spatial multiplexing in correlated fading via the virtual channel representation

856 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 5, JUNE 2003

Spatial Multiplexing in Correlated Fading via theVirtual Channel Representation

Zhihong Hong, Member, IEEE, Ke Liu, Student Member, IEEE, Robert W. Heath, Jr., Member, IEEE, andAkbar M. Sayeed, Senior Member, IEEE

Abstract—Spatial multiplexing techniques send independentdata streams on different transmit antennas to maximally exploitthe capacity of multiple-input multiple-output (MIMO) fadingchannels. Most existing multiplexing techniques are based on anidealized MIMO channel model representing a rich scatteringenvironment. Realistic channels corresponding to scattering clus-ters exhibit correlated fading and can significantly compromisethe performance of such techniques. In this paper, we study thedesign and performance of spatial multiplexing techniques basedon a virtual representation of realistic MIMO fading channels.Since the nonvanishing elements of the virtual channel matrixare uncorrelated, they capture the essential degrees of freedomin the channel and provide a simple characterization of channelstatistics. In particular, the pairwise-error probability (PEP)analysis for correlated channels is greatly simplified in the virtualrepresentation. Using the PEP analysis, various precoding schemesare introduced to improve performance in virtual channels. Uni-tary precoding is proposed to provide robustness to unknownchannel statistics. Nonunitary precoding techniques are proposedto exploit channel structure when channel statistics are known atthe transmitter. Numerical results are presented to illustrate theattractive performance of the precoding techniques.

Index Terms—Beamforming, correlated channels, diversity, pre-coding, space–time coding, spatial multiplexing, virtual channelrepresentation.

I. INTRODUCTION

I NFORMATION theoretic studies [1], [2] indicate thatmultiple antennas at the transmitter and receiver, so-called

multiple-input multiple-output (MIMO) systems, can dra-matically increase the capacity and diversity in wirelesscommunication systems. Over the last few years, severaldistinct bandwidth-efficient communication techniques in-cluding space–time coding [3]–[5] and spatial multiplexing[1], [6] have been developed to exploit the potential of MIMOsystems. Spatial multiplexing focuses on the rate advantagewhereas space–time coding focuses on the diversity advantage

Manuscript received May 10, 2002; revised October 25, 2002. Thiswork was supported in part by the Office of Naval Research under Grant#N00014-01-1-0825 and in part by the National Science Foundation underGrants CCR-9875805 and CCR-0113385. This paper was presented in partat the 40th Annual Allerton Conference on Communication, Control, andComputing, IL, October 2–4, 2002.

Z. Hong, K. Liu, and A. M. Sayeed are with the Department of Electricaland Computer Engineering, University of Wisconsin-Madison, Madison,WI 53706 USA (e-mail: [email protected]; [email protected];[email protected]).

R. W. Heath, Jr., is with the Department of Electrical and Computer En-gineering, The University of Texas at Austin, Austin, TX 78712-1804 USA(e-mail: [email protected]).

Digital Object Identifier 10.1109/JSAC.2003.810361

of MIMO systems. Thus far, most of MIMO studies heavilyuse a statistical channel model that is an idealized abstractionof spatial propagation characteristics and assumes independentand identically distributed (i.i.d.) fading between differenttransmit–receive antenna pairs. This idealized channel modelallows tractable and elegant capacity analysis and space–timecode design. In practice, however, the channel coefficientsbetween different transmit–receive antenna pairs exhibit cor-relation due to clustered scattering in realistic environmentsand the relatively small antenna spacing. In such realisticconditions, the capacity of MIMO channels can be substantiallylower depending on the level of correlation [7], [8]. Therefore,a channel model that accurately captures the characteristics ofthe propagation environment is needed for realistic capacityassessments, as well as for designing space–time modulationand coding techniques that are matched to channel statistics.Parametric physical models (see, e.g., [9]) based on array pro-cessing techniques that explicitly model signal copies arrivingfrom different directions provide one such approach. However,the nonlinear dependence of these models on physical channelparameters, such as angles of departure/arrival, makes themrather difficult to be incorporated in transceiver design, explicitcapacity calculations, and space–time code design.

In this paper, we propose a framework for spatial mul-tiplexing in correlated MIMO channels using avirtualrepresentation for MIMO channels that has been introducedrecently [8]. The virtual representation captures the essence ofphysical modeling without its complexity, provides a tractablelinear channel characterization and offers a simple and trans-parent interpretation of the effects of scattering and arraycharacteristics on channel capacity and diversity. The virtualrepresentation corresponds to a fixed coordinate transformationwith respect to spatial basis functions defined by fixed virtualangles of arrival/departure. The resulting virtual channel matrixrepresents the channel by beamforming in fixed directions. Inthe context of space–time coding, the most attractive featureof the virtual channel matrix is that different scattering clus-ters correspond to different nonvanishing submatrices withapproximately uncorrelated entries. Analogous to the i.i.d.idealized statistical model, the virtual channel matrix capturesthe essential degrees of freedom in correlated MIMO channelsvia the powers in its nonvanishing uncorrelated entries. Thus,the virtual channel representation provides a powerful tool forstudying the impact of correlated fading on modulation andcoding for MIMO channels. It also provides a natural frame-work for combining beamforming ideas from array processingwith space–time coding techniques.

0733-8716/03$17.00 © 2003 IEEE

HONG et al.: SPATIAL MULTIPLEXING IN CORRELATED FADING VIA THE VIRTUAL CHANNEL REPRESENTATION 857

Our approach to spatial multiplexing in correlated MIMOchannels is motivated by the analysis of the pairwise-errorprobability (PEP) in the virtual channel representation. As wewill see, the key problem in correlated channels is that someparticular codeword error vectors may lie in the channel nullspace thereby increasing the PEP. The virtual channel matrix,due to its uncorrelated nonvanishing entries, clearly exposesthe interaction between the signal space and the channel thatcauses such degradation in spatial multiplexing performance.Based on our analysis, precoding the transmitted codewordsemerges as a simple and effective way for dealing with cor-related fading. When the channel statistics are unknown atthe transmitter, precoding viaunitary matrices is proposed forrotating the transmitted codewords to avoid collisions betweenthe codeword error vectors and the channel null space. Whenchannel statistics are known at the transmitter, the structure ofnonvanishing entries in the virtual channel correlation matrixis exploited to develop precoding techniques vianonunitarymatrices thatrotate and scale the codeword vectors to avoidcollisions with the channel null space, as well as to matchtransmitted signal power to the relative channel power indifferent spatial dimensions.

Precoding techniques have been investigated by several re-searchers in related contexts. Precoding to rotate the signal con-stellation and improve robustness of spatial multiplexing is con-sidered in [10] for a polarized channel and in [11] in the presenceof spatial correlation for the special case of two transmit andtwo receive antennas. Both above schemes consider diagonalprecoding matrices and both assume knowledge of the (nondi-agonal) channel correlation matrix at the transmitter. Linear pre-coding is considered in [12] for space–time coded system withknown fading correlations and in [13] for transmit diversity withrandom-fading channels. The key advantage of our precodingschemes is due to the uncorrelated nature of the virtual channelmatrix: the virtual channel correlation matrix is diagonal regard-less of the correlation structure of the original channel matrix.As we will see, this greatly simplifies PEP analysis and also of-fers direct insights for matching signaling schemes to channelcharacteristics. In a sense, the proposed precoding techniqueseffectively combine beamforming ideas with space–time codingvia the virtual channel representation.

The paper is organized as follows. In Section II, we pro-vide a brief review of the virtual channel representation. InSection III, we provide a general framework for PEP analysisin correlated MIMO channels based on the virtual channel rep-resentation. Section IV focuses on the performance of spatialmultiplexing techniques in correlated channels and discussesthe mechanisms underlying the degradation of performancein such channels. Motivated by the analysis in Sections IIIand IV, precoding techniques are proposed in Section V thateffectively account for correlated fading depending on whetherchannel statistics are known at the transmitter or not. Numericalresults illustrating the performance of precoding techniques arepresented in Section VI. Section VII presents some concludingremarks.

The following notation is used throughout the paper. We usefor conjugate, for transpose, for conjugate transpose, trfor the trace operator, vec for stacking columns of a ma-

Fig. 1. A schematic illustrating the virtual representation of a physicalscattering environment. The virtual representation corresponds to beamformingin fixed directions determined by the resolution of the arrays.

trix, for Kronecker product [14], tr for theFrobenius norm, and to denote expectation with respect torandom variable when it is not clear from the context. We useboldface lowercase letters to denote vectors and boldface upper-case to denote matrices. is the th element of the vectorwhile is the element in theth row and th column of thematrix .

II. V IRTUAL CHANNEL REPRESENTATION

We consider a narrowband MIMO system with transmitantennas and receive antennas. In the absence of noise,we have , where is the -dimensional transmittedvector, is the -dimensional received vector, and is the

channel matrix coupling the transmit and receive an-tennas. The idealized statistical model corresponding to a richscattering environment assumes that the elements ofare i.i.d.complex Gaussian random variables. However, the elements of

are correlated in realistic environments and the statistics ofare dictated by the scattering and array characteristics, such asangular spreads of scattering clusters and antenna spacing [7],[8]. In this paper, we use the virtual channel representation in-troduced in [8] to capture the statistical structure of correlatedfading channels imposed by clustered scattering environments.

A schematic illustrating the virtual channel representation isshown in Fig. 1. Consider one-dimensional uniform linear ar-rays (ULAs) of antennas at both the transmitter and receiverfor simplicity and assume that far-field assumptions apply. Thechannel matrix can then be described via the array steering andresponse vectors given by

(1)

where represents the angle variable and is related to the phys-ical angle (measured with respect to the horizontal axis asillustrated in Fig. 1) as , where is the wave-length of propagation and is the antenna spacing. The vector

represents the signal response at the receiver array dueto a point source in the direction . Similarly, repre-sents the array weights needed to transmit a beam focused in the


direction of . Note that the steering and response vectors in(1) are periodic in with period 1.

Parameterized physical models representvia signal prop-agation over multiple paths (see, e.g., [8] and [9])

(2)

where is the fading gain, represents the angle of arrival(AoA), and represents the angle of departure (AoD) asso-ciated with theth path. The virtual representation, on the otherhand, exploits the finite dimensionality of the signal space1 touse spatial beams infixed, virtual directions (as illustrated inFig. 1) to capture the effect of the scattering environment [8].The virtual channel representation can be expressed as

(3)where ( ) and

( ) are full-rankmatrices defined by the fixed virtual angles and .Uniform sampling of in the principal period ( )is a natural choice for virtual spatial angles

(4)

which yields unitary matrices and . is andiscrete Fourier transform (DFT) matrix and is anDFT matrix. Therefore, is unitarily equivalent to andcaptures all channel information.

Realistic propagation environments can be modeled via a su-perposition of scattering clusters with limited angular spreads(see, e.g., [8] and [15]). The virtual channel matrix pro-vides an intuitively appealing “imaging” representation for suchenvironments: different clusters correspond to different nonva-nishing submatrices of . Furthermore, it is shown in [8] thatthe nonvanishing elements of are approximately uncorre-lated under the usual assumption of uncorrelated physical scat-tering. The virtual channel matrix clearly reveals the capacityand diversity afforded by a given scattering environment. Thecapacity multiplier provided by a cluster is determined by thesize/rank of the corresponding submatrix and depends on thenumber of virtual angles that lie within the angular spread ofthe cluster. The number of nonvanishing entries in each subma-trix determines the diversity afforded by the cluster and dependson the nature of scattering within the cluster.

A valuable representation of (3) is obtained by stacking thecolumns of as

vec vec

(5)

1Due to finite number of antennas and, thus, finite array apertures.

using the identity vec vec . Letdenote the correlation matrix of and the corre-

lation matrix of vec . and are related by

(6)

where are the diagonal entries of .Note that due to the approximately uncorrelated nature of theelements of , is approximately diagonal. We assume

to be exactly diagonal in this paper.2 Furthermore,may have some zero elements on the diagonal correspondingto the vanishing elements in due to clustered scattering.We note that and are unitarily equivalent since theKronecker product of two unitary matrices is also unitary.Thus, (6) is an eigendecomposition of and (5) is the corre-sponding Karhunen–Loeve-like representation for each channelrealization. Therefore, the nonvanishing diagonal elements of

that capture the power in the nonvanishing elements ofalso determine the eigenvalues of. As we will see later, thisis a very powerful property of the virtual channel matrix fromthe viewpoint of PEP calculations.

III. PEPIN SPATIAL MULTIPLEXING SYSTEMS

In this section, we review spatial multiplexing and derive thePEP for spatial multiplexing in correlated channels via the vir-tual representation.

A. Review of Spatial Multiplexing

Spatial multiplexing is a modulation technique for MIMOcommunication systems in which independent streams of dataare multiplexed in space and subsequently demultiplexed at thereceiver [1], [6]. During every discrete-time symbol period, theencoder multiplexes complex symbols from aunit energy constellation to form a complex vector codeword.The components of this vector codeword are modulated, up-con-verted and launched into the channel.

Neglecting symbol timing errors and frequency offsets, thereceived signal vector after matched filtering and sam-

pling can be written as

(7)

where is the vector realization of i.i.d. complex circularlysymmetric additive white Gaussian noise (AWGN) processeswith distribution , where denotes the iden-tity matrix of dimension and is the total signal power. Thechannel is assumed to be perfectly known at the receiver (viatraining symbols, e.g.,) butunknown to the transmitter. As willbecome clearer, limited information about thestatisticsof at

2The approximation gets better with larger number of antennas and/or largearray apertures [8].


the transmitter can be fruitfully exploited. We assume the use ofthe maximum-likelihood (ML) decoder at the receiver.

B. Pairwise-Error Probability (PEP)

For random channels the metric of interest is typically theaverage probability of error. Exact calculation of the symbol orbit-error probability for spatial multiplexing systems, however,is difficult [11], [16]. One solution commonly employed [17]is to upper bound the desired error probability using the unionbound and the PEP.

Let denote the probability that isdecoded at the receiver erroneously as for a given . Letthe error vector and define the error corre-lation matrix of as . Using theChernoff bound to upper bound the PEP andthe fact that has complex normal distribution with zero meanand covariance , it can be shown that [18]

(8)

This is strictly true only for nonsingular. As we pointed out,is often singular or nearly singular due to clustered scattering. Toavoid difficulties with the singular distribution, we can proceedwith the above derivation assuming that and thencan let go to zero to arrive at the result in (8). A more formalderivation of (8) is presented in [16] and [19].

Substituting (6) into (8) using the virtual representation, fol-lowing the identity and using the dis-tributive property of the Kronecker product, we have

(9)

where ,rank and denotes theth eigenvalue of .

C. Signaling in the Virtual Channel

Define the DFT of the codeword error correlation matrix as

(10)

which is rank-1 and is the DFT of the errorvector . From (9), we see that the PEP is governed by theinteraction between and . The virtual channelrepresentation, thus, suggests the DFT as a natural precodingmatrix so that the codewords directly interact with the scatteringcharacteristics captured by . That is, we consider transmittedsignals of the form so that the channel (7) becomes

(11)

Fig. 2. Signaling in the virtual channel. (a) System depiction withA asa precoder andA as a postcoder.s is the multiplexed symbol vector.(b) Equivalent representation of (a).s directly interacts withH .

where and similarly for . The above equa-tion says that at the transmitter, we first apply a DFT () tothe transmitted codewords before launching them onto thechannel and at the receiver, we first apply a DFT () to thereceived vector before decoding . Thus, (11) corresponds tosignaling and reception directly in the virtual (Fourier) domainas illustrated in Fig. 2. Our subsequent development is in thecontext of (11). In this context, the transmit and receive antennascorrespond to the virtual transmit and receive elements (corre-sponding to beams in virtual directions) and the matrixbecomes the codeword error correlation matrix ( ) corre-sponding to the actual transmitted vectors [not their DFT as in(10)].

D. Rank and Eigenvalue Characterization

According to the rank and determinant criteria [3], [20], theerror rate performance is a function of the rank and the productof the eigenvalues of . The diversity advantage is deter-mined by the rank which we can bound as

(12)

since is a rank-1 matrix. In realistic channels corre-sponding to clustered scattering, is not diagonal but isstill diagonal due to the properties of the virtual representation[8]. Note that the maximum rank of is and, thus,the absence of coding across time limits spatial multiplexing toa diversity advantage (defined as the minimum rank of all pos-sible ) that is at most equal to . Space–time codingcan increase the rank of to thereby restoring thediversity loss. In addition to the rank of , the nature ofscattering can also reduce the diversity advantage. This is be-cause some diagonal elements of may be zero or near-zerodepending on the scattering geometry and antenna spacing [8].However, due to the inherent difference in the ranks of and

, could have many vanishing (or small) di-agonal entries and still yield the maximum diversity advantage( ). It all depends on the interaction between (channelstatistics) and (code error properties) in (9).

Let diag be thediagonal decomposition of in terms of matrices,

, where is the th column of.

Theorem 1: Explicit characterization of eigenvalues of. The eigenvalues of are given by

(13)


Proof: We suppress the superscript ( ) for simplicityof notation. First note that for ,

. Furthermore, the nonvanishing eigen-values of a product of matrices are not changed by changing theorder of matrices. Now

(14)

where note that matrices in the last equalityare diagonal matrices. Thus, the eigenvalues of are givenby the diagonal entries of the matrix in (14) which are given in(13).

The above theorem has a very insightful interpretation: theth eigenvalue is equal to a weighted sum of the powers of all

virtual transmit antennas coupled to theth virtual receive an-tenna (via the th row of ). The weighting is given by themagnitude squares of the error vector components, ,corresponding to different virtual transmit antennas. Note thatthe th eigenvalue may be zero if either for all

( th row of is zero) or if for those values of forwhich . Rearranging the order of nonzero eigen-values, the PEP in (9) can be bounded at high signal-to-noiseratios (SNRs) as

(15)and, thus, the diversity gain is and the coding gain is

. Note that both the diversity and codinggains depend on both the error codeword properties, as well aschannel characteristics as evident from Theorem 1. For given

, to maximize diversity gain we would like to have as manyelements of error vectors be nonvanishing for which thecorresponding is nonzero, where denotes the setof all possible codeword error vectors. To maximize codinggain, in addition, we would like each to be as large aspossible over the entire set.

IV. SPATIAL MULTIPLEXING IN THE VIRTUAL CHANNEL

In this section, we use the PEP analysis in the previous sec-tion to get further insight into the performance of spatial multi-plexing in correlated channels.

Achieving the upper bound in (12) assumes that column spaceof is not contained in the null-space of .However, due to the unconstrained nature (relative to generalspace–time coding) of codeword error vectors in spatial multi-plexing and due to the vanishing diagonal entries of in clus-tered scattering, full intersection between the column space of

and the range space of cannot be guaranteed.This results in loss of diversity gain which we quantify in thissection.

Exploiting the diagonal nature of and the structure of thecodeword error vectors in spatial multiplexing, we can obtainan exact expression for the diversity advantage that holds in avariety of situations in which the same constellation is used atall virtual transmit antennas.

Theorem 2: Exact diversity advantage. For spatial multi-plexing using the same constellation at each virtual transmitantenna, the minimum rank of over satisfies

(16)

where is the indicator function and is a threshold valuefor determining the essentially nonvanishing entries of.

Proof: First note thatby the distributive property

of Kronecker products. Thus, the rank of is deter-mined by the rank of . Recall thatdiag . Thus, we have

(17)

Clearly, the worst set of error vectors are of the form in whichis nonzero in only one element (unit error vector). In the case

that the same constellation is used on all virtual antennas, allsuch error vectors are possible. Suppose thatis only nonzeroin the th element. From (17), the rank of for suchan error vector is determined by the nonvanishing diagonal ele-ments in which is exactly the expression correspondingto the index in (16). Thus, the minimum rank correspondsto with the smallest number of nonvanishing elements.And this is achievable since all such unit error vectors are pos-sible. This yields the equality in (16).

Theorem (16) gives a convenient and explicit expression forthe worst-case diversity advantage as a function of the scatteringenvironment. The actual diversity gain can be larger dependingon the interaction between the channel and the error codewordvectors. The proof of the theorem yields useful insights in thisregard. In particular, it shows that if theth virtual transmit an-tenna is not coupled at all to the receivers— (the

th column of is identically zero)—no symbols should betransmitted on that virtual transmit antenna; that is shouldbe zero. This is because those transmissions are not observableat the receiver at all. This is important in practice since some ofthe virtual transmit elements might be weakly coupled to the re-ceivers and should not be used for transmission. In such cases,data should only be multiplexed on the virtual transmit antennascorresponding to relatively strong columns of . As our nu-merical results demonstrate, accounting for such weakly cou-pled virtual transmit antennas can result in significant improve-ment in performance. A related important insight is to avoid uniterror codeword vectors. Unfortunately, since the symbols at dif-ferent virtual transmit antennas are independent in spatial mul-


tiplexing, all such error vectors are possible. As we discuss inthe next section, applying a precoding transform to the trans-mitted vectors is an effective way to avoid such error vectors.More importantly, since we cannot introduce dependencies be-tween transmitted symbols in spatial multiplexing, such a pre-coding approach is the only way of avoiding unit error codewordvectors.

V. UNITARY AND NONUNITARY PRECODING

In this section, we leverage the insights from PEP analysisto propose unitary and nonunitary precoding that improve therobustness of spatial multiplexing in correlated channels.

A. Unknown Channel Statistics—Unitary Precoding

Our PEP analysis shows that the key source of performancedegradation is the existence of unit error vectors and the van-ishing diagonal entries of due to clustered scattering. Fur-thermore, may not be available at the transmitter. In suchcases, it is of interest to develop techniques that minimize theoccurrence of unit error vectors and also make spatial multi-plexing robust to the vanishing diagonal elements of.

Designing general space–time coding schemes that are robustto correlated channels is an interesting open problem. In spa-tial multiplexing, since the transmitted vectors are not spatiallycoded, we are left with the option of applying a precoding trans-form to the transmitted vectors. In this paper, we focus on linearprecoding transforms. Let be a precoding matrixthat is applied to the output of the spatial multiplexer. Insteadof transmitting codeword , we transmit in Fig. 2. Thereceiver observes

(18)

The goal of this section is to find , without knowledge ofor , to improve the error-rate performance of the communi-cation link.

Due to lack of knowledge of , we choose to be unitarysince it does not change the spatial distribution of power. A sim-ilar approach has been taken in [10], in which a precoding matrixis used to improve the robustness of spatial multiplexing sys-tems in polarization channels. Their precoder assumes knowl-edge of a nondiagonal , parameterized by the cross-polar-ization discrimination, but otherwise serves to precondition thetransmitted signal vectors. In our case, the precoder is designedto be robust to the nonvanishing diagonal entries of.

From the structure of the virtual representation, a DFT pre-coder may sound tempting; that is [we remove the inFig. 2(a)] so that . In this case, the trans-mitted codewords undergo a DFT due to the channel propa-gation before encountering . However, such a DFT precoderis not attractive since it maps a constant error vector of the form

into the unit vector which is detri-mental to the PEP from Theorem 2. However, with some modi-fication a DFT precoder may be useful, as elaborated below.

From the proof of Theorem 2, diversity advantage is compro-mised when there are cancellations in the product .Furthermore, such cancellations are particularly acute for unit

error vectors. We define arobust unitary precoderas one thatsatisfies the conditions in the following definitions. Let(notbe confused with the in Theorem 2) be a design parameter.

• Define . A robustunitary precodersatisfies for

and for all , for some .• An optimal robust unitary precoderis one that maximizes

the threshold over the space of candidate precoders; thatis

(19)

An optimal precoder may not be unique.It is clear from Theorem 2 that a robust unitary precoder en-

sures that all elements of are sufficiently nonvanishing forall . Then, from (13) in Theorem 1, we see that all nonva-nishing elements in contribute to the eigenvalues and, thus,contribute to the diversity gain by maximizing the rank. [Notethat is replaced with in (13) with precoding.] An “op-timum” precoder makes the modulus of all elements ofas large as possible to maximize the magnitude of eigenvaluesin (13). This contributes to both the diversity gain and codinggain.

Finding an optimum robust unitary precoder is difficult sincedepends on the number of transmit antennas, as well as the

specific type of constellation. Optimization is difficult since ro-bust unitary precoders satisfy a max–min criterion, where boththe maximum and minimum are effectively taken over discretespaces. However, a unitary can be represented using a re-duced set of coefficients. We use this fact to propose two dif-ferent unitary precoders.

Diagonal Precoder: The simplest class of unitary matricestake the form

diag (20)

Clearly, by design and there are onlyfree parameters since we can assume the first coefficient is onewithout loss of generality. However, this precoder does not solvethe problem of avoiding unit error codeword vectors. In con-junction with a DFT precoder, though, this simple precoder maybe effective. Suppose that we transmit signals of the form

. From (7) and (18), the channel equation in this casebecomes . The intuition behindthis precoder is that the matrix (which is analogousto an all pass filter) will smear the error vectors over all frequen-cies (virtual angles) so that all elements of [whichreplaces in (13) in this case] will be nonvanishing for bothunit error vectors, as well as constant error vectors (which causeproblems in a DFT precoder).

For reasonable , we propose the following exhaustivesearch technique.

Search 1: Uniformly quantize the phase values toand let be the set of candidate pre-

coding matrices. Choose the such that (19) is satisfied.We consider an example of a spatial multiplexing system with

QPSK symbols and . The optimal precoding ma-trix is found and the simulation results are shown in Fig. 4 in


Section VI, where we show significant improvements with pre-coding. Given the performance improvements, a natural ques-tion is whether a nondiagonal unitary can even further im-prove performance.

General Unitary Precoder:Consider a general unitary pre-coding matrix . Since the criterion in (19) does not readilyadmit a closed form solution, a numerical search is needed. Oneapproach is to characterize a unitary matrix using a canonicalrepresentation, say using the Givens rotations, as in [11]. Wethen quantize the angles of the rotations to enumerate a familyof candidate precoding matrices and optimize. For large num-bers of antennas this becomes computationally intensive due tothe number of parameters that need to be quantized. In this case,we propose the following design based on a random search.

Search 2: Generate a set of random complex ma-trices from the complex Gaussian distribution. Letbe the setof orthogonal matrices constructed from the QR decompositionof the random matrices [21]. Choose the such that (19)is satisfied.

Clearly, Search 2 is not guaranteed to be optimal. We find,however, that it is relatively easy to find a good as we willshow in Section VI

B. Known Channel Statistics—Nonunitary Precoding

Robust precoders do not exploit knowledge of the channelstatistics and structure at the transmitter—the precoding ma-trix is designed to make the system robust to channel structurein clustered scattering environments. We now present a generalnonunitary precoding design which can incorporate knowledgeof channel structure at the transmitter. The second-order sta-tistics of the channel, captured by , typically change moreslowly than the channel realizations and, thus, it may be pos-sible to convey back to the transmitter.

Consider a nonunitary precoding matrix

(21)

where is a unitary precoding matrix (as in the previoussection) and is chosen to satisfy the power constraint

. Generally, if for ,is a nonunitary matrix. When , becomes

unitary. could be interpreted as a scaling or power allocationmatrix. Since we have the knowledge of at the transmitter,instead of the robustness criterion, we seek athat minimizesthe maximum PEP (9) for all possible error vectors. Letand

denote sets of candidate and matrices, respectively.We define an optimal nonunitary precoding matrix as one thatminimizes the maximum PEP of codeword error

(22)

where now . The eigen-values in (13) now become

(23)

Fig. 3. Geometric representation of effects of precoding: a) original errorconstellation; b) unitary precoding; c) nonunitary precoding.

The use of unitary or nonunitary precoding can be motivatedby considering the geometry of signal constellations andchannel statistics. An example of signal error vector constel-lation is illustrated in Fig. 3(a). Since may be sparse, thechannel matrix has a certain “null space”, captured by thevanishing diagonal elements of in the context of PEP. Ifan error vector lies in this channel null space, it cannot bedetected at the receiver and significantly compromises the PEP.In order to improve performance, unitary precoding rotates theerror vector constellation to some angle to avoid error vectors(particularly unit error vectors) lying in the channel null space.This corresponds to making as many elements ofin (23)nonvanishing as possible over all . This contributesprimarily to diversity gain and some to coding gain. Withoutknowledge of channel statistics at the transmitter, unitaryprecoding can provide robust performance since it tries tominimize the projection of error vectors onto the channel nullspace. When the transmitter has knowledge of, nonunitaryprecoding not only prevents an error vector from lying in thechannel null space, but it also increases the projection of allthe error vectors onto the channel range space via judiciouschoice of . This corresponds to making as largeas possible in (23) over all . This increases the codinggain compared with unitary precoding. A schematic illustratingthe effects of unitary and nonunitary precoding is shown inFig. 3(b) and (c). A similar channel geometry explanation wasalso given in [11], where only a particular 22 system withdiagonal precoding was considered.

To further convey the intuition of unitary and nonuni-tary precoding, consider a concrete example of a

of theform

(24)

where stands for nonvanishing elements with variance 0.02 andstands for nonvanishing elements with unit variance. As we

mentioned earlier, nonvanishing elements of correspond tothe nature of scattering. In this example, is weakly sup-ported by due to low power in . Similarly, ,that corresponds to the third-virtual transmit angle and ,only couples with one virtual receive angle and, thus, there isno diversity gain for this symbol. Therefore, we would expectthe reliability of transmission of and to be poor. In


order to improve performance, we could spread the informa-tion transmitted by and over other transmit angles(corresponding to and with more power). Uni-tary precoding is designed to achieve this goal. On the otherhand, couples with receivers so weakly that it may not beused for transmission at all. Therefore, instead of equally dis-tributing transmit power among all the virtual transmit angles,it is natural to allocate more transmit power to the more reli-able transmit angles if sucha priori information is available atthe transmitter. The diagonal weighting matrixin nonunitaryprecoding is designed to achieve this goal. It is also clear that inthe extreme case of a fully populated , which is analogousto an i.i.d. channel [8], neither unitary precoding nor nonunitaryprecoding would be needed.

A numerical search is generally needed to obtain. Whenthe dimension is large, the search for that jointly optimizes

and may be very computationally intensive. An alterna-tive suboptimal solution is the following.

Search 3: Let be an optimal robust unitary pre-coder obtained, for example, using Search 1 or Search 2 inSection V-A. Let denote the space of matrices obtained,for example, by quantization and normalization. Choose thethat satisfies (22) for the given and known .

Using a that is an optimal robust precoder has a numberof important advantages. First, if is not available, then thetransmission is still robust. Second, there are fewer parametersto optimize in than in . Since the channel statistics changeover time on a relative slower time scale than changes in channelrealizations, the optimization of can be done in real-time.We now give another optimization criterion forbased on mu-tual information.3 Assume that the codewords are complexGaussian distributed, , (the transmitted signal isnow , and ). Therefore,it can be shown using Jensen’s and Hadamard inequalities, theaverage mutual information betweenand is [2]

(25)

where , . The op-timal then maximizes (25) under the power constraint. Thisis a standard waterfilling problem [23] and the solution is

(26)

here denotes the positive part ofand is chosen so that.

C. Virtual Transmit Antenna Selection

The mutual information based power allocation procedurein (26) explicitly illustrates an important point that is implicitin (22). In some cases, a particular virtual transmit anglewill contribute a negligible amount of power to the receiver[e.g., the first virtual antenna in (24)] making this virtualangle infeasible—the power budget may not be sufficient to

3In [22], it is argued that maximizing mutual information also minimizes PEP.

Fig. 4. No precoding versus diagonal precoding as a function of the numberof nonvanishing elements ofR .

allocate any power to this virtual antenna. This problem can bealleviated by applying a technique known as stochastic antennasubset selection [24]. The idea is to allow only a subset of

of the virtual angles to be used. The size of canbe determineda priori from and then Search 3 can beperformed over candidate subset. Alternatively, the number andlocations of nonzero in (26) can be used to determine theactive virtual transmit antennas and then Search 3 can be usedto find for these antennas.

VI. NUMERICAL RESULTS

A. Unitary Precoding Without Channel Statistics

In this experiment, we consider a spatial multiplexing systemwith quaternary phase-shift keying (QPSK) symbols and

. We consider the performance with and withoutprecoding as a function of the number of nonzero diagonal ele-ments of . We denote with nonzero elements as ,

, , ,, . We used

100 000 Monte Carlo simulations to estimate the averagesymbol-error rate for each .

First consider the case of no precoding as illustrated in Fig. 4.The symbol-error rate is about 0.5 for and since thesymbol stream transmitted from the second virtual antenna islost in the null-space of the channel. For the case of, thereis a first-order diversity, since the minimum rank is one as fol-lows from Theorem 2. Finally, corresponds to a richchannel and yields a second-order diversity gain and maximumcoding gain.

Now we show how even diagonal precoding improves the di-versity advantage in sparse-scattering environments. By quan-tizing the angles with , we used Search 1 to find thefollowing optimal diagonal precoding matrix with

(27)

The performance improvement obtained using this precoderis illustrated in Fig. 4 and compared with the case of no pre-


Fig. 5. Diagonal, general unitary, and nonunitary precoding as a function ofthe number of nonvanishing elements ofR .

coding. Even with a single nonzero element, as predicted byTheorem 2, there is a first-order diversity gain since both datastreams get coupled to the nonvanishing due to rota-tion. As fills out with 2, 3, and 4 nonzero diagonal en-tries there is a second-order diversity advantage as predicted byTheorem 2. Since each additional nonzero element increases thereceived power, there is an increase in coding gain as predictedby Theorem 1.

Now consider a general unitary precoder. Using Search 2,we searched over 10 000 random 22 matrices and found thefollowing optimal precoding matrix with :

(28)

From Theorem 1, we expect that there is a marginal codinggain compared with precoder (27) due to the marginal differencein . In Fig. 5, we compare the precoder (28) to (27) and find thatthey have comparable performance. This is possible due to thesuboptimality of the search procedure and also due to the factthat our design criterion is not exactly motivated by the PEP. Thecomparison shows that the diagonal precoder jointly used withDFT has the advantage of less intensive search and comparableperformance to the general unitary precoder.

B. Nonunitary Precoding With Channel Statistics

To illustrate how nonunitary precoding can further improvethe interaction between the codeword and the channel, we com-pare a unitary precoder with a nonunitary precoder that exploitsknowledge of . First, we search for unitary precoder thatyields the in (28). Then we use (22) to search forfor each

, . For and , the optimal from Search 3is trivial. The optimal for is . Thecomparison of unitary precoding and nonunitary precoding isshown in Fig. 5. Nonunitary precoding outperforms unitary pre-coding by up to 2 dB when is sparse ( , ). For denserscattering ( ), the improvement of nonunitary precoding overunitary precoding is apparent though reduced. For the fully pop-ulated channel ( ), the nonunitary precoding matrix and

Fig. 6. Effects of unitary and nonunitary precoding on 2� 2 virtual channelswith two nonzero elements.

unitary precoding are identical since the optimalin this caseis .

C. Impact of Precoding on Degenerate Virtual Channels

We have shown that precoding can improve the performanceof spatial multiplexing in correlated channels. Now we use anexample to explicitly illustrate the diversity and coding advan-tage due to precoding using Theorem 1 and 2. We use three dif-ferent possible 2 2 representations for : row degenerate,column degenerate, and diagonal

(29)

For , Theorem 2 shows that the minimum diversityadvantage of this system is one and the only eigenvalue isgiven by Theorem 1 as , which is determinedby the chosen constellation. Clearly, the precoder cannotimprove performance since both the diversity advantage andthe minimum eigenvalue are fixed. Without precoding,provides no diversity gain by Theorem 2. However, with pre-coding, Theorem 1 reveals thathas two nonzero eigenvalues

, so the diversity gain is increased to twoand the degenerate column suggests a trivial power allocationscheme. The coding gain is further improved by using nonuni-tary precoding. Similarly, for , without precoding, diversityadvantage is one according to Theorem 2 and this is improvedto two by precoding according to Theorem 1. The codinggain is given by . Theorem 1shows that we also improve the coding gain by maximizing theeigenvalues by our search criterion. The comparison of threevirtual representation with and without precoding is shown inFig. 6 and confirms our intuition and analytical results.

D. Virtual Transmit Antenna Selection Based on Second-OrderChannel Statistics

In this experiment, we conclude by comparing all the pro-posed precoding approaches in a spatial multiplexing system


Fig. 7. Effects of unitary, nonunitary precoding, and subset selection on the4� 4 virtual channel in (24).

with QPSK symbols and for the channel withthe virtual representation in (24).

Consider two scenarios: with and without subset selection. Inthe first case, all four virtual transmit antennas are used (). In the second case, based on the weak coupling of virtual

transmit antenna , we use only cor-responding to and .

Using Search 1 with , we found optimal diagonalprecoding matrices for and . Using Search 2with 10 000 candidate matrices, we found optimal general uni-tary precoding matrices for and . For thenonunitary precoding scenario,was determined by the powerallocation scheme (26). The error-rate curves for all these sce-narios are shown in Fig. 7. The case with no precoding pro-vides minimal diversity gain as expected. Unitary precodingwithout channel statistics provides diversity gain as predictedby Theorem 2. Due to the weak coupling of the first virtualangle, the power on is wasted. Subset selection fixes thisproblem, but at the expense of rate since now three symbols andnot four are sent in each time period. Interestingly, nonunitaryprecoding with power allocation has dramatically improved per-formance both with and without subset selection. The reasonis that in either case the power is distributed only on the threemajor virtual angles. Without subset selection, due to the pres-ence of unitary precoding, the information stream that wouldhave been transmitted on the first virtual antenna is mixed onthe remaining three antennas and can still be decoded. Finally,note that the allocation (26) is obtained by maximizing the mu-tual information with a Gaussian codebook; it does not nec-essarily optimize the error performance when restricted to fi-nite alphabet constellations. Thus, better precoder optimizationsbased on may even further improve performance.

VII. CONCLUSION

We have investigated the design of spatial multiplexing tech-niques for correlated fading channels via the virtual channelrepresentation. We derived bounds for the PEP in spatial mul-

tiplexing using the virtual channel matrix . The approxi-mately uncorrelated nature of greatly simplified this anal-ysis. In particular, we derived exact expressions for the min-imum rank of the matrix governing the PEP, as well as exactcharacterization of its eigenvalues. The analysis in the virtualrepresentation yields many useful insights on the interaction be-tween the codeword space and the channel and its effects onperformance. In particular, a key insight to avoid performancedegradation in correlated channels is to avoid unit error code-word vectors and to exploit virtual transmit antennas that arestrongly coupled to the channels. These insights are leveraged todevelop effective multiplexing techniques for correlated MIMOchannels. Unitary precoding is proposed to make spatial multi-plexing robust to channel correlation when channel statistics areunknown at the transmitter. When channel statistics are knownat the transmitter, we showed that power allocation coupled withunitary precoding is a simple and effective means for exploitingthis information. Currently, we are investigating extension of thework for space–time trellis coding and linear dispersion codingfor correlated MIMO channels.

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Zhihong Hong (S’98–M’02) received the B.S.degree from Tsinghua University, Beijing, China,in 1994, and the M.S. and Ph.D. degrees, in 1998and 2002, respectively, from North Carolina StateUniversity, Raleigh, all in electrical engineering.

From 1999 to 2002, he was a Research Assistantat the Center for Advanced Computing and Commu-nication (CACC), North Carolina State University.Since February 2002, he has been with the Universityof Wisconsin at Madison, as a Postdoctoral ResearchAssociate. His research interests include advanced

coding and modulation in wireless communications, multicarrier transmissions,and wireless networks.

Ke Liu (S’99) received the B.S. degree in electricalengineering from Tsinghua University, Beijing,China, in 1999, the M.S. degree in electricalengineering, and the M.A. degree in mathematics,both from the University of Wisconsin at Madison,in 2001 and 2002, respectively, where he is currentlyworking toward the Ph.D. degree in electricalengineering.

His research interests include information theory,signal processing and coding in wireless communi-cations and wireless networks.

Robert W. Heath, Jr. (S’96–M’01) received the B.S.and M.S. degrees from the University of Virginia,Charlottesville, in 1996 and 1997, respectively, andthe Ph.D. degree from Stanford University, Stanford,CA, in 2002, all in electrical engineering.

In October 1998, he was part of the founding teamof Iospan Wireless Inc., formerly Gigabit WirelessInc., San Jose, CA. From 1999 to 2001, he servedas a Senior Consultant for Iospan Wireless Inc. InJanuary 2002, he joined the Electrical and ComputerEngineering Department, The University of Texas at

Austin, where he serves as an Assistant Professor as part of the Wireless Net-working and Communications Group. His research lab, the Wireless SystemsInnovations Laboratory, focuses on the theory, design, and practical implemen-tation of wireless systems. His current research interests are coding, modula-tion, equalization, and resource allocation for MIMO wireless communicationsystems. He has published over 25 refereed conference and journal papers andhas four patents issued, all in the area of wireless communications, signal pro-cessing, and communication theory.

Akbar M. Sayeed (S’89–M’97–SM’02) receivedthe B.S. degree from the University of Wisconsin atMadison, in 1991, and the M.S. and Ph.D. degreesfrom the University of Illinois, Urbana-Champaign,all in electrical and computer engineering, in 1993an 1996, respectively.

While at the University of Illinois, he was aResearch Assistant in the Coordinated ScienceLaboratory and was also the Schlumberger Fellowin signal processing from 1992 to 1995. From1996–1997, he was a Postdoctoral Fellow at Rice

University, Houston, TX. Since August 1997, he has been with the Universityof Wisconsin, where he is currently an Assistant Professor in electrical andcomputer engineering. His research interests are in wireless communications,sensor networks, statistical signal processing, wavelets, and time-frequencyanalysis.

Dr. Sayeed received the NSF CAREER Award in 1999 and the ONR YoungInvestigator Award in 2001. He served as an Associate Editor for the IEEESIGNAL PROCESSINGLETTERSfrom 1999 to 2002.

spatial multiplexing in correlated fading via the virtual channel representation

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