Correlation and Capacity ofMeasured Multi-user MIMO Channels

Florian Kaltenberger, David Gesbert, Raymond KnoppInstitute Eurecom

2229, Route des Cretes - B.P. 19306904 Sophia Antipolis, France

Marios KountourisWireless Networking and Communications Group

The University of Texas at AustinAustin, TX 78712, USA

Abstract-In multi-user multiple-input multiple-output (MU­MIMO) systems, spatial multiplexing can be employed to increasethe throughput without the need for multiple antennas andexpensive signal processing at the user equipments. In theory,MU-MIMO is also more immune to most of propagation limita­tions plaguing single-user MIMO (SU-MIMO) systems, such aschannel rank loss or antenna correlation. However, in this paperwe show that this is not always true. We compare the capacityand the correlation of measured MU-MIMO channels for bothoutdoor and indoor scenarios. The measurement data has beenacquired using Eurecom's MIMO Openair Sounder (EMOS).The EMOS can perform real-time MIMO channel measurementssynchronously over multiple users. The results show that inmost scenarios MU-MIMO provides a higher throughput thanSU-MIMO also in the measured channels. However, in outdoorscenarios with a line of sight, the capacity drops significantlywhen the users are close together, due to high correlation at thetransmitter side of the channel. In such a case, the performanceof SU-MIMO and MU-MIMO is comparable.

I. INTRODUCTION

We study the downlink (or broadcast) channel of a widebandmulti-user multiple-input multiple-output (MU-MIMO) systemin which there are multiple antennas at the base-station (BS)and possibly multiple antennas at the user equipment (UE).

Information theory reveals that if the channel is fully knownat the transmitter and the receiver, the optimum transmitstrategy for the MU-MIMO broadcast channel involves atheoretical pre-interference cancellation technique known asdirty paper coding (DPC) combined with an implicit userscheduling and power loading algorithm [I], [2]. Comparedto a single-user MIMO (SU-MIMO) time division multipleaccess (TDMA) system, DPC can bring a theoretical perfor­mance gain of up to rnax(min(1\l/N, K), 1) in an independentand identically distributed (ij.d.) Rayleigh fading channel,where 1\1 and N is the number of transmit antennas andreceive antennas respectively and K is the number of users[3]. However, DPC is very computationally expensive and thussimpler, sub-optimal transmit strategies have been proposed.

The assumption of an i.i .d. channel is often justified usingthe argument that the users are spatially separated and thus thesignals arriving at different users will be independent even inthe presence of a line of sight (LOS) component [2]. However,it was shown in [4] that the throughput in the measured

This research was supported by the project PACAM with SFR. the ECunder FP7 Network of Excellence project NEWCOM++ and Eurecom.

978-1-4244-2644-7/08/$25.00 ©2008 IEEE

channels is worse than the one in i.i.d. channels. In thispaper we investigate the spatial correlation of measured MU­MIMO channels for both outdoor and indoor scenarios. Wealso compare the performance of different linear MU-MIMOprecoding schemes, such as zero-forcing (ZF) and regularizedinversion (also called MMSE precoder) [5].

Realistic MU-MIMO channel measurements have been ob­tained using Eurecom's MIMO Openair Sounder (EMOS)[6]. The EMOS can perform real-time channel measurementssynchronously over multiple users moving at vehicular speed.For this paper, we have used four transmit antennas andfour users with one antennas each. The measured channelsare used to calculate the capacity offline, assuming a perfectfeedback channel. To the best of our knowledge, no suchcomparison based on real MU channel measurements has beenreported. Real indoor channel measurements have been usedin [7] for the evaluation of the proposed MU-MIMO scheme.Real outdoor channel measurements have been used in [8] tostudy limited feedback. However, the channel measurementswere obtained with one receiver at different times and notsynchronously as in our measurements.

The paper is organized as follows. We introduce the sig­nal model in Section II. The performance metrics for theevaluation of the channel measurements are presented in III.In Section IV we describe the EMOS in some more detailand explain how the channel measurements are performed. InSection V the measurement campaign is described and resultsarc discussed. We tinally give conclusions in Section VI.

II. SYSTEM MODEL

We consider a multi-user, multi-antenna downlink channelin which a BS equipped with 1\1 antennas communicates withK ::; M terminals, each equipped with N antennas. Thereceived signal Yk ,m',fJ E CNx 1 of the k-th user at time mand frequency q is mathematically described as

Yk.rn ,q == Hk;m,qXrn,q + llk,rn,q for k == 1, ... , K (I)

where H k,rn ,q E eNx!vI represents the k-th user channelresponse at time m and frequency q, X m',fJ E eMx1 is thevector of transmitted symbols at time m and frequency q, andllk,rn.q E eN x 1 is ij.d. circularly symmetric additive complexGaussian noise with zero mean and variance (J2, Vk. Weassume that each of the receivers has perfect and instantaneous

(5)

(6)

where Wk is obtained by normalizing the k-th column of W.Assuming equal power allocation over the users, the achiev­

able sum rate is given by

where the second-term in (4) represents the multi-user inter­ference. We assume that each user will decode S :::; N streamsthat constitute its data. The goal of linear precoding is to

design {Wk }f=l based on the channel matrix knowledge, soa given performance metric is maximized for each stream.

J) Zero-Forcing Precoding (Channel Inversion): For easeof exposition, we assume N == 1 and we define H ==[hi ... hI] T. The unit-norm beamforming vector of user kis denoted as Wk E Cl\lxl, k == 1, ... ~ K.

A standard suboptimal approach providing a promisingtradeoff between complexity and performance is zero-forcingprecoding, also known as channel inversion. In ZF, the pre­coder is designed to achieve zero interference between theusers, Le., hkwj == 0 for j =I k. The ZF precoding matrix isgiven by the Moore-Penrose pseudoinverse of H

K (P 2)R ZF =~ log2 1 + Ka2 IhkWkl '

When the channel is ill-conditioned, at least one of the singularvalues of (HHH) -1 is very large, resulting in a very low SNR

at the receivers. Note also that ZF precoding-in contrast toZF (least-squares) equalization at the receive side which causesnoise enhancement when the channel is nearly rank deficient­incurs an excess transmission power penalty. Therefore, thecapacity of channel inversion with no user selection does notincrease linearly with 1\1, unlike the optimum capacity.

2) MMSE Precoding (Regularized Channel Inversion): Forrank-deficient channels, the performance of ZF precoding canbe improved by a regularization of the pseudo-inverse, whichcan be expressed as:

(3)

Csc (H1 ~ ... ~ H K ~ P) ==

K II + Hk: (Lf=l ~j) H[11Inax E log2 ~

~,,;;>O,L::t<~l tr(~,,)S;P ~:=1 II + H~, (Lj# ~j) H[11(2)

III. CAPACITY AND CORRELATION

To analyze the measured MU-MIMO channels we use thefollowing performance metrics: sum rate capacity under OPC,sum rate capacity under MU-MIMO linear precoding, sum ratecapacity under SU-MIMO TOMA, correlation at the receiver,correlation at the transmitter.

knowledge of its own channel. The transmitter is subject toan average power constraint, i. e., 1E{x;;'.qXrn .q} :::; P, whichimplies that the total transmit power is not dependent on thenumber of transmit antennas. For notation convenience, in thefollowing sections we drop the time and frequency indices.

where the maximization is over the set of all positive semidef­inite transmit covariance matrices I: k . k == 1~ .... K. Theobjective function of the maximization in (2) is a concavefunction of the covariance matrices, making it very difficultto deal with. Fortunately, due to the MAC-BC duality, thesum rate capacity of the MIMO BC is equal to the sum ratecapacity of the dual MAC with power constraint P

A. Dirty Paper Coding

From the results in [1], [9], [10], the sum capacity ofthe MU-MIMO downlink channel can be expressed by the

following maximization:

where each of the matrices Q'l is a poSltlve semidefinitecovariance matrix. Since (3) involves the maximization of aconcave function, efficient numerical algorithms exist. In thispaper, we use the specialized algorithm developed in [II] tocalculate CSC(H1.... ~ H K . P).

It has been shown [12] that the sum rate capacity given

in Equation (3) is actually achieved by using OPC. However,npc is very complex and difficult to implement. Thus we alsostudy linear precoding schemes in the next section.

B. Linear Precoding Sum Rate

Let 8k E eN x 1 denote the k-th user transmit symbol

vector. Under linear precoding, the transmitter multiplies thedata symbol for each user k by a precoding matrix W k ECAl x N so that the transmitted signal is a linear function

x == ~~=1 W k81.~. The resulting received signal vector foruser k is given by

YI.: == Hk,WI.~8k + E Hk W j 8j + nl.:~ (4)j#k

(7)

where (3 is a regularization factor. The above scheme isoften referred to as Minimum Mean Square-Error (MMSE)precoding due to the analogous with MMSE beamformingweight design criterion if the noise is spatially white. Theachievable throughput is given by

~ ( \hkw k\2 )R MMSE == L..J, log2 1 + . . .' 2.. 2 ~ (8)

k=l Lj#l.~lhkwjl +Ka IPwhere Wk is the normalized k-th column of the precoder givenin (7).

Similarly to MMSE equalization, a non-zero ;3 value resultsin a measured amount of multi-user interference. The amountof interference is determined by /3 > 0 and an optimaltradeoff between the condition of the channel matrix inverseand the amount of crosstalk ought to be found. In practice,

the regularization factor is commonly chosen as f3 == Ala 2IPmotivated by the results in [5] that show that it approximatelymaximizes the SINR at each receiver, and leads to linear

TABLE IEMOS PARAMETERS

(b) Powerwave Antenna

(d) Panorama Antennas

(a) Server PC with PLATON boards

(c) Dual-RF CardBus/PCMCIA Card

B. Sounding Signal

The EMOS is using an OFDM modulated sounding se­quence. One transmit frame is 2.667 ms long and consistsof a synchronization symbol (SCH), a broadcast data channel(BCH) comprising 7 OFDM symbols, a guard interval, and48 pilot symbols used for channel estimation (see Fig. 2).The pilot symbols are taken from a pseudo-random QPSKsequence defined in the frequency domain. The subcarriersof the pilot symbols are multiplexed over the four transmitantennas to ensure orthogonality in the spatial domain. TheBCH contains the frame number of the transmitted frame thatis used for synchronization among the UEs.

Fig. 1. EMOS base-station and user equipment [6]

C. Channel Estimation Procedure

Each UE first synchronizes to the BS using the SCH. Itthen tries to decode the data in the BCH. If the BCH can bedecoded successfully, then the channel estimation procedureis started. The channel estimation procedure consists of two

antenna (part no. 7760.00) composed of four elements whichare arranged in two cross-polarized pairs (see Fig. 1(b». TheUEs consist of an ordinary laptop computer with Eurecom'sdual-RF CardBus/PCMCIA data acquisition card (see Fig.1(c» and two clip-on 3G Panorama Antennas (part no. TCLIP­DE3G, see Fig. 1(d». The platform is designed for a fullsoftware-radio implementation, in the sense that all protocollayers run on the host PCs under the control of a Linux realtime operation system.

(10)

(11)

R Tx == JE{HHH},

RRx == JE{HHH}.

Parameter ValueCenter Frequency 1917.6MHz

Bandwidth 4.8 MHzBS Transmit Power 30dBm

Number of Antennas at BS 4 (2 cross polarized)Number of UE 4

Number of Antennas at UE 2Number of Subcarriers 160

D. Transmit and Receive Correlation

The transmit and receive correlation matrices of the MU­MIMO channel H are defined as

C. Time Division Multiple Access SUln Rate

The capacity of a single user k is given by

CSU-MIMO(Hk~P) == max log2I I + HkQkHfl·Qk 2::0,tr(Qk )sP

(9)The maximum is achieved by choosing the covariance matrixQk to be along the eigenvectors of the matrix HkH{: andby choosing the eigenvalues according to the water fillingprocedure [13]. The maximum sum rate capacity is achievedby transmitting to the user with the largest single-user capacity.However, in this paper we assume that all users are servedfairly proportional in a round robin fashion, i. e., we treat eachH k as a different realization.

For the measurements we calculate the expectation by takingthe mean of the channel over all frequencies q and all framesrn in one measurement.

R Tx and RRx give insight to what extent the signals leavingthe different transmit antennas and the signals arriving at thedifferent receive antennas respectively are correlated. Note thatin general RTx and RRx do not fully characterize the secondorder statistics of the channel [14]. However, they are moreintuitively and sufficient for the purpose of this paper.

IV. THE EMOS MULTI-USER PLATFORM

A. Hardware Description

The Eurecom MIMO Openair Sounder (EMOS) is basedon the OpenAir hardware/software development platform atEurecom. The platform consists of a BS that continuouslysends a signaling frame, and one or more UEs that receivethe frames to estimate the channel. For the BS, an ordinaryserver PC with four PLATON data acquisition cards (see Fig.I(a» is employed along with a Powerwave 3G broadband

capacity growth with AI. The performance of MMSE iscertainly significantly better at low SNR and converges tothat of ZF precoding at high SNR. However, MMSE doesnot provide parallel and orthogonal channels and thus powerallocation techniques cannot be performed in a straightforwardmanner.

Fig. 2. Frame structure of the OFDM Sounding Sequence.

. . . . . .. - MU-MIMO OPC 4U outdoor far- SU-MISO TOMA 4U outdoor far- - - MU-MIMO OPC 4U indoor- - - SU-MISO TDMA 4U indoor, , , , , , , MU-MIMO OPC 4U outdoor near, , , , , , , SU-MISO TDMA 4U outdoor near

0.3

0.2

0.1

U-S 0.5

inside standard passenger cars which were beeing driven alongthe routes shown in Fig. 3, keeping a large distance. In thesecond measurement, the cars were parked close together on aparking space indicated in the figure. The third measurementwas conducted indoors in the neighboring building. The indoorscenario is characterized by strong reflections (the buildingsis actually located behind the main lobe of the antenna) andthus there is no LOS. The users were all in the same room,moving around slowly.

Fig. 4. eOF of the sum rate of SU-MIMO TOMA compared to MU-MIMOwith ope for all three measurements (meas. no. I = "outdoor far", meas. no.2 = "outdoor ncar", meas. no. 3 = "indoor"). The average SNR is fixed toIOdB for each user.

°01-.....-.t!t::...._....~L..--8L..--1.t::O===1I::2

===1I::4

===1I::6

==:::::.J18

bits/sec/Hz

Multiuser Capacity for M=4, N=1, and SNR=1OdB

0.6

C. Discussion

0.9

0.7

0.8

0.4

It can be seen from Fig. 4 that MU-MIMO DPC as wellas SU-MISO TDMA do not show a very high variabilitywith respect to the three different measurements. However,the linear MU-MIMO precoding schemes (see Fig. 5) are verysensitive to the channel conditions. Especially the performanceof the ZF precoder drops significantly in the outdoor scenariowhere the users are close together. In the indoor scenarioand the other outdoor scenario where all users are well

B. Results

For all evaluations in this paper, we use only the firstantenna at the UEs. Further, to ensure a constant averageSNR of 10 dB at the UEs, the channel of every user isnormalized over the whole measurement run (about 50 sec).Firstly, we compare the performance of MU-MIMO usingDPC, ZF precoding, and MMSE precoding as well as SU­MIMO TDMA based on the empirical cumulative densityfunction (CDF) of the sum rate (Equations (3), (6), (8), and(9)). The results are plotted in Figures 4 and 5. Secondly wecompare the transmit and receive correlation matrices R Tx

and RRx (cf. Equations (10) and (II)). In Fig. 6 the matricesare represented graphically using different shades of gray toindicate the absolute value of the matrix entry (white = nocorrelation, black =high correlation).

Guard Interval(8 OFDM Symbols)•

steps. Firstly, the pilot symbols are derotated with respect tothe first pilot symbol to reduce the phase-shift noise generatedby the dual-RF CardBuslPCMCIA card. Secondly, the pilotsymbols are averaged to increase the measurement SNR. Theestimated MIMO channel is finally stored to disk. For a moredetailed description of the channel estimation see [6].

D. Multi-user Measurelnent Procedure

In order to conduct multi-user measurements, all the UEsneed to be frame-synchronized to the BS. This is achieved bystoring the frame number encoded in the BCH along with themeasured channel at the UEs. This way, the measured channelscan be aligned for later evaluations. The frame number is alsoused to synchronize the data acquisition between UEs. Onemeasurement run (fi Ie) starts every 22.500 frames (60 sec) andis exactly 18.750 frames (50 sec) long.

V. MEASUREMENTS AND RESULTS

A. Measurement Description

For the presentation in this paper we selected three differentrepresentative measurement runs. Two of the measurementswere conducted outdoors in the vicinity of the Eurecominstitute. The scenario is characterized by a semi-urban hillyterrain, composed by short buildings and vegetation with apredominantly present LOS. Fig. 3 shows a map of the envi­ronment. The BS is located at the roof of Eurecom's southmostbuilding. The antenna is directed towards Garbejaire, a smallnearby village. Tn the first measurement the DEs were placed

Fig. 3. Map of the measurement scenario. The position and the openingangle of the BS antenna are also indicated. In the first measurement the userswere driving in cars along the indicated routes (the colors show the receivedsignal strength in dBm along the routes). In the second and third measurementthe users were close together at the indicated positions.

outdoor far indoor outdoor near

Multiuser Capacity for M=4, N=l, and SNR=10dB

Fig. 5. CDF of the sum rate of MU-MIMO with ZF and MMSE precoding forall three measurements (meas. no. I = "outdoor far", meas. no. 2 = "outdoorncar", meas. no. 3 = "indoor"). The average SNR is fixed to J OdB for eachuser.

0.1

0.2

0.3

0.5

0.4

2 3 4

234234

2342 3 4

234

Fig. 6. Absolute values of the Tx and Rx correlation matrices.

§:~....:.....,.. :~ :ai...................i.:.~ 3 3 3

4 '. 4 4

§:rsJ:~ :~~ 3 3 3

444

are as orthogonal as possible. Otherwise it is not worth todo MU-MIMO linear precoding, since its performance is thencomparable to SV-MIMO TDMA.

REFERENCES

181614128 10bits/sec/Hz

I- I - MU-MIMO ZF 4U outdoor far- MU-MIMO MMSE 4U outdoor far- - - MU-MIMO ZF 4U indoor

I : -: I - - - MU-MIMO MMSE 4U indoor, , , , , , , MU-MIMO ZF 4U oU1door near, , , , , , , MU-MIMO MMSE 4U outdoor near

0.1

0.3

0.2

0.4

~

80.5

0.6

09/(·..·..·.. ·

0.8 ... /.. ~

0.7 'r'" .

separated, the performance of the linear MU-MIMO schemesis comparable.

These facts can be explained by looking at the channelcorrelation in the different scenarios (Fig. 6). It can be seenthat in all the scenarios, almost no correlation can be measuredat the receive side between the different users. However, at thetransmit side, the situation is different in all three scenarios.In the outdoor case with spatially separated users, there isjust a little correlation between cross polarized componentsof the transmit antenna. In the indoor scenario, there isslightly, but not significantly more correlation. Tn the outdoorscenario where the users are located close together however,the correlation rather strong.

When the channel is strongly correlated it means that thechannel matrix is ill-conditioned. Thus at least one of thesingular values of (HHH) -1 is very large, resulting in a verylow SNR at the receivers, when ZF precoding is used. TheMMSE precoder can alleviate this problem, but still suffersfrom the high correlation at the transmitter.

VI. CONCLUSIONS

We have presented capacity and correlation analysis ofmeasured MU-MIMO channels. The data was acquired usingEurecom's MU-MIMO channel sounder EMOS. We haveshown that linear precoding schemes are very sensitive tothe spatial correlation of the channel. In particular, the per­formance of a ZF precoder drops significantly in outdoorscenarios, when the users are close together. It was foundout that this drop in performance is due to the strong channelcorrelation at the transmit side in those scenarios. This findingis contradicts the common assumption of the MU-MIMOchannel being i.i.d. However, it is true, that in all the measuredMU-MIMO channels, there is no correlation at the receiveside. We conclude that to ensure good performance of MU­MIMO it is essential to do user selection at the base station.The users should be selected in such a way that their channels

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