 2πλ ∆T into μ quantity, the tracking equation

(19) reduces to:

fk, = fk, −1 + μ ImΩ(RT,k,fk, −1) (22)

As regards the computation of the step-size μ, notice thatif fk, −1 is near the broadside line, the frequency update isconstrained so that

¯fk, − fk, −1

¯< (3 ÷ 4)σα where σα

is the standard deviation of the DOD. Slowly varying spatialfrequencies might allow the use of only 1 bit of feedback forupdating the main DOD information. In this case, only thesign of fk, − fk, −1 is significant.

C. Optimization of βIn this work the minimum user’s spatial separation β is

found by maximizing the sum-throughput averaged over anumber of frames (recall that each frame is composed by Wbursts):

β = arg maxS(β)=k1,...,kM

E[1

W

PMm=1 γ(ρkm , ρkm)], (23)

where γ(ρkm , ρkm) = γ(ρkm) + (W − 1) γ(ρkm) and theexpectation is made with respect to the (block) fading that im-pairs subsequent frames. The SINRs indexes km refer to usersbelonging to the selected subset S = S(β) = k1, ..., kM (Sis a function of β from (12)). Notice that, for the k-th user,the average throughput within square brackets in (23) is theweighted sum of the spectral efficiency experienced during thefirst burst of the frame (from worst-case SINR measurements11

ρkm) and of the spectral efficiency during the remaining W−1bursts (from real SINRs ρkm). For propagation settings as infigure 6 and 7, we found that β = sin(20 deg) is the optimalsolution (see figure 5 in solid lines) and holds regardless of thenumber of active users (as far as L ranges between L = 3 andL = 12). The optimal value can be intuitively motivated bynoticing that the optimal solution would require the scheduledusers to be separated at least by the beampattern main-lobeaperture [33] to minimize the interference. As a final remark,we observe that β should be chosen not too large to have asufficient number of candidate subsets for scheduling; on thecontrary, a small value for β would reduce the effectiveness ofthe worst-case SINR bounds thus decreasing the sum-spectralefficiency for burst 1 (see dashed lines in figure 5).

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Stefano Savazzi received the M.Sc. and thePh.D degree (with honors) in TelecommunicationEngineering from the Politecnico di Milano in 2004and 2008, respectively. He was a Visiting Researcherat the Signals and Systems department, UppsalaUniversity, Sweden in 2005 and at Department ofElectrical and Computer Engineering, University ofCalifornia San Diego (UCSD), CA, US in 2007. Heholds a patent on the work developed for his M.S.thesis. His current research interests include signalprocessing aspects for digital wireless communica-

tions and, more specifically, channel estimation, antenna array processing forMIMO-OFDM systems and cooperative networking.

Monica Nicoli received the M.Sc. degree (withhonors) and the Ph.D. degree in Telecommunica-tion Engineering from Politecnico di Milano, Italy,in 1998 and 2002, respectively. During 2001 shewas a visiting researcher at Signals and Systems,Uppsala University, Sweden. Since 2002, she isAssistant Professor at Dipartimento di Elettronica eInformazione, Politecnico di Milano. Her researchinterests are in the area of signal processing forcommunication systems, including multi-user multi-antenna multi-carrier systems, turbo processing, ra-

dio localization, cooperative communications, channel aware resource alloca-tion. She received the Marisa Bellisario Award in 1999.

Mikael Sternad (S’83-M’88-SM’90) was born inEskilstuna, Sweden, in 1957. He received his Ph.Din automatic control from Uppsala University in1987 and is currently professor in Automatic Controlat Uppsala University. He has since 2000 beenleader of the Wireless IP project, a collaborationbetween Uppsala University, Chalmers University ofTechnology and Karlstad University on 4G wirelesssystems based on adaptive techniques. He has beenactive within the EU WINNER projects, working onchannel prediction, adaptive transmission, deploy-

ment techniques, multiple access and over-all system design, leading the workon MAC layer design. Having authored around 110 papers and five bookchapters and with a background in robust and adaptive estimation, control andfiltering, his main current research interests are the many trade-offs and designissues encountered in wireless systems design and the use of signal processingand control algorithms in communication systems. He is also involved inacoustic signal processing, in particular pre-compensation of room acousticsand car audio systems. He is co-founder and was chairman of the board(2001-2005) of Dirac Research, a company working in this field.