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404 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005 Performance Analysis of Linear Precoding Based on Field Trials Results of MIMO-OFDM System Hemanth Sampath, Member, IEEE, Vinko Erceg, Senior Member, IEEE, and Arogyaswami Paulraj, Fellow, IEEE Abstract—We use field trial results obtained from a mul- tiple-input multiple-output (MIMO) orthogonal frequency-divi- sion multiplexing (OFDM) wireless system with two transmitter and three receiver antennas (2 3), to first validate the prop- erties of the transmit correlation matrix in a macro-cellular environment. We find that approximately 20% of the locations have well-defined transmit correlation matrices. Furthermore, the eigenvectors of the transmit correlation matrix vary slowly over distance with 60% of the locations having eigenvector variation of less than 1 dB over a distance of 20 m. Next, we quantify the performance of the optimal statistical linear precoding (OSLP) [4], [5] and statistical one-dimensional (1-D) eigenbeamforming (SEB) based on transmit correlation matrices, and the 1-D eigen- beamforming (EB)-based on perfect channel knowledge at the transmitter. We find that the OSLP and SEB schemes obtain array gain over the Alamouti scheme at lower signal-to-noise ratio (SNR) with a median gain of 2.0 (1.5) dB at the 1.0-(3.5) km cell-radii. However, the SEB scheme (unlike the OSLP scheme) looses diver- sity order at higher SNR that leads to a performance loss. The EB scheme provides the best performance over the Alamouti scheme, at the expense of increased feedback requirements. Index Terms—Antenna arrays, beamforming, correlation, eigenvalues and eigenfunctions, feedback, multiple-input mul- tiple-output (MIMO) systems, orthogonal frequency-division multiplexing (OFDM). I. INTRODUCTION A BROADBAND wireless channel has impairments such as time, space, and frequency-selective fading. Space-time coding [9], [10] can provide improved link reliability in a mul- tiple-input multiple-output orthogonal frequency-division mul- tiplexing (MIMO-OFDM) system, when no channel knowledge is available at the transmitter. When channel knowledge is made available at the transmitter, additional spatial processing can be employed at the transmitter to boost system performance. When instantaneous channel knowledge is available at the transmitter, a one-dimensional (1-D) eigenbeamformer (EB) can be employed that transmits symbols on the strongest eigen- vector of each channel realization [12]. The drawback is that such a scheme requires an accurate and fast feedback from receiver to transmitter, which may be difficult to achieve in practice. This motivated the study for using linear precoding techniques based on channel statistics at the transmitter. If only Manuscript received August 15, 2003; accepted January 28, 2004. The editor coordinating the review of this paper and approving it for publication is H. Xu. The authors completed this work while at Iospan Wireless, Inc. H. Sampath is with Qualcomm Inc., San Diego, CA 92121 USA (e-mail: [email protected]). V. Erceg is with Broadcom Inc., San Diego, CA 92128 USA (e-mail: [email protected]). A. Paulraj is with the Department of Electrical Engineering, Stanford Univer- sity, Stanford, CA 94305 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.844151 Fig. 1. Transmit spatial processing for a space-time coded system, at tone . the transmit antenna correlation matrix is known at the trans- mitter, it has been shown that the performance is improved when symbols are transmitted on the eigenvectors of the transmit correlation matrix [4]–[8]. The optimal statistical linear pre- coder (OSLP) that minimizes an upper-bound of the average pairwise error probability (PEP) employs a water-pouring power allocation on the eigenvectors of the transmit correlation matrix [4], [5]. Such a precoder was shown to also minimize a tight upperbound on symbol error rate [6]. In comparison, a statistical 1-D eigenbeamformer (SEB) pours power only on the strongest eigenvector of the transmit correlation matrix [1]. In this letter, we will use field trial results obtained from an Alamouti coded 2 3 MIMO-OFDM wireless system in the downlink (DL) to 1) study the properties of the transmit cor- relation matrix and 2) quantify the performance of the OSLP, EB, and SEB schemes over the Alamouti scheme. II. SYSTEM MODEL Consider a MIMO-OFDM system with carriers (tones), transmit antennas, and receive antennas. It is well- known that by the use of inverse fast Fourier transform (IFFT) and FFT at the transmitter (Tx) and receiver (Rx), respectively, and by the use of cyclic prefix, OFDM converts a broadband fre- quency selective MIMO channel into multiple narrowband flat fading MIMO channels at different carriers (tones). The channel response at each tone is a multiplicative matrix channel, , of size . Fig. 1 shows a space-time block coded system with a spa- tial processor, at tone . The spatial processing can include an OSLP, EB, or SEB scheme. A space-time code block is denoted by a matrix , where is the block index, is the tone-index, is the block-size across the time domain, and is the block-size across the spatial domain . We assume the channel matrix to be constant over a space-time code block, but can vary over multiple code-blocks. The space-time codeword is then processed by an spa- tial processing matrix— , or , where the subscripts indicate the spatial processor type. The pro- cessors and are determined by averaging 1536-1276/$20.00 © 2005 IEEE

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Page 1: Performance analysis of linear precoding based on field trials results of MIMO-OFDM system

404 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005

Performance Analysis of Linear Precoding Based on FieldTrials Results of MIMO-OFDM System

Hemanth Sampath, Member, IEEE, Vinko Erceg, Senior Member, IEEE, and Arogyaswami Paulraj, Fellow, IEEE

Abstract—We use field trial results obtained from a mul-tiple-input multiple-output (MIMO) orthogonal frequency-divi-sion multiplexing (OFDM) wireless system with two transmitterand three receiver antennas (2 3), to first validate the prop-erties of the transmit correlation matrix in a macro-cellularenvironment. We find that approximately 20% of the locationshave well-defined transmit correlation matrices. Furthermore, theeigenvectors of the transmit correlation matrix vary slowly overdistance with 60% of the locations having eigenvector variationof less than 1 dB over a distance of 20 m. Next, we quantify theperformance of the optimal statistical linear precoding (OSLP)[4], [5] and statistical one-dimensional (1-D) eigenbeamforming(SEB) based on transmit correlation matrices, and the 1-D eigen-beamforming (EB)-based on perfect channel knowledge at thetransmitter. We find that the OSLP and SEB schemes obtain arraygain over the Alamouti scheme at lower signal-to-noise ratio (SNR)with a median gain of 2.0 (1.5) dB at the 1.0-(3.5) km cell-radii.However, the SEB scheme (unlike the OSLP scheme) looses diver-sity order at higher SNR that leads to a performance loss. The EBscheme provides the best performance over the Alamouti scheme,at the expense of increased feedback requirements.

Index Terms—Antenna arrays, beamforming, correlation,eigenvalues and eigenfunctions, feedback, multiple-input mul-tiple-output (MIMO) systems, orthogonal frequency-divisionmultiplexing (OFDM).

I. INTRODUCTION

ABROADBAND wireless channel has impairments such astime, space, and frequency-selective fading. Space-time

coding [9], [10] can provide improved link reliability in a mul-tiple-input multiple-output orthogonal frequency-division mul-tiplexing (MIMO-OFDM) system, when no channel knowledgeis available at the transmitter. When channel knowledge is madeavailable at the transmitter, additional spatial processing can beemployed at the transmitter to boost system performance.

When instantaneous channel knowledge is available at thetransmitter, a one-dimensional (1-D) eigenbeamformer (EB)can be employed that transmits symbols on the strongest eigen-vector of each channel realization [12]. The drawback is thatsuch a scheme requires an accurate and fast feedback fromreceiver to transmitter, which may be difficult to achieve inpractice. This motivated the study for using linear precodingtechniques based on channel statistics at the transmitter. If only

Manuscript received August 15, 2003; accepted January 28, 2004. The editorcoordinating the review of this paper and approving it for publication is H. Xu.The authors completed this work while at Iospan Wireless, Inc.

H. Sampath is with Qualcomm Inc., San Diego, CA 92121 USA (e-mail:[email protected]).

V. Erceg is with Broadcom Inc., San Diego, CA 92128 USA (e-mail:[email protected]).

A. Paulraj is with the Department of Electrical Engineering, Stanford Univer-sity, Stanford, CA 94305 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2005.844151

Fig. 1. Transmit spatial processing for a space-time coded system, at tonem.

the transmit antenna correlation matrix is known at the trans-mitter, it has been shown that the performance is improved whensymbols are transmitted on the eigenvectors of the transmitcorrelation matrix [4]–[8]. The optimal statistical linear pre-coder (OSLP) that minimizes an upper-bound of the averagepairwise error probability (PEP) employs a water-pouringpower allocation on the eigenvectors of the transmit correlationmatrix [4], [5]. Such a precoder was shown to also minimizea tight upperbound on symbol error rate [6]. In comparison, astatistical 1-D eigenbeamformer (SEB) pours power only onthe strongest eigenvector of the transmit correlation matrix [1].

In this letter, we will use field trial results obtained from anAlamouti coded 2 3 MIMO-OFDM wireless system in thedownlink (DL) to 1) study the properties of the transmit cor-relation matrix and 2) quantify the performance of the OSLP,EB, and SEB schemes over the Alamouti scheme.

II. SYSTEM MODEL

Consider a MIMO-OFDM system with carriers (tones),transmit antennas, and receive antennas. It is well-

known that by the use of inverse fast Fourier transform (IFFT)and FFT at the transmitter (Tx) and receiver (Rx), respectively,and by the use of cyclic prefix, OFDM converts a broadband fre-quency selective MIMO channel into multiple narrowband flatfading MIMO channels at different carriers (tones). The channelresponse at each tone is a multiplicativematrix channel, , of size .

Fig. 1 shows a space-time block coded system with a spa-tial processor, at tone . The spatial processing can include anOSLP, EB, or SEB scheme. A space-time code block is denotedby a matrix , where is the block index, isthe tone-index, is the block-size across the time domain, and

is the block-size across the spatial domain .We assume the channel matrix to be constant over aspace-time code block, but can vary over multiple code-blocks.The space-time codeword is then processed by an spa-tial processing matrix— , or ,where the subscripts indicate the spatial processor type. The pro-cessors and are determined by averaging

1536-1276/$20.00 © 2005 IEEE

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005 405

over fast fading channel realizations and, hence, are assumed tobe stationary over time1.

Assuming an OSLP scheme at the transmitter, the receivedsignal at tone is given by

(1)

where is the received space-time codewordand is the noise matrix. We assume that

, wheredenotes the complex conjugate transpose. Equations sim-

ilar in spirit to (1) can be written for EB and SEB schemes. Inthis letter, we assume the average transmit symbol constellationenergy to be unity, i.e., and

, i.e., the OSLP scheme is powerpreserving. We assume for the rest of the letter that .

III. CHANNEL MODEL

The MIMO channel at tone can be written as (see, e.g.,[11])

(2)

where is the transmit antenna covariance ma-trix, is the receive antenna covariance ma-trix, is an matrix of independent complexGaussian elements having zero mean and unit variance,is the line-of-sight (LOS) constant channel matrix, and isa phase term to account for transceiver motion (with time) andreceiver measurement phase noise error. The channel -factoris the ratio of the powers of the LOS component (fixed) to thenonline-of-sight (NLOS) or scatter component.

The correlation coefficient between the transmit antennasand is given by the th element of , denotedas , where we have assumed . Similarly, wehave for the receive antennas. The transmit correlationmatrix normalized across the receive antennas is computed as

(3)

where denotes expectation over time . Using(2) in (3), and noting that and

, we get

(4)

Let us define the singular value decomposition (SVD) aswhere and

1It will be seen later (from field trial results), that the statistics slowly varyover distance due to shadow fading and movement of scatterers in a suburban en-vironment. This can be handled by updating the linear precoder and the channelstatistics at a slower rate, assuming stationarity over short distance.

are orthogonal eigenmatrices and is adiagonal matrix containing the nonzero singular-valuesarranged in a decreasing order from top-left to bottom-right. Forhigh -factor, the transmit correlation matrix tracks the LOSchannel matrix, i.e., .For low -factor, the transmit correlation matrix tracks thetransmit covariance matrix, i.e., . For ani.i.d. Rayleigh-fading channel with zero -factor, we have

.

IV. TRANSMIT SPATIAL PROCESSING

A. Optimal Statistical Linear Precoder

Suppose the codeword is transmitted and themaximum likelihood (ML) decoder in the receiver chooses thenearest distinct codeword . The code error matrix canbe written as . From [9],the pairwise error probability (PEP) for a given isupper-bounded as

(5)

where is the exponential function and is the traceof a matrix. The performance of the space-time coding schemeis determined by the minimum distance error matrix,

,where denotes the determinant of a matrix. For orthog-onal space-time block codes, we have that , where

assuming a unit average constellation en-ergy and an -QAM constellation. Furthermore, assuming thechannel conditions are such that there is no receive correlationand no LOS channel component ( and ), wecan write an upper bound for the worst-case PEP averaged overall channel realizations as [9]

(6)

The that minimizes the average worse case PEP can hencebe obtained by solving the following optimization problem:

(7)

subject to (8)

where (8) constrains the total power across transmit an-tennas at each OFDM tone , and denotes the trace ofthe matrix.

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406 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005

The solution of the optimization problem posed in (7) is givenfrom [4], [5] as2

(9)

(10)

where the sign means and is a constantthat is computed from the trace constraint and is the noisevariance.

We see that the OSLP processor transmits symbols on theeigenmodes of the transmit covariance matrix. The power allo-cation policy is a form of the well-known water-filling solution,for which iterative algorithms exist to compute the power oneach eigenvector. The power allocation policy in (10) assumesno receive correlation or LOS channel component. However, aswill be seen from field trial results, this power allocation policyprovides considerable gains in a non-LOS environment wheremajority of users have low factors and low receive antennacorrelations.

B. Statistical 1-D EB

The SEB scheme transmits symbols on the strongest eigen-mode of the transmit correlation matrix. Hence, we can write

(11)

where the is the dominant eigenvector obtainedfrom the first column of the eigenmatrix .

C. 1-D EB

The EB scheme transmits symbols on the strongest eigen-mode of each channel realization . Let us definethe singular value decomposition (SVD):

, where andare and orthogonal eigenmatrices,respectively, and is a matrix containingthe singular-values arranged along the leadingdiagonal, in a decreasing order from top-left to bottom-right.The nondiagonal terms in the matrix are equal to zero.Hence, we can write

(12)

where the is the dominant eigenvector obtained fromthe eigenmatrix .

V. FIELD TRIALS

A. Description

The performance of OSLP, SEB, and EB were verifiedusing channel measurements from the commercial 2 3MIMO-OFDM Airburst communication system in a test-mode.The system employed a total of 1024 tones spanning a 2-MHzbandwidth. In the test-mode, the base-station (BS) transmittedknown training signals and the receiver sampled the receivedsignals over tones spaced 72 kHz in frequency and OFDMsymbols spaced every 50 ms. The transmitter antennas were

2For the readers of papers [4] and [5], we note that a scaling factor of 4 wasabsorbed in the definition ofJ . Furthermore, in [4] and [5], the � term was er-roneously omitted in the derivation steps of the optimal power allocation policy.As a result, the scaling factor 4� does not appear in the final expression for� (m) in [4] and [5].

located on the rooftop (approximately 17-m high) of an officebuilding in the San Francisco Bay area and the receiver wasplaced at the top of a 3-m-high mast mounted on a van. Movingmeasurements at low vehicular speeds were taken to covervarious receiver locations for three fixed values of distancebetween transmitter and receiver (1.0 and 3.5 km) over a 120sector. The environment can be characterized as suburbanwith residential blocks (1–3 floors), commercial buildings(2–5 floors), moderate trees (10–20 m high), and slightly hillyterrain. The transmit and receive linear arrays, both equippedwith directional antenna elements, were approximately pointedtoward each other. At the transmitter, two spatially separatedantennas with vertical polarization was used. The transmitantennas each had a gain of 16 dBi with a 3-dB beamwidth(azimuth) of 120 and were spaced 16 wavelengths apart. Atthe receiver, three spatially separated antennas were used withslant 45 polarization. The receive antennas were spaced0.75 wavelengths apart and each had a gain of 12 dBi with a3-dB beamwidth of 90 . The total transmit power and EIRPwere 35.5 and 51 dBm, respectively. The transmission was atthe 2.683-GHz (MMDS band) center frequency.

B. Channel Model Validation

Moving channel measurements were used to compute -fac-tors, transmit correlation matrices, and antenna correlation co-efficients. These parameters were computed using 50 consecu-tive channel realizations recorded over a 2.5-s time interval anda spatial distance of 3–7 m depending on mobile speed. Themeasurements were repeated at several locations at the 1.0- and3.5-km cell-radii. Measurements indicate that the LOS compo-nent is small for most locations, with the median -factor lessthan 0 dB across the cell. Only 10% of users at 1.0 (3.5) km have

-factors greater than 9 (2) dB.Let us define the condition number of the 2 2 transmit

correlation matrix to be, where and are the max-

imum and minimum eigenvalue, respectively. Fig. 2 showsthe cumulative distribution function (CDF) of the conditionnumber of the transmit correlation matrix and the CDFof the transmit correlation coefficient , at the 1.0-and 3.5-km cell-radii. As observed from (4), the transmit corre-lation matrix and, hence, condition number measured includesthe effect of the LOS component. However, the correlationcoefficient is based on the transmit covariance matrix and doesnot include the LOS component. From Fig. 2, we see that morethan 20% of the users have condition numbers greater than12(6) dB and correlation coefficient greater than 0.55 (0.4) atthe 1.0- (3.5-) km cell-radii. Indeed, the transmit correlationmatrices are more well-defined at close-in distances to BS dueto less scattering in the environment.

Next, we verify the eigenvalue and eigenvector variationof with distance. Let the initial reference measure-ment of be at location (obtained by averaging 50channel realizations recorded over 2.5 s and 3–7 m spatialdistance around location , depending on the mobile speed).Let the dominant eigenvector and condition number of ,measured at a distance meters from the location begiven as and , respectively. The (dominant)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005 407

Fig. 2. CDF of transmit correlation coefficient and condition number of transmit correlation matrix.

eigenvector variation between locations and is given as. The condition number

variation is given as ,where denotes the norm.

Fig. 3 shows the distributions dB anddB obtained at the 1.0- and 3.5-km cell-ra-

dius. We see that the dominant eigenvector of the correlationmatrix varies slowly with distance, with more than 60% of loca-tions having eigenvector variation and eigenvalue variation ofless than 1 dB across a spatial distance of 20 m (which is muchlarger than the channel coherence distance). This means thateigenvectors can be estimated and tracked during mobile motionwith limited feedback spanning several channel coherence timesin a macro-cellular environment. However, we see that the condi-tion number varies faster compared to the eigenvector, with lessthan 40% of users having an eigenvalue variation of less than 1dB across a spatial distance of 20 m, implying a faster feedback ofeigenvalues is required for good linear precoding performance.We can attribute this variation to shadowing effects of buildingsand trees in a macro-cellular environment. Finally, we also notethat the eigenvectors are more stationary at the 3.5-km cell-radiusthan at 1 km. This could be because for a unit spatial distancetraversed by the mobile, the angular distance relative to the BSchanges more rapidly at close-in distances to the BS.

C. Performance Improvement Over Alamouti Scheme

TheAlamouticodeisanorthogonalspace-timeblockcodewithand provides maximum diversity for a system with

transmit antennas. In the first OFDM symbol time slot,thesymbols and aresenton the two transmitan-tennas. In the second OFDM symbol time slot, symbols

and aresentonthetwotransmitantennas.Thesize2 2space-time codeword is given as

(13)

Assuming the linear ML (Alamouti) receiver and a sta-tionary channel over OFDM symbols, the instantaneousreceive SNR for the Alamouti coded system is given as

.For a system employing a SEB processor, the receive SNR is

given as . The SEB gain over theAlamouti scheme is given as

(14)

For a system employing a EB processor on eachchannel realization, the receive SNR is given as

. The EB gain over theAlamouti scheme is given as

(15)

For a system employing OSLP processor, the receive SNRis given as , whereis chosen according to (9) and (10). The OSLP gain over anAlamouti coded system is given as

(16)

We note that the (and, hence, ) is a function ofthe modulation order and noise variance , unlike

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408 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005

Fig. 3. Eigenvector variation and condition number variation of the transmit correlation matrix with distance.

Fig. 4. Theoretical Performance in flat-fading channel: -�- Alamouti, -�-OSLP, -�- SEB, -+- EB.

and . The performance of all schemes is dependant on theeigenvalue distribution of the transmit correlation matrix.

We first describe the performance of the above schemes usingsymbol-error rate (SER) versus SNR simulations on a flat fadingchannel with and QAM modulation, as well as

and -QAM modulation (see Fig. 4). We assume100 000 independent channel realizations. We define

. It is seen that for high and lower SNRs (higherSERs), the OSLP and SEB schemes give identical performancegain (array gain) over the Alamouti scheme. For example, in the

-QAM scenario, both schemes obtain 1.6-dB gain at10 SER. This is because in such scenarios, both the schemes

pour power only on the strongest eigenvector of the transmitcorrelation matrix. It can be verified via simulations that up to10 dB performance gain can be obtained as .However, for lower and higher SNRs (lower SERs), the SEBscheme loses diversity while the OSLP schemes preserves di-versity order obtained by the Alamouti space-time code. This isreadily observed from the slope of the SER versus SNR plotsfor both schemes. For example, in the -QAM sce-nario, the SEB scheme loses 3 dB over the Alamouti scheme at10 SER, wheras no performance loss is reported for the OSLPscheme. This is because as , the OSLP scheme pourspower across all eigenvectors of the transmit correlation matrix(thereby preserving diversity order), unlike the SEB scheme thatpours power only on one of the eigenmodes and loses potentialdiversity benefits from the other eigenmodes. The OSLP perfor-mance is always better than or equal to the Alamouti scheme,across all channel scenarios, unlike the SEB scheme. Finally,we note that the EB scheme on each channel realization pro-vides the maximum performance improvement, at the expenseof increased feedback requirement.

We corroborate these observations using field trial results.Figs. 5 and 6 show the CDF of , and

computed at the 1.0- and 3.5-km cell-radii. Our re-sults were obtained by normalizing the channel energy across50 channel realizations to be unity for each OFDM tone andtransmit-receive antenna pair. At the 1.0-km cell-radius, we as-sume a representative modulation order of 64 QAM and SNRsof 15 dB (low SNR) and 25 dB (high SNR). At the 3.5-kmcell-radius, we assume a representative modulation order of 4QAM and SNRs of 0 dB (low SNR) and 10 dB (high SNR).These numbers are typical for the system parameters given inSection V.A.

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005 409

Fig. 5. SNR gain over Alamouti scheme at the 1.0-km cell-radius.

Fig. 6. SNR gain over Alamouti scheme at the 3.5-km cell-radius.

From Figs. 5 and 6, we see that for lower SNRs (higherSERs), the OSLP and SEB schemes obtain array gain overthe Alamouti scheme. For example, for the dB,4-QAM scenario, the median gains dB.For the dB, 64-QAM scenario, the median gains

dB. At higher SNRs, the SEB schemeloses diversity order when compared to the OSLP scheme. Thisis noted by the difference in slope of distribution com-pared to distribution. For example, in the dB,4-QAM scenario, dB whereasdB for 1% of the locations. Similarly, for the dB,64-QAM scenario, dB whereasdB for 1% of the locations. Ideally, we must havedB, but in practice we see that a small fraction of locationshave dB that can be attributed to eigenvalue andeigenvector estimation errors during measurement.

We can observe that the EB scheme based on instantaneouschannel knowledge gives the best performance with gains of 1–3dB across the cell, depending on the eigenvalue distribution ofchannel realizations. Finally, we note that the spatial processinggains decrease with increasing cell-radius since the channelshave lower spatial correlations and factors at larger distances

from the BS due to increased scattering in the environment. Inother words, the performance gains drop as the channel matrixelements become IID.

VI. CONCLUSION

We used field trial results obtained from a MIMO-OFDMwireless system with two transmit and three receiver antennas(2 3), to validate the properties of the transmit correlation ma-trix in a macro-cellular environment and quantify performanceof transmit spatial processing schemes. We found that approx-imately 20% of the locations have well-defined transmit corre-lation matrices. For 60% of the locations, eigenvectors of thetransmit correlation matrix vary slowly over distance. We foundthat the OSLP and SEB schemes obtained array gain over theAlamouti scheme at lower SNR with a median gain of 2.0 (1.5)dB at the 1.0- (3.5-) km cell-radii. However, the SEB scheme(unlike the OSLP scheme) loses diversity order at higher SNRand leads to performance loss. This makes the OSLP schemeattractive for wireless systems with low-rate feedback link. TheEB scheme provides the best performance with a median gainof 2.6 (2.3) dB over the Alamouti scheme at the 1.0- (3.5-) kmcell-radii, at the expense of increased feedback requirements.

ACKNOWLEDGMENT

The authors would like to thank J. Tellado, J. Dring, R. Kr-ishnamoorthy, S. Catreux, P. Soma, and D. Baum for their rolein the measurement campaign; and D. Gesbert, D. Gore, and R.Nabar for valuable discussions on linear precoding derived inmanuscripts [4] and [5].

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