optimal transmitter eigen-beamforming and space-time block ... · zhou and giannakis: optimal...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 7, JULY 2003 1673

Optimal Transmitter Eigen-Beamforming andSpace–Time Block Coding Based on

Channel CorrelationsShengli Zhou, Member, IEEE,and Georgios B. Giannakis, Fellow, IEEE

Abstract—Optimal transmitter designs obeying the water-fillingprinciple are well-documented, and widely applied, when the prop-agation channel is deterministically known and regularly updatedat the transmitter. Because channel state information (CSI) may becostly or impossible to acquire in rapidly varying wireless environ-ments, we develop in this paper statistical water-filling approachesfor stationary random fading channels. These approaches requireonly knowledge of the channel correlations that do not necessitatefrequent updates, and can be easily acquired. Applied to a multipletransmit-antenna paradigm, our optimal transmitter design turnsout to be an eigen-beamformer with multiple beams pointing toorthogonal directions along the eigenvectors of the channel’s cor-relation matrix, and with proper power loading across the beams.The optimality pertains to minimizing a tight bound on the symbolerror rate. The resulting loaded eigen-beamforming outperformsnot only the equal-power allocation across all antennas, but alsothe conventional beamformer that transmits the available poweralong the strongest direction. Coupled with orthogonal space–timeblock codes, two-dimensional (2-D) eigen-beamforming emerges asa more attractive choice than conventional one-dimensional (1-D)beamforming with uniformly better performance, without rate re-duction, and without complexity increase.

Index Terms—Beamforming, channel correlation, space–timeblock codes, transmit diversity.

I. INTRODUCTION

M ULTIANTENNA diversity is well motivated for wire-less communications through fading channels. Although

receive-antenna diversity has been widely applied in practice,in certain cases, e.g., cellular downlink, multiple-receive an-tennas may be expensive or impractical to deploy, which en-deavors transmit-diversity systems. Equipped with space–timecoding at the transmitter, and intelligent signal processing atthe receiver, multiantenna transceivers offer significant diversityand coding advantages over single-antenna systems [1], [28],[29]. Our attention in this paper is thereby mainly focused on

Manuscript received September 26, 2001; revised March 10, 2003. Thiswork was supported by the National Science Foundation Wireless Initiativeunder Grant ECS–9979443, the National Science Foundation under GrantCCR–0105612, and by the ARL/CTA under Grant DAAD19-01-2-011. Thematerial in this paper was presented in part at the IEEE Global CommunicationsConference, San Francisco, CA, November/December 2000, and the IEEEInternational Conference on Communications, New York City, April/May2002.

The authors are with the Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]; [email protected]).

Communicated by R. Urbanke, Associate Editor for Coding Techniques.Digital Object Identifier 10.1109/TIT.2003.813565

application scenarios dealing with single receive—but multipletransmit—antennas.

Multiantenna systems can further enhance performance andcapacity, when perfect or partial channel state information(CSI) is made available at the transmitter [19]. For slowlytime-varying wireless channels, this amounts to feeding back tothe transmitter the instantaneous channel estimates [19], [30].But when the channel varies rapidly (relative to the speed of thefeedback channel), it is costly, yet not always meaningful, toacquire instantaneous CSI at the transmitter, because optimaltransmissions tuned to previously acquired information becomeoutdated quickly. Designing optimal transmitters based onstatistical information about the underlying stationary randomchannels, is thus well motivated.

Through field measurements, ray-tracing simulations, orusing physical channel models, the transmitter can acquire suchstatistical CSIa priori [25]. For certain applications, such asfixed wireless, the spatial fading correlations can be determinedfrom such physical parameters as antenna spacing, antennaarrangement, angle of arrival, and angle spread [25]. Likewise,for systems employing polarization diversity, second-orderchannel statistics will involve the correlation between differ-ently polarized transmissions [4]. Alternatively, the receivercan estimate the channel correlations by long-term averagingof the channel realizations, and feed them back reliably tothe transmitter through a low data rate feedback channel (thisis referred to as covariance feedback in [14], [15], [30]). Inapplications involving time-division duplex (TDD) protocols,the transmitter can also obtain channel statistics directly sincethe forward and backward channels share the same physical(and statistically invariant) channel characteristics even whenthe time separation between the forward and the backwardlink is long enough to render the deterministic instantaneouschannel estimates outdated. In frequency-division duplex(FDD) systems with small angle spread, the downlink channelcovariance estimates can be also obtained accurately from theuplink channel covariance through proper frequency calibrationprocessing [18].

Based on channel covariance information, optimal trans-mitter design has been pursued in [14], [15], [30] based on acapacity criterion, which specifies the theoretical maximumrate of reliable communication achievable in the absenceof delay and processing constraints (see also [20] when nosuch information is available at the transmitter). Focusing onsymbol-by-symbol detection, optimal precoding was designedin [6] to minimize the symbol error rate (SER) for differential

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1674 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 7, JULY 2003

binary phase-shift keying (BPSK) transmissions, and in [10]for phase-shift keying (PSK) based on channel estimation error,and conditional mutual information criteria.

Building on our work in [10], we design here optimaltransmit-diversity precoders for widely used constellationsbased only on the channel’s second-order statistics. Ourperformance-oriented designs rely on four criteria: a tightupper bound on SER, the exact SER, the conditional mutualinformation, and the mean-square channel estimation error.Common to all optimal precoders is the form of a generalizedbeamformer with multiple beams pointing to orthogonal direc-tions along the eigenvectors of the channel’s correlation matrix;hence, the name, optimal transmitter eigen-beamforming. Theoptimal eigen-beams are power-loaded according to a spatialwater-filling principle.

To increase the data rate without compromising the perfor-mance, we also develop parallel transmissions equipped withorthogonal space–time block coding (STBC) [1], [8], [28]across optimally loaded eigen-beams. Wedding optimal pre-coding with orthogonal STBC leads to a two-dimensional (2-D)eigen-beamforming which turns out to enjoy uniformly betterperformance than the conventional one-dimensional (1-D)beamforming without rate reduction, and without complexityincrease. When the channel coefficients are independent andidentically distributed (i.i.d.), or when channel knowledge isnot available at the transmitter, the optimal solution reduces tothe orthogonal STBC. Hence, as a byproduct, we also establishthe optimality of orthogonal STBC of [1], [8], [28] in terms oftheexact minimum SERfor i.i.d. channels.

The rest of this paper is organized as follows. Section IIdescribes the discrete-time baseband equivalent system model.Section III develops optimal eigen-beamformers under differentcriteria. The optimal transmitter design for channel estimationis presented in Section IV. Section V is devoted to jointlyexploiting orthogonal STBC and optimal eigen-beamforming.Numerical results are presented in Section VI, and conclusionsare drawn in Section VII.

Notation:Bold upper (lower) letters denote matrices (columnvectors); , , and denote conjugate, transpose, andHermitian transpose, respectively; stands for the absolutevalue of a scalar and the determinant of a matrix, and forthe Euclidean norm of a vector; stands for expectation,

for the trace of a matrix; stands for the real part ofa complex number, and for the imaginary part;denotes the sign of a real number, andthe integer floor;denotes the identity matrix of size; denotes an all-zeromatrix with size ; stands for a diagonal matrixwith on its diagonal; denotes theth entry of a vector; and

denotes the th entry of a matrix.

II. SYSTEM MODEL

Fig. 1 depicts the block diagram of a transmit diversity systemwith a single receive and transmit antennas. At theth

transmit antenna, the information-bearing signal isfirst spread by the code

Fig. 1. Discrete-time baseband equivalent model.

of length to obtain the chip sequence

After spectral shaping by the transmit filter (not shown inFig. 1), the continuous-time signal

is transmitted through theth antenna, where is the chipduration. The transmission channels are flat faded (frequencynonselective) with complex fading coefficients .The received signal in the presence of additive white Gaussiannoise is thus given by

After receive-filtering with that is matched to , issampled at to yield the discrete time samples

. Selecting and to possess the square-rootNyquist- property avoids inter symbol interference, and al-lows one to express as

(1)

where , and . To cast(1) into a convenient matrix–vector form, we define thevectors

and

the channel vector

and the spreading code matrix1

Although the channel is allowed to vary, we assume itconstant over chips; thus, , and

1The spreading matrixCCC can be viewed (and will be invariably referred to)as a precoder, or, as a beamformer.

ZHOU AND GIANNAKIS: OPTIMAL TRANSMITTER EIGEN-BEAMFORMING AND SPACE–TIME BLOCK CODING 1675

TABLE IPARAMETERS OFTHREE CONSTELLATIONS, WITH Efjs jg = E

(1) can be rewritten as: . Because wewill focus on symbol-by-symbol detection, we omit the symbolindex , and subsequently deal with the input–output model

(2)

At the receiver, we first acquire2 the channel to enable max-imum ratio combining (MRC) using

(3)

The MRC receiver is known to maximize the signal-to-noiseratio (SNR) at its output [22]. Furthermore, slicing the MRCoutput yields the desired symbol esti-mate ; e.g., with BPSK, we obtain .

For a given precoder , (3) specifies the optimal receiverin the sense of maximizing the output SNR. The question thatarises is how to select an optimal precoder. The precoder de-sign relying on exact knowledge of, or noisy estimates ob-tained via feedback, becomes challenging (if not impractical) toacquire when the channel varies fast. We are thus motivated todesign the precoder based only on knowledge of the channel’ssecond-order statistics: namely, and

instead of (or ) itself. We will first optimize forthe configuration of Fig. 1. Due to spreading, this multiantennatransmitter only transmits one information symbol everytimeslots. Such a redundant transmitter was also studied in [6], andhas its own merits for “power-limited” (e.g., spread-spectrummilitary communication) systems, where spectral resources arenot at a premium but low transmission power is desired. To en-able operation in “bandwidth-limited” scenarios, we will com-bine our optimum low-rate precoder with orthogonal STBC inSection V.

III. STATISTICAL WATER FILLING FOR OPTIMAL

EIGEN-BEAMFORMING

Throughout this paper, we adopt the following assumptions:

a0) the channel is complex Gaussian3 distributed, withzero-mean, and covariance matrix ;

2Similar to [6], [15], [30], we temporarily assume here that channel estimatesat the receiver are error free. Imperfect channel estimates will be considered inSection IV.

3All Gaussian random vectors in this paper are circularly symmetric.

a1) the noise is zero mean, white, complex Gaussianwith each entry having variance per real andimaginary dimension, i.e., ;

a2) the channel and the noise are uncorrelated:.

a3) only channel correlation information isavailable at the transmitter.

Notice that assumptions a0)–a3) have also been adoptedby [14], [15], [30], where they are referred to as covariancefeedback. The more complicated scenarios where a0)–a2) areviolated require further investigation, and will not be pursuedanalytically here; however, we will test by simulations therobustness of replacing in a3) by the sample correlationestimate .

Different from the optimal transmitter designs based on ca-pacity criteria [14], [15], [19], [30], we will investigate the un-coded system (2), and our performance metric will be the SER.Notice that error-correcting codes developed for single-antennatransmissions (termed scalar codes in [9], [20]) can be appliedas outer codes in our system; and the uncoded SER criterion willstill provide a good indicator for the coded bit-error rate (BER)as well. In the following, we will first derive a closed-form SERexpression, that will facilitate our optimal precoder design.

The signal to noise ratio at the MRC receiver output for afixed channel realization is

(4)

where is the average energy of the under-lying signal constellation. Since the SER depends on theSNR differently for different constellations, we will considerthree widely used constellations: -ary phase-shift keying( -PSK), -ary amplitude modulation ( -AM), and square

-ary quadrature amplitude modulation (-QAM), as listedin Table I.

Because the channels are random, the average SER should beconsidered by averaging over all possible channel realizations.To arrive at a closed-form average SER expression, we will firstsimplify (4). Toward this goal, we will diagonalize usingits spectral decomposition

(5)


where is unitary, and denotes the th eigenvalue ofthat is nonnegative: . Without loss of generality, weassume that ’s are arranged in a nonincreasing order:

.

Using (5), we prewhiten to , so that , and the

entries of are i.i.d. with unit variance . Let usdefine

(6)

The SNR of (4) then reduces to

(7)

Since is nonnegative definite, it can be decomposed as, where contains

the nonnegative eigenvalues of. Because is unitary,the vector has i.i.d. entries, with covariance matrix

. We denote the th entry of as . The SNR in(7) can thus be further simplified to

(8)

Notice that the SNR expression in (8) coincides with thatof the MRC output for independent channels [26], with

denoting the th subchannel’s SNR. Averagingover the Rayleigh-distributed , closed-form SER expres-sions for different signal constellations have been found in [3],[26], [27]. For convenience, we list them here as well

(9)

(10)

(11)

where , and is the mo-ment generating function of the probability density function(pdf) of evaluated at [26, eq. (24)], and theconstellation-specific is given by

for PSK (12)

for AM (13)

for QAM (14)

Notice that in (11) is slightly different, but equivalentto its counterpart in [26, eq. (46)]. When is Rayleigh dis-

tributed, the moment-generating function assumesthe following simple form [26]:

(15)

Our ultimate goal is to minimize the SER in (9), (10), or (11),with respect to . However, direct optimization based on theexact SER turns out to be difficult because of the integrationinvolved. Instead, we will seek alternative criteria to designthe optimal , that will enable simple closed-form precodersolutions.

A. SER Bound Criterion

Using the definite integral form for the Gaussian-function,the well-known Chernoff bound can be easily expressed as

(16)

by observing that the maximum of the integrand occurs at[26]. Likewise, in (15) peaks at , and

thus an upper bound on the SERs in (9)–(11) can be obtained ina unifying form

(17)

where , and takes on constellation-specificvalues as in (12), (13), or (14).

The upper bound in (17) can also serve (within a scale) as alower bound of the SER (this bound is not available in [3], [26],[27]). Indeed, observing that

(c.f. (15)), we have, e.g., for -PSK, that

(18)

The parameter can be either evaluated through numer-ical integration, or by the closed-form expression for the indefi-nite integral , , as tabulated in [11, p. 148]. Theparameters and for -AM and -QAM constel-lations can be obtained similarly. For we canverify that for QPSK (4-PSK), and

for 16-QAM.Equations (17) and (18) imply that the exact SER can be

bounded tightly in the small interval ,with taking different values depending on the constellation,and the number of antennas. Hence, is a good SERindicator, and one can predict the SER pretty well through

without carrying out the numerical integration in (9),


(10), or (11). We are now well motivated to choose thethatminimizes the tight upper bound on SER instead of minimizingthe SER itself.

The bound in (17) is also proportional to the Chernoff boundon the pairwise symbol error probability (PEP) that is knownto dictate system performance at high SNR. To relate the PEPwith the , suppose that a symbolis transmitted, butis erroneously decoded as. For a given channel, the PEP isgiven by (c.f. [22, Sec. 5.2-6])

(19)

Equation (19) can be interpreted as the SER expres-sion for BPSK with the instantaneous SNR being

Setting for BPSK, and taking the same steps weused to derive (17), we obtain the Chernoff bound averaged overRayleigh-fading channels as

(20)

where is defined in Table I. Substituting the value offor the different constellations in Table I, it is straightforward toverify that for the classes of constellations considered, we have

(21)

PEP (22)

Equation (22) shows clearly that minimizing the PEPis equivalent to minimizing the . Both amount tochoosing the precoder that maximizes the following objec-tive function:

(23)

For notational brevity, we define , from whichis uniquely determined by . We then simplify

(23) to

(24)

Without any constraint, maximizing leads to thetrivial solution that requires infinite power to be transmitted

. A practical constraint that takes into account

limited budget resources is the average transmitted power,which is expressed as

Without loss of generality, we assume that and; i.e., the total transmitted power is per

symbol. We are thus faced with the following constrainedoptimization problem:

subject to (25)

According to Hadamard’s inequality [7, p. 502], is max-

imized when the matrix is diag-onal; otherwise, we can always construct another matrixby nulling the off-diagonal entries in such that

, without altering the power constraint. Becauseshould be diagonal, we write it explicitly as

(26)

where . Since is a monotoni-cally increasing function, we can equivalently optimize the costfunction that will turn out to be moreconvenient. The equivalent constrained optimization problem issimplified to

subject to (27)

Differentiating the Lagrangian with respect to ,and equating it to zero, we obtain

(28)

where denotes the Lagrange multiplier. From (28), we cansolve for

(29)

Plugging (29) into of (27), we could obtain , which in turndetermines through (29), . However, this solu-tion may not guarantee , , for a given power budget

. We thus need to set negative’s to zero. With the special

notation , we set from (29).Suppose that the given power budget supports nonzero

’s; the value of can be determined easily, as will be clear

soon. Next, we substitute into the power constraint.This yields


and we arrive at the optimal loading

(30)

The nonincreasing order of the eigenvalues implies that, as confirmed by (30). We first set ,

and test if . The entry in (30) withimposes the following lower bound on the required SNR:

If is not large enough to afford optimal power allocationacross all beams, causing , then we should turn offthe th beam by setting , and set ; andso, we will find the desired .

The practical power loading algorithm can be summarized inthe following steps.

Step 1: For , calculate based only onthe first channel eigenvalues as

(31)

Step 2: With the given power budget leading toin the interval , set

and obtain according to (30) with

Having specified the optimal , we have found the optimalin (26). Because we have proved that a diagonal is

optimal, we deduce that the optimal must have orthogonalcolumns. This along with the definition imply thatthe optimal and can be factored as

�� (32)

where the columns of�� are orthonormal, and the diagonal en-tries of are given by (30).

Equation (32) provides a general optimal precoder for ran-dom channels for a given transmit-power budget. We summarizeour results thus far in a theorem.

Theorem 1: Suppose a0)–a3) hold true. The optimum receivefilter is given by (3), and the optimum precoding matrix

�� has and formed as in (5), (30), and(26) with�� an arbitrary orthonormal matrix. Optimalityin refers to maximum SNR, while optimality in per-tains to minimizing a tight upper bound on the average SER, orthe average pairwise error probability.

The optimal power loading in (30) was derived in [6]onlyfor differentialBPSK signals by minimizing the exact averageSER. Compared to [6], we not only provide general solutionsfor commonly used signal constellations, but we also allow forcoherent detection at the receiver.

B. Optimally Loaded Eigen-Beamforming Interpretation

When the optimal precoder in (32) is viewedas a weight matrix, it can be interpreted as a generalized (ortime-varying) beamformer. Different from conventional beam-forming that transmits all available power along the channel’sstrongest direction (implemented via the first row of ),here orthogonal basis beams along the eigenvectors of thechannel covariance matrix are employed; thus, the nameeigen-beamforming. To see this more clearly, suppose first that�� and . This choice corresponds to trans-mitting different basis beams at different time slots, and canbe viewed as a time-varying beamformer pointing towardone eigen-direction per time slot. With a general��, those

eigen-beams are multiplexed together at each time slot.Specifically, the antenna-steering vector at theth time slot(the th row of ) is �� , where is the

th column of . The matrix takes care of power loadingacross all eigen-beams. Notice that more power is distributedto stronger channels because in (30) ; and,furthermore, is constant for any .Thus, the power allocation obeys the water-filling principle[7]. To distinguish our design that relies on second-orderchannel statistics from conventional water filling that utilizesdeterministic (or instantaneous) CSI, we can think of it asstatistical water filling.

When the system operates at a prescribed power, it is clear that only basis beams

are used per time slot. With optimal loading on thosebeams,the of (17) can be simplified to

(33)

Following the terminology of [29], we call the diversityorder, because it determines the slope of the average SERas a function of the SNR. Full diversity is achieved when

. Based on (31) and (33), one can easily determine whatdiversity order should be used to achieve the best performancefor a given power budget with known channel correlations

. Specifically, we deduce from Theorem 1, and(31)–(33).

Corollary 1: The optimal diversity order is, whenfalls in the interval of , where is definedin (31).

Corollary 2: Full diversity schemes are not SER-bound op-timal across the entire SNR range; their optimality is ensuredonly when the SNR is sufficiently high: .


If the transmit power in Corollary 2 satisfies, then we infer from (30) that . In this

case, power is approximately equi-distributed to each antenna.Notice that apart from requiring it to be orthonormal, so far

we left the matrix�� unspecified. To fully exploit thediversity offered by antennas, the spreading factor must sat-isfy ; otherwise, the matrix in (7) loses rank, and isforced to have zero eigenvalues. On the other hand, the choice

does not gain anything in terms of optimizing (25) rel-ative to the minimum choice . It is thus desirable tochoose as small as possible to minimize bandwidth expan-sion, or equivalently, increase the transmission rate. When thedesired diversity order is, as in Corollary 1, we can reduce the

matrix�� to an fat matrix �� , where�� is any orthonormal matrix, without loss of optimality.This way, we can achieve rate for a diversity transmissionof order .

Alternatively, one cana priori force the matrix (and, thus,��) to be fat with dimensionality , which corresponds tosetting , deterministically. Optimal powerloading can then be applied to the remainingbeams. We willterm this scheme (with and�� chosen beforehand to be )a -dimensional (-D) eigen-beamformer. As a consequence ofTheorem 1 and Corollary 1, we then have the following.

Corollary 3: With , the -D : eigen-beamformer achieves the same average SER performance asan - dimensional eigen-beamformer, when

.

Two interesting special cases arise from Corollary 3. The firstis conventional 1-D eigen-beamforming with , [14], [15],[19], [30]. As asserted in Corollary 3, the 1-D eigen-beamformerwill be optimal when ;i.e., when the first and second eigenvalues are disparate enough,or when is sufficiently low. In general, this observationagrees with [19], [30], [14], [15], where the optimal scenariofor 1-D beamforming is specified based on a capacity criterion,instead of the SER criterion adopted here.

The more interesting case is 2-D eigen-beamforming, whichcorresponds to . The 2-D beamformer is optimal when

. Notice that theoptimality condition for 2-D eigen-beamforming is less restric-tive than that for 1-D beamforming, since , and

does not depend on the disparity between the first twoeigenvalues and . Compared to the 1-D beamformer, the2-D eigen-beamformer is optimal over a larger SNR range, orequivalently, over a larger set of fading channels with the givenSNR. Notice that rate loss occurs when . However, aswe shall see in Section V, 2-D eigen-beamforming achieves thesame rate as 1-D beamforming, and subsumes the latter as a spe-cial case.

The SNR threshold in (31) depends also on, and thuson the underlying signal constellation. As decreases, fulldiversity is first discarded for constellations with a smaller,which corresponds to larger signal constellations. The lowest

is achieved by BPSK for which .Based on the capacity criterion, the optimal capacity-

achieving input distribution has been specified in [15], [30],

that amounts to transmitting independent complex Gaussianinputs along the eigenvectors of . However, the powerallocation across different directions is only solved by numer-ical optimization. Our water-filling power allocation along theeigen-beams is obtained in closed form; it is thus simple toimplement, for a given transmit power and signal constellation.

C. I.I.D. Fading Channels

In this subsection, we consider , which corre-sponds to i.i.d. channels, or lack of information at the transmitter[19], [30]. In this case, the optimal loading in (30) correspondsto having . In words, equal power loadingacross all beams is optimal for i.i.d. channels. The precoder

�� now becomes proportional to an orthonormalmatrix.

Recall that for general Rayleigh channels, (30) was obtainedby minimizing an upper bound on the SER. We will show thatfor i.i.d. channels, equi-power loading possesses stronger opti-mality in terms of minimizing theexactSER. Toward this objec-tive, let us express for an arbitrarythe integrand of (9), (10),or (11) via the unifying objective function

(34)

Comparing (34) with (17), we infer that one can reach (34)by replacing in (17) with . Substituting in(30) by , we can readily obtain the optimal loading that min-imizes . Specifically for , we find from (30)that . Therefore, equi-power loading achieves theminimum , regardless of the value of. Integratingover , we obtain the minimum SER in (9), (10), or (11) withequi-power loading, as summarized in the following.

Theorem 2: Under a0)–a3), and if the channel vector has co-variance matrix , equi-power loading across all

antennas is optimal in the sense of achieving exactly the min-imum SER.

D. Conditional Mutual Information Criterion

In this subsection, we will rely on a particular conditional mu-tual information criterion to design the optimum. Let us sup-pose is transmitted. Recalling (2), and interchanging the rolesof and , we can view as the “input” to the “channel” .We then seek precodersthat maximize the mutual information

between the Gaussian inputand the output , con-ditioned on . This viewpoint implies that we are looking for theoptimal that best matches the channelwhen a single symbol

has been transmitted; and it is different from finding the op-timal input distribution conditioned on channel realizations—aproblem that is considered in [14], [15], [19], [30]. Mutual infor-mation expressions between Gaussian vectors are well known(see, e.g., [2, Theorem 1], [24], and references therein).

Lemma 1: Consider the finite-dimensional vector model, where , and satisfy a1) and a2). The conditional


mutual information between and , , is maximizedwhen is Gaussian as per a0), and is given by

(35)

Based on (5), we can simplify (35) to

(36)

For PSK signals, we have deterministically for eachconstellation point. For AM and QAM constellations, we willmaximize theminimumconditional mutual information, subjectto the power constraint

subject to (37)

Notice the similarity between (37) and (24). Mimicking thesteps used to derive (30), we find the optimal loading as

(38)

Comparing (38) with (30) and using Table I, we deduce that(38) coincides with (30) only when real (BPSK or AM) constel-lations are used, in which case . Therefore, forreal constellations, the optimal loading of (30) also maximizesthe minimal conditional mutual information . How-ever, for 2-D complex constellations (e.g.,-PSK or -QAMwith ), these two criteria lead to different solutions, since

. Because (30) is directly related to the SER,loading based on (30) is better justified than (38) in practice.

E. Multiple Receive Antennas

The main focus of this paper is on multiple transmit and asingle receive antennae. In this subsection only, we extend ourresults to multiple receive antennas. Similar to [15], we will as-sume that the channel vectors observed on different receive an-tennas are mutually uncorrelated, but have the same covariancematrix; i.e.,

a4) with denoting the channel vector corresponding tothe th receive-antenna, it holds that

where is the number of receive antennas.

It has been verified in [25] through ray-tracing simulationsthat the channel models obeying a4) are appropriate when thebase station (BS) is unobstructed, and the subscriber unit (SU)

is surrounded by rich local scatterers. It turns out, that the an-tenna spacing at the SU is much smaller (one or two orders ofmagnitude) than that at the BS, to yield uncorrelated channelsamong different antennas. Since a4) offers a good approxima-tion for certain applications, our analysis in this subsection willrely on it.

The received signal (2) at theth antenna is now. Collecting into an vector

, it is straightforward to verify that the MRCoutput SNR is

which includes (4) as a special case corresponding to .Following the same steps used to derive (8) and based on a4),we can decompose as

where are zero-mean Gaussian random vari-ables with unit variance. We can then obtain an upper boundon the SER as (c.f. (17))

(39)

Equation (39) implies that minimizing is equivalentto minimizing , and the optimal precoder coincideswith that in the single-receive antenna case. As in Theorem 2,equi-power loading is also optimal in the sense of minimizingthe exact SER, when the channels among multiple transmit andmultiple receive antennas are i.i.d., or, when no channel knowl-edge is available at the transmitter (the case where space–timecoding has been advocated).

IV. OPTIMAL BEAMFORMING FORCHANNEL ESTIMATION

In the previous section, we assumed that the channels areperfectly known at the MRC receiver to derive a closed-formSER expression. In practical systems, however, the channelsneed to be estimated through, e.g., training symbols. Trainingsequences are usually placed at the beginning of each burst forquasi-static channels, which assumes that the channel remainsconstant during each data transmission burst, but is allowed tochange independently from burst to burst [12]. Alternatively,training symbols can be inserted throughout the transmission,which constitutes the popular pilot symbol-assisted modulation(PSAM) [5]. Acquiring the channel based on the periodicallyembedded pilots, PSAM is particularly suitable for transmis-sions over rapidly (and thus continuously) fading environments.

Specifically, the transmitter inserts a known symbol ineach short frame consisting of symbols; i.e.,

For each time slot, the receiver can estimate the channelbased on nearest received blocks containing pilotsymbols at


This is possible by exploiting the channel’s spatial and temporalcorrelations (see [5] for details in a single-antenna setup).

It is thus of interest to seek the precoderthat optimizes thechannel estimation task at the receiver, when the channel co-variance information is available at the transmitter and remainsinvariant for sufficiently long time. For simplicity, we assumethat only the spatial correlation matrix isavailable at the transmitter. The general case with known spa-tial–temporal correlations goes beyond the scope of this paper.

Due to the lack of temporal channel correlations at thetransmitter, we optimize based on the followingsuboptimaltwo-step channel estimation method:

S1) Estimate based on

S2) For each block index, interpolate usingnearest channel estimates obtained from S1),for

The optimal combining weights are determined by thechannel’s temporal (or spatiotemporal) correlationsthrough, e.g., Wiener filtering [5]. If the channelscan be viewed as constant during a data transmissionburst, as in the static-channel model of [12], thencould be obtained by averaging ’s to decreasethe estimation error.

Because temporal correlations are not assumed available, S2)becomes irrelevant to the transmitter. We will thus choosetominimize the estimation error in S1) based on . We are nowfaced with a channel estimation problem based on a single pilotsymbol . Dropping symbol index, we simplify the systemmodel as .

Because pilots with small amplitude lead to poor estimationaccuracy, amplitude variation of pilot symbols is not desirable.We thus draw the pilot symbol from a PSK constellation, andlet , which does not necessarily equal. Sincecontains unknowns, we need at least time slots to estimateit. For parsimony, we will focus on square matrices ,in this section.

Under a0)–a2), the received block has zero mean:Relying on the received block , and letting

, , the linear minimummean-square error (LMMSE) estimator for is [17, pp.389–390]

(40)

and has covariance matrix

(41)

Note that the LMMSE estimator in (40) does not require thechannel (or even the noise ) to be Gaussian distributed as re-quired in a0) and a1). However, under a0) and a1), the LMMSE

estimator coincides with the (not necessarily linear) MMSE es-timator [17, Ch. 12]. Using a2), we have and

, which allow us to write (40) as

(42)

Here we want to select the that minimizes the mean-squareerror of the channel estimator (c.f. (41))

(43)

where the last step results from the matrix inversion lemma [13,p. 565].

Based on (5), we can rewrite (43) as

(44)

We are now set to select that minimizes in (44),under the same power constraint as in (25). We prove in theAppendix that the optimal precoder has the same form as in(32), i.e., �� , except for the power loading matrix

, whose diagonal entries are now specified as

(45)

where . Comparing (45) with (38), we find themidentical for PSK signals if . This is not surprising,since given the known pilot symbol , the maximum condi-tional mutual information between the channel vectorand thechannel output has been achieved by using the optimal designin (38); thus, conveys maximum information aboutto yielda reliable estimate for it.

Notice, however, that the optimal loading in (45) is differentfrom (30) for all constellations other than BPSK . Thissuggests that the optimal precoderduring the training phaseshould be chosen different from the data transmission phase,unless the underlying constellation is BPSK. As we discussedearlier (c.f. the two-step loading algorithm following (31)), wemay need to avoid distributing power to the weaker beams fora given finite power . Specifically, if based on the threshold

, the ratio falls into theinterval , we nullify in (45). Insuch cases, the channel coefficients are determined only up to

beams. The nice thing is thatfor all the considered signal constellations. Thus, if we main-tain , all beams employed for data transmission areestimated through the pilot symbols.


V. EIGEN-BEAMFORMING AND ORTHOGONAL STBC

In the system model (2), we transmit only one symbol overtime slots (chip periods), which essentially amounts to repe-

tition coding (or a spread-spectrum) scheme. As we mentionedin Section II, this is useful for “power-limited” (e.g., military)communication systems, where bandwidth is not at a premiumbut low transmission power is desired [6], [10]. For “bandwidth-limited” systems, on the other hand, it is possible to mitigatethe rate loss by sending symbols, , simul-taneously. The rate will then increase to symbols persecond per hertz. However, our objective here is to increase thedata ratewithout compromising the performance. This wouldrequire elegant symbol separation at the receiver that does notincur optimality loss. But let us suppose for the time being thatthe separation is indeed achievable, and each symbol is essen-tially going through a separate channel identical to the one wedealt with in Section III. The optimum precoder for willthen be

�� (46)

where the diagonal entries of are determined by (30). Be-cause the factor in (46) is common , designing sep-arable precoders is equivalent to selecting separable��

matrices. Fortunately, this degree of freedom can be afforded byour design in Section III because so far�� ’s are only requiredto have orthonormal columns.

Thanks to the orthogonal design of STBCs [1], [8], [28], mul-tiplexed data transmissions can be demultiplexed at the receiverwithout performance loss, as will be clear soon. Our combiningof optimal eigen-beamforming with STBC will treat real andcomplex symbols separately in the following two subsections.

A. Real Symbols

For convenience, we first list the results of [8], [28]:

Definition 1: For real symbols , and ma-trices �� each having entries drawn from , thespace–time-coded matrix

�� (47)

is termed a generalized real orthogonal design (GROD) in vari-ables of size and rate , if either one oftwo equivalent conditions holds true:

i) [28] or

ii) the matrices �� satisfy the following conditions[8]:

�� (48)

We refer the readers to [8] for the specific forms on the ma-trices �� . Based on GROD, we can assign to each symbol

the precoder in (46), and form the transmitted space–timematrix as

�� (49)

The corresponding input–output block relationship is (c.f. (2)). The receiver consists of

parallel detectors , with the th detector outputgiven by

(50)

For real symbols, the desired output (first term in the right-handside of (50)) is real. On the other hand, the interference term(the sum in (50)) is imaginary [8], since ,for , due to the design of (48) and (46). Then, canbe estimated through the real part of, which coincides withthe real part of the MRC output if only is transmitted as inSection III.

A rate- GROD exists for any number oftransmit antennas [8], [28]. Hence, for real constellations,we can achieve optimal performance without rate reduction,simply by space–time coded blocks of symbols as in (49). Inso doing, we do not increase complexity either, since eachsymbol is essentially going through a separate channel, andsymbol-by-symbol detection is performed right after MRC, asin (50). By proper power loading via , the -dimensionaloptimal eigen-beamformer can reach any diversity order

, with no rate penalty. Although the -dimensionaleigen-beamformer is overall optimal with no penalty for realsymbols, that is not always the case with complex symbols, aswe will see next.

B. Complex Symbols

Let and denote the real and imaginary part of,respectively. The following orthogonal STBC designs areavailable for complex symbols [8], [28].

Definition 2: For complex symbolsand matrices �� each having entries drawnfrom , the space–time coded matrix

�� (51)

is a GCOD in variables of size and rate ,if either one of two equivalent conditions holds true:

i) [28] or

ii) the matrices �� satisfy the conditions [8]:

��

��

��

(52)

For complex symbols , we define two pre-coders corresponding to�� as �� and

. The transmitted space–time coded matrixis now

(53)

At the th detector output, the decision variable is formed by

(54)


Fig. 2. The 2-D eigen-beamformer,u := [UUU ] .

where has variance ; and the secondequality in (54) can be easily verified by using (52) (see [8],[9] for a detailed derivation). Notice that the SNR of (54) is thesame as the MRC output for the single-symbol transmissionstudied in Section III; thus, the optimal loading in (30) enablesspace–time block-coded transmissions to minimize the upperbound on SER, but with rate . Relative to single-symboltransmission, the complexity of (54) is slightly increasedbecause it requires two MRC modules. Since this complexityincrease is negligible relative to the complexity associated withdecoding the error-correcting outer codes, which is always ap-plied in practical systems, the space–time-coded transmissionof (53) entails comparable complexity to the single-symboltransmission of (2).

For complex symbols, a rate–GCOD only exists for .It corresponds to the well-known Alamouti code [1]

space

time(55)

For , rate– orthogonal STBC exist, while for, only rate- codes have been constructed [28], [8].Therefore, for complex symbols, the -dimensional eigen-

beamformer of (53) achieves the optimal performance with norate loss only when , and pays a rate penalty when

. To trade off the optimal performance for a constantrate- transmission, it is possible to construct a 2-D eigen-beam-former with the Alamouti code applied to the strongest 2-Deigen-beams. Specifically, we construct a space–timematrix

(56)

The 2-D eigen-beamformer achieves the optimal performanceas the -dimensional eigen-beamformer when

, as specified in Corollary 3. The implementation of the2-D eigen-beamformer is depicted in Fig. 2.

Notice that the optimal scenario for 1-D beamforming wasspecified in [14], [15] from a capacity perspective. The interestin 1-D beamforming stems primarily from the fact that it allowsfor scalar coding with linear preprocessing at the transmit

antenna array, and, thus, relieves the receiver from the com-plexity burden required for decoding the capacity-achievingvector coded transmissions [14], [15], [19], [30]. Because eachsymbol with 2-D eigen-beamforming goes through a separatebut better conditioned (with diversity order) channel, thesame capacity-achieving scalar code applied to an 1-D beam-former can be applied to a 2-D eigen-beamformer, but for twoparallel streams (see also [9] on how to achieve the maximumachievable coded diversity using scalar codes instead of vectorcodes). Therefore, 2-D eigen-beamforming outperforms 1-Dbeamforming even from a capacity perspective, since it canachieve the same coded BER with less power. Notice that if

has only one nonzero entry , the 2-D eigen-beamformerreduces to the 1-D beamformer, with and transmittedduring consecutive time slots, as confirmed by (55) and Fig. 2.This leads to following conclusion.

Corollary 4: The 2-D eigen-beamformer includes the 1-Dbeamformer as a special case and outperforms it uniformly,without rate reduction and complexity increase.

Corollary 4 suggests that the 2-D eigen-beamformer is moreattractive than the 1-D beamformer, and deserves more atten-tion. It is also worthwhile recalling that the 2-D eigen-beam-former is overall optimal for systems employing onlytransmit antennas.

Utilizing channel correlations at the transmitter, our op-timal transmissions implement a combination of STBC andeigen-beamforming via (49) and (53). Orthogonal space–timeblock-coded transmissions are sent along beams that arepointing to orthogonal directions along the eigenvectors of thechannel correlation matrix, and are power-loaded according tothe water-filling principle. We summarize our result in thesetwo subsections as follows.

Theorem 3: Under assumptions a0)–a3), the optimal trans-mission consists of orthogonal STBC across the power-loadedbeams formed along the eigenvectors of the channel’s correla-tion matrix, constructed as in (49) for real symbols, or asin (53) for complex symbols, with the optimal loading in(30); the optimality pertains to minimizing an upper bound onthe SER.


C. Discussion and Comparisons

The combination of orthogonal STBC with beamforming hasalso been studied in [16], [31]. The major distinction betweenthis approach and [16], [31] lies in the fact that feedback of in-stantaneous channel estimates is used in [16], [31]. Thus, appli-cation of [16], [31] is for slowly fading channels. Because onlycovariance information is required by our approach, it is suit-able for fast-fading random channels.

When , we have also found in Section III-C thatloading with is optimal in the exact SERsense, which leads to and .Therefore, we have the following.

Corollary 5: Under a0)–a3), orthogonal space–time blockcodes are optimal in terms of achieving the exact minimumSER, when the channels are i.i.d.

Orthogonal STBC is known to be optimal for i.i.d. chan-nels in terms of maximizing the SNR under an eigenvalue con-straint [8]. Different from [8], we here use an SER that dependson SNR differently for different constellations. More impor-tant, the eigenvalue constraint4 employed in [8] limits the max-imum power along each particular direction. Since increasingthe power along each direction always increases the system per-formance, it is not surprising that under the eigenvalue con-straint, the optimal solution of [8] turns out to suggest equalpower loading for all directions, all approaching the specifiedpower limit: . The latter will hold true evenfor correlated channels, and the optimal solution will alwaysbe equal power loading, if one adopts the eigenvalue constraintof [8]. In contrast, we here use the average power constraint,and the optimal solution reduces to equal power allocation onlyfor i.i.d. channels, but necessitates judicious and constellation-specific power loading for correlated channels.

Under the average power constraint, a union bound on theSER for linear STBC has been derived in [23], where it is shownthat orthogonal STBC is optimal in terms of minimizing theunion bound within the class of unitary, linear space–time blockcodes; this class corresponds to requiringfor real symbols, and

for complex symbols. The result of [23] is independent ofchannel statistics, and holds true for each channel realization.Here, we show that orthogonal STBC is optimal in terms ofminimizing the exact SER only for i.i.d. Rayleigh channels. Ourresults show that for correlated fading channels with covariance

, the optimal solutions (49) and (53) are no longer withinthe class of unitary, linear block codes. However, the optimaldesigns here belong to the class of linear block codes, sincethe transmitted symbols on each antenna come from a linearmapping of the real and imaginary parts of the informationsymbols. This discussion serves as an example to assert that:

Corollary 6: Orthogonal STBC is not SER optimal in thegeneral class of linear block codes for non-i.i.d. fading channels.

4In our notation, this constraint of [8] requiresmax f � f for someprescribed boundf .

VI. NUMERICAL RESULTS

We consider a uniform linear array with antennasat the transmitter, and a single antenna at the receiver. We usethe rate– STBC from [8] with and . Weassume that the side information including the distance betweenthe transmitter and the receiver, the angle of arrival, and theangle spread are all available at the transmitter. Letbe thewavelength of a narrow-band signal, the antenna spacing,and the angle spread. We assume that the angle of arrival isperpendicular to the transmitter antenna array (“broadside” asin [25]). Thus, using the result of [25, eq. (6)] for small anglespread, we can simplify the correlation coefficient between theth and the th transmit antenna as follows:

(57)

Our tests will focus on two channels: Channel 1 hasand ; while Channel 2 has lower spatial correlationswith and . Notice that can be calculatedfrom the radius of the local scatterers and the distance betweenthe transmitter and the receiver [25].

We will present simulations for two constellations: quater-nary phase-shift keying (QPSK) and 16-QAM. Results for otherconstellations can be obtained similarly. In all the plots, the SNRis defined as the total transmitted power divided by the noisepower: SNR .

Figs. 3 and 4 show the optimal power allocation among dif-ferent beams for Channels 1 and 2, for both QPSK and QAMconstellations. At low SNR, the transmitter prefers to shut offcertain beams, while it approximately equates power to all an-tennas at sufficiently high SNR to benefit from diversity. No-tice that the choice of how many beams are retained depends onthe constellation-specific SNR thresholds. For QPSK, we canverify that 10.2 dB, and 37.5 dB for Channel1, while 15.0 dB, and 8.1 dB for Channel 2.Since , the threshold for 16-QAM is

7.0 dB higher for QPSK; we observe that 7.0 dBhigher power is required for 16-QAM before switching to thesame number of beams as for QPSK.

With Channel 1, Figs. 5 and 6 depict the exact SER, andthe SER upper bound for optimal power loading, equal powerloading (that has the same performance as plain STBC withoutbeamforming), and 1-D beamforming. Since Channel 1 ishighly correlated, only beams are used in the consideredSNR range for optimal loading. Therefore, the 2-D eigen-beam-former is overall optimal for Channel 1 in the considered SNRrange, and its performance curves coincide with those ofthe optimal loading. Figs. 5 and 6 confirm that the optimalallocation outperforms both the equal power allocation and the1-D beamforming. The difference between optimal loadingand equal power loading is about 3 dB as SNR increases, sincetwo out of four beams are so weak that the power allocated tothem is wasted. The differences between the upper bound andthe exact SER in Figs. 5 and 6 not only confirm our theoreticalanalysis of (18), but also justify our approach that pushes downthe upper bound to minimize the exact SER.

On the other hand, Channel 2 is less correlated than Chan-nel 1, and all four beams are used at high SNR. Equal


Fig. 3. Optimal versus equal power loading: Channel 1.

Fig. 4. Optimal versus equal power loading: Channel 2.

power loading approaches the optimal loading when SNR issufficiently high, but is inferior to both the 2-D eigen-beam-forming and the optimal loading at low to medium SNR,as confirmed by Figs. 7 and 8. It is also shown that 2-Deigen-beamforming outperforms 1-D beamforming uniformly,and the difference is quite significant at moderate to highSNR. By checking the eigenvalues of Channel 2, we find that

. Notice that the first twoeigenvalues are not disparate enough, and the 1-D beamformeris only optimal when 8.0 dB for 16-QAM.On the other hand, the 2-D eigen-beamformer achieves opti-mality up to 15.1 dB for 16-QAM, as seenin Fig. 8. This observation corroborates the importance of 2-D

eigen-beamforming relative to 1-D beamforming, and suggeststhat 2-D eigen-beamformer deserves further research attention.

Next, we check the robustness of the loading algorithm withrespect to finite-sample effects that introduce estimation error inthe channel correlation matrix. We will discuss two simple waysto estimate at the receiver (or at the transmitter in a TDDmode). The first is based on the channel estimates. We firstsample every blocks, where is chosen sufficientlylarge such that and are uncorrelated. Col-lecting channel vectors , we can estimate


Fig. 5. SER versus SNR: Channel 1, QPSK.

Fig. 6. SER versus SNR: Channel 1, 16-QAM.

Alternatively, we can estimate directly by collecting re-ceived blocks , which correspond to the pilot sym-bols ( is square in this case, as discussed in Section IV). The

can then be estimated by the sample average

from which we obtain the channel correlation matrix estimate

(58)

For simplicity, we assume that the transmitter does not haveCSI at the beginning and employs equal power allocation for

channel estimation; thus, ��. After estimating thechannel correlation matrix at the receiver using (58) withblocks , this statistical CSI is fed back to the trans-mitter to initialize power allocation. Based on , the trans-mitter calculates the optimal according to Theorems 1 or 3.With deployed for data transmission, the closed-form SERexpressions in (9), (10), or (11) are still applicable by assumingthat the channel estimatesat the receiverare noise free, whichprovides an upper bound (achievable when the channel estima-tion accuracy increases) for the overall system performance. Weplot the SER performance for the optimal loading based onin Figs. 9 and 10 for Channels 1 and 2, respectively. We observe


Fig. 7. SER versus SNR: Channel 2, QPSK.

Fig. 8. SER versus SNR: Channel 2, 16-QAM.

that the optimal loading based on leads to a system perfor-mance very close to the ideal case (perfectly known), andoutperforms the equal power loading, even when the correlationestimates are formed based only on indepen-dent samples. Therefore, transmitter eigen-beamforming basedon channel correlations is robust to estimation errors. The latterimplies that the optimal eigen-beamformers designed with sta-tistical CSI offer an attractive choice in fast-fading applications.

VII. CONCLUSION

In this paper, we have proposed optimal transmitter designsbased only on the channel’s second-order statistics, for a setup

involving a single receive and multiple transmit antennas. Theoptimal precoders for both data transmission and channel esti-mation turn out to be generalized beamformers with multiplebeams pointing to orthogonal directions along the eigenvectorsof the channel’s correlation matrix, and with different powerloading across the multiple beams obeying a spatial (statistical)water-filling principle. Orthogonal space–time block codes arenaturally coupled with the proposed transmitter eigen-beam-formers, to increase the data rate without compromising the per-formance. A 2-D eigen-beamformer subsumes the conventional1-D beamformer as special case, and outperforms it uniformlywithout rate reduction, and without complexity increase. Simu-


Fig. 9. Loading based on^RRR : Channel 1.

Fig. 10. Loading based on^RRR : Channel 2.

lations confirm the superiority of optimal loading with respectto equal power loading, and the conventional beamforming.

APPENDIX

PROOF OF(45)

We first introduce the following two lemmas.

Lemma 2: For an positive-definite matrix , it holdsthat .

Proof: Because the matrix is positive definite,its eigen-decomposition yields �� , where

is unitary, and�� is diagonal with positive diagonal entries. We can readily verify that

��

and

��

Since , we can apply the inequalities


and

to obtain that .

Lemma 3 [21, Lemma 7]:For an positive-definitematrix with , it holds true that

, where the equality holds if and only if isdiagonal.

We deduce from Lemma 2 that under the power constraint,in (44) achieves its minimum

(59)

We next proceed to find the optimal (equivalently, ) thatsatisfies the equality in (59). The diagonal entries of , de-noted as , satisfy . Because the ma-trix is positive definite, it follows fromLemma 3 that

(60)where the equality holds if and only ifis diagonal. The inequality (60) implies that for any nondiag-onal solution of (44), we can reduce the cost function in(44), by constructing a diagonal which retains only thediagonal entries from , without altering the power con-straint. Therefore, to minimize in (44), the matrix

should be diagonal of the form given by (26).Consequently, we can reformulate our constrained optimiza-

tion problem as

subject to (61)

Applying the Lagrange multiplier method, we form the La-grangian , and take derivatives with respect to

to obtain

(62)

where denotes the Lagrange multiplier. Solving for from(62), we find

(63)

and plugging back to the power constraint in (61), we obtain. Substituting into (63), we reach the optimal solution of (45).Plugging (26) and (45) back into (59), we can verify that in-

deed the minimum value has been achieved if ; thus,the condition of (32), (26), and (45) on is sufficient and nec-essary to minimize the objective function of (44) subject to thepower constraint.

ACKNOWLEDGMENT

The authors would like to thank Prof. M.-S. Alouini of theElectrical and Computer Engineering Department at the Uni-versity of Minnesota for pointing their attention to [6] and [26].

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