optimal antenna diversity signaling for wide-band systems utilizing

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002 341

Optimal Antenna Diversity Signaling for Wide-BandSystems Utilizing Channel Side InformationEko N. Onggosanusi, Akbar M. Sayeed, Member, IEEE, and Barry D. Van Veen, Member, IEEE

Abstract—Optimal multi-antenna wide-band signaling schemesare derived for multipath channels assuming perfect channel stateinformation at the transmitter. The scheme that minimizes the bit-error probability in the single-user case is a rank-one space–timebeamformer which focuses the signal transmission in the direc-tion of the most dominant channel mode. Several suboptimal vari-ations are discussed for multiuser applications. The optimal sig-naling scheme given channel statistics at the transmitter is also de-rived. The optimal scheme in this case is a full-rank space–timebeamformer that transmits on all channel modes. Analysis andsimulation results are used to compare the schemes proposed inthis paper. Finally, we discuss the optimal signaling scheme when adelayed version of the channel state is available at the transmitter.It is shown that in this case the optimal scheme is a rank-1 beam-former when the channel variations are sufficiently slow and is afull rank beamformer in a sufficiently fast fading channel.

Index Terms—Antenna arrays, diversity signaling, feedback,multipath, spread spectrum communication systems, transmitbeamforming.

I. INTRODUCTION

SPATIO-TEMPORAL diversity, combining multiple an-tennas and temporal signaling, has emerged as a key

technology in state-of-the-art systems. For example, an antennaarray is required at the base station in the third-generationWCDMA standard [1]. Receive and transmit antenna diversityare utilized for uplink and downlink applications, respectively.In receive diversity, multiple copies of the transmitted dataare processed to combat channel fading. Transmit diversity,on the other hand, utilizes a predesigned signaling schemeto send multiple copies of the data for the same purpose.Use of antenna arrays at both the base station and mobile isenvisioned in future wireless communication systems [2]. Inthis case, transmit and receive diversity techniques can be usedsimultaneously to enhance system performance.

The use of time-division duplexing, where uplink and down-link transmission are interleaved in time, as well as feedbackchannels for frequency-division duplexing, allow the transmitterto obtain channel information. This information can be used

Paper approved by M.-S. Alouini, the Editor for Modulation and DiversitySystems of the IEEE Communications Society. Manuscript received December12, 2000; revised May 21, 2001. This work was supported in part by the NationalScience Foundation under Award ECS-9979448.

E. N. Onggosanusi is with DSPS & R&D Center, Texas Instruments, Inc.,Dallas TX 75243 USA (e-mail: [email protected]).

A. M. Sayeed and V. D. Van Veen are with the Department of Electrical andComputer Engineering, University of Wisconsin-Madison, Madison, WI 53706USA (e-mail: [email protected]; [email protected]).

Publisher Item Identifier S 0090-6778(02)01367-3.

to design an efficient signaling scheme at the transmitter. Sev-eral methods for transmit signal design using channel informa-tion are found throughout the literature. For example, a filter-bank-based scheme for maximizing data rate in multicarrier sys-tems is described in [3]. The use of channel side information(CSI) at the transmitter for narrow-band systems is discussed in[4]. Block signaling design to achieve maximum signal-to-noiseratio (SNR) and diversity gain for flat fading channels can befound in [5] and [6].

This paper focuses on the design of optimal diversity sig-naling schemes that minimize the bit error rate (BER) given CSIat the transmitter. We show that when perfect channel state infor-mation is available, the BER-optimal solution is characterizedby a rank-1 structure: the same signature code is transmittedthrough each antenna and weighted according to the channelstate. For a single-path channel, we show that the signature codeis arbitrary and the optimal antenna weights focus the transmis-sion at the most dominant spatial channel mode. For a mul-tipath channel, however, we show that the optimal signaturecode is a discrete-time sinusoid with frequency equal to thechannel frequency having maximum gain. This optimal schemecan be viewed as adaptive frequency hopping with spatial beam-forming. A beamforming solution for flat fading channels wasalso suggested in [4] and [7] when perfect or sufficiently accu-rate CSI is available at the transmitter.

In multiuser scenarios, strictly enforcing optimality for eachuser may result in two or more users having the same maxi-mizing frequency and hence the same signature code at a par-ticular instant. One way to resolve this problem is to use thesubdominant frequencies for other users having the same maxi-mizing frequency as the first user. Another alternative approachis to assign distinct fixed signature codes to different users.Then, beamformer weights are chosen to minimize the BER foreach user given the sub-optimal code. We term this structure thesuboptimal beamforming (space-only optimization) scheme.

When only channel statistics are available at the transmitter,we show that BER is minimized by transmitting over allthe channel modes. For a given statistics, distinct linearlyindependent signature codes are chosen for different transmitantennas to minimize the average BER. Similar results can befound in [8]. This structure is termed the multicode scheme,which is related to the block signaling scheme in [5]. Wealso analyze the effect of delayed channel state informationat the transmitter and demonstrate that the BER-optimalsolution deviates from the rank-1 scheme and is analogous to amulticode scheme when the channel fading is sufficiently fastrelative to the delay. Previous work related to this issue can befound in [7], where analytical comparison between rank-one

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342 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 2, FEBRUARY 2002

beamforming (termed closed-loop transmit diversity) and amulticode (termed open-loop transmit diversity) scheme in thepresence of delay was given. Similar conclusions were reachedregarding the best scheme given the channel fading rate.

The rest of the paper is organized as follows. The channelmodel is outlined in Section II, followed by BER analysis forthe coherent receiver in Section III. Diversity signaling designgiven perfect CSI is discussed in Section IV. Section V coverssignaling design when only the channel statistics are available.The optimal scheme given delayed channel state information isdiscussed in Section VI. Analysis and simulation comparing dif-ferent schemes are given in Section VII, followed by concludingremarks in Section VIII.

We use the following notation. Superscripts, , and in-dicate matrix transpose, matrix conjugate transpose, and com-plex conjugation, respectively. Uppercase boldface denotes amatrix while lowercase boldface indicates a vector.denotesthe identity matrix. denotes a complexcircular Gaussian vector with mean and covariance ma-trix . Expectation is denoted as Eand the Euclidean normof vector is denoted as . The symbol denotes the Kro-necker product and vec is formed by stacking the columnsof matrix into a vector [9], [10]. is the column vector

with 1 located in the th row. The diagonal matrixgenerated by the vector is denoted by . The max-imum eigenvalue and the corresponding eigenvector of matrix

is denoted by and , respectively. The fol-lowing assumptions are used.

1) The systems operate in a Rayleigh fading channel [11].We primarily consider single-user systems. The resultsalso apply to multiuser systems in which multiuser inter-ference can be approximated as white noise.

2) Noise-free measurements of channel coefficients andmultipath delays are available at the transmitter andreceiver.

II. CHANNEL AND SIGNAL MODEL

Consider a system with transmit and receive antennas.The transmitted signal undergoes a frequency selec-tive -input, -output fading channel with delay spread of.The signal at the -th receive antenna can be written as

(1)

where is the signal transmitted via theth antennaand is the channel impulse response representingthe coupling between theth transmit th receive antenna.We assume that the additive noise process is temporallyand spatially white zero mean complex Gaussian. That is,E .

In this paper, we focus on utilizing the available spatio-tem-poral degrees of freedom for diversity only—that is, a singlebit stream is transmitted on all antennas. Thus, ,

, where is the data symbol transmitted

over all the elements within 1 symbol duration and, where is the th transmit

antenna signature waveform. For direct sequence wide-bandsystems, is the waveform corresponding to thethelement length signature code . Let denote the(two-sided) signal bandwidth. Analogous to a direct-sequencespread spectrum system, we represent as

(2)

where and is a unit-energy waveform of duration. We sample at the rate to enable discrete-time

processing without loss of information. Let

(3)

Hence, contains samples of the received signal at theth an-tenna over one symbol duration, whileis an matrixcontaining the signature codes from all transmit antennas. Now,define as the time-shift matrix correspondingto the path delay . We assume the delay is cyclic, so that

is circulant. For example, when and ,

. While the actual delay corresponds to

a linear shift, negligible error is introduced by this assump-tion for sufficiently large provided that . Further-more, if achip-levelcyclic prefix is introduced, the cyclic shiftis exact [11]. Since is essentially bandlimited to , it suf-fices to look at the sampled version of the channel impulse re-sponses at , . By defining

, ,, vec ,

and vec , we have from(1) and (2)

(4)

where .

III. COHERENTRECEIVER

For simplicity, consider coherent BPSK modulation( ) over a wide-band multipath channel with

. Here denotes the energy in one symbol.Hence, intersymbol interference (ISI) is negligible andsymbol-by-symbol detection suffices. The channel state isassumed to be constant within a symbol duration (slow fading).The maximum likelihood detector [11] in this case is

(5)

ONGGOSANUSIet al.: OPTIMAL ANTENNA DIVERSITY SIGNALING FOR WIDE-BAND SYSTEMS UTILIZING CSI 343

where is the coherent test statistic corresponding to thethreceive antenna obtained from matched filtering and maximumratio combining. Using (4), can be written as

(6)

where is an estimate of the channel coefficient vector.

A. BER Analysis

Assuming perfect estimates of channel coefficientsand multipath delays at the receiver, substitution of (4)into (6) gives

(7)

Given and assuming equally likely symbols, thein-stantaneousBER corresponding to (5) is1

(8)

where

(9)

is the received SNR gain. The instantaneous BER assesses thesystem performance for a particular channel realization. The av-erage BER reflects the system performance over a longer timescale and is defined as E BER , wherethe expectation is taken over all possible channel realizations.Either the instantaneous or average BER will be used to designthe diversity signaling scheme, depending upon the nature of theCSI available at the transmitter.

IV. DIVERSITY SIGNALING GIVEN CHANNEL STATES

In this section, we discuss the design of signature codes(see Fig. 1) to minimize BER given the

knowledge of channel states . In practice, estimates ofchannel states may be available at the transmitter either via afeedback channel or indirect measurement. A feedback channelfrom the receiver to the transmitter is used in frequency-divi-sion duplexing (FDD) systems. Indirect channel measurementis applicable in time-division duplexing (TDD) since theuplink and downlink channels are identical. Provided that theswitching time between uplink and downlink is significantlysmaller than channel coherence time, accurate measurement

1Q(x) = (1=p2�) e du.

Fig. 1. Diversity signaling considered in this paper.

can be obtained. In this section, we assume that channel statesare perfectly known at the transmitter. The optimal signalingscheme is derived for a general multi-antenna system in an

-path channel. A variation of the optimal scheme that providesmore flexibility for multi-access scenarios is also given.

A. The Optimal Diversity Signaling Scheme: Rank-1Space–Time Beamformer (Space-Time Optimization)

We are interested in finding the signature code matrixthatminimizes the instantaneous BER given . Combining(8) and (7) and using the fact that is decreasing with ,the optimal solves

(10)

where the trace constraint ensures that sum of energy in allcodes is unity. The optimum solutionis found as follows. Let

such that vec andvec . Note that trace vec vec

. Applying the identity vec vectwice [9], we have

vec (11)

Hence, defining vec , we may write(10) as (12), shown at the bottom of the next page. The solutionin (12) is unique up to a multiplicative factor.

Valuable insight to nature of the solution to (12) is obtainedby considering the single- and multiple-path cases separately.

Theorem 1: For a single-path ( ) channel, the optimalset of signature codes is , , where

is any unit-norm length-N signature code and

(13)

(12)


Fig. 2. The optimal diversity signaling structure given the channel states:common signature code for all antennasc followed by a beamformerw.

Proof: For a single-path channel,. Also, . Hence,

where the second equality follows from the eigenstructure ofKronecker product [9]. The proof is completed by noting thatany length- unit-norm vector is an eigenvector of .

The instantaneous BER obtained using the signature codes inTheorem 1 is

As depicted in Fig. 2, all the transmit antennas share thesamelength- signature code followed by the beamformer

. Thus, the optimal diversity signaling scheme isdecomposed into temporal and spatial elements, represented by

and , respectively. The signature codeis arbitrary sincethe channel is frequency-independent for . This allowsdifferent signature codes to be assigned to different users tominimize multi-acess interference and each user separatelycomputes its optimum beamformer based upon the most recentchannel states. The single path model may be applicable to amulti-antenna OFDM system in which the channel associatedwith each subcarrier is frequency-nonselective.

For a multipath channel ( ), the optimal solution givenin (12) can be further simplified by exploiting the circulant prop-erty of , which results in the following theorem.

Theorem 2: For a multipath channel ( ), the optimal setof signature codes is , , where

(14)

(15)

(16)

... (17)

Proof: See Appendix A.The techniques we apply to obtain the optimal solution are

similar to those used in [12] to derive a space–time channel(block) diagonalization. Notice that theth element of isthe complex conjugate of the frequency response of the channelbetween the th receive and th transmit antenna at frequency

. Also, for , simplifies to , where. Lastly we observe that the distinct

signature codes generated by all possible values ofare or-thogonal.

The optimum solution in Theorem 2 results in the minimuminstantaneous BER

BER (18)

with and given in (15) and (17), respectively. In afrequency-selective channel, the minimum BER is obtainedby transmitting the signal at the (discretized) frequency corre-sponding to the maximum channel gain. This is reflected in thechoice of signature code. Note that, at any particular instant,the transmission is focused to a particular frequency within theavailable bandwidth. However, over longer time intervals wherethe channel state varies significantly, the transmitted signal islikely to traverse the entire bandwidth. Once the maximizingfrequency is chosen, the spatial beamformerfocuses thesignal on the dominant spatial channel mode at that frequencyto minimize instantaneous BER. We term this structure theoptimal space–time beamforming scheme. In essence, focusingthe transmission to the dominant spatio-temporal channelmode is a generalization of selection diversity. We note that thespace–time signaling structure depicted in Fig. 2 applies to boththe optimal solutions in Theorem 1 ( ) and 2 ( ). For

, is arbitrary.

B. Suboptimal Beamforming Scheme (Space-OnlyOptimization)

In some cases, one may be restricted to use a suboptimaltemporal signature code. An example of such situation is inmulti-user scenario, where strictly enforcing optimality for eachuser may result in two or more users having the same maxi-mizing frequency and hence signature codeat a particularinstant. One way to resolve this problem is to use subdominantfrequencies for other users that have the same maximizing fre-quency as the first user. This approach does not result in mul-tiuser interference (MUI) even in frequency-selective channels.However, it requires coordination among users and temporalsignaling for each user must be adapted based on the channelstate information.

In practical CDMA systems, Gold or Walsh–Hadamard codesare used for temporal (user-specific) signature codes. Here, one


is restricted to use a fixed suboptimal codein frequency se-lective channels. In this case, CSI can still be used to performspace-only optimization. Given, the spatial beamformeris chosen to minimize the BER. In this case, in (10) canbe written as , where the optimization is performed over

. Analogous to (11), we have

vec

(19)

where the last equality follows from Identity 2 at the end of Ap-pendix A. From (10) and (19), the optimal spatial beamformergiven the signature codeis given by (20), shown at the bottomof the page, and the resulting instantaneous BER can be writtenas shown at the bottom of the page. For a single-path channel,the performance of this suboptimal scheme is identical to theoptimal one since the optimal signature codeis arbitrary, asshown in Theorem 1. Note that the above analysis and optimiza-tion ignore MUI, which in general exists when spreading codessuch as Gold or Walsh–Hadamard are used in frequency-selec-tive channels. Hence, the solution in (20) only minimizes thesingle-userperformance bound of the system. This serves as agood approximation when the number of users and spreadinggain are large and/or power control is used, as MUI can be ap-proximated as additional Gaussian noise [13]. In general, whena multiuser detector is used, the optimal spatial beamformer de-pends on the type of multiuser detector. This problem is beyondthe scope of this paper.

V. DIVERSITY SIGNALING GIVEN CHANNEL STATISTICS

When only channel statistics are available at the transmitter,the appropriate performance measure is theaverage BER.For Rayleigh fading channels, the second-order statisticscompletely characterize the channel. The advantage of usingchannel statistics is that they vary much slower in time than thechannel states and thus can be measured more accurately at thetransmitter in the TDD case, or require less frequent feedbackin the FDD case.

To obtain an analytical expression for the average BER, it isuseful to write in (8) as , where

. For Rayleigh fading, ,where is the channel covariance matrix. Assuming that nopair of channel coefficients are completely correlated and nochannel coefficient has zero energy, is positive definite. Itcan be shown via KL expansion [14] forthat

E

E

E (21)

where are the eigenvalues of the matrix

. Since is nonnegative definite and Hermitiansymmetric, it follows that is nonnegative definite. Using theidentity [15], itcan be shown that

(22)

Notice that the eigenvalues of depend upon the channel co-variance matrix and the signature waveform correlation struc-ture . Since is fixed, we choose such that minimizes(22). The following theorem addresses this issue.

Theorem 3: Under the constraint trace , where isa constant, is a sufficient condition tominimize the average BER in (22).

Proof: See Appendix B.This is consistent with the fact that maximum diversity gain is

obtained when all independent diversity channels have the sameaverage energy [11]. The constraint trace maintainsthe average received power constant.

The above theorem suggests that, to minimize BER, the setof signature codes should be chosen such that

(23)

Without loss of generality, we choose . Sinceis positive-definite and hence nonsingular, it can be inferredfrom (23) that the optimal code correlation matrixmust be

(20)

BER


nonsingular. Hence, from (7), it is easy to see thatmust befull column rank, unlike the optimal rank-one solution forwhen CSI is available ( ). This implies that, when onlychannel statistics are available at the transmitter, the signaturecodes used for different transmit antennas should be linearly in-dependent. We refer to this particular structure as themulticodescheme. Similar idea of signaling design based upon the channelstatistics can be found in [8].

When all the channel coefficients are independent andidentically distributed (i.i.d.) so that , theproblem is reduced to choosingsuch that .For a single-path channel, any set of orthogonal codes canbe used, such as Walsh–Hadamard codes. However, for amultipath channel ( ), the code correlation matrixresulting from Walsh–Hadamard codes is quite different from

. A set of Gold sequences is one choice that resultsin [11]. Hence, choosing to be a set of Goldsequences will result in near-optimal performance.

Since some channel coefficients can be correlated or havedifferent average energy, the channel covariance matrixcanbe any Hermitian positive definite matrix. In thiscase, using codes that result in may result inperformance loss compared to the optimal codes. In many cases,there is no set of codesthat satisfy (23) exactly. A special casewhen an exact solution exists is when since for any

semi-unitary matrix and Hermitian and positivedefinite, we may write . Hence, since

(24)

satisfies the optimality condition in (23).

VI. OPTIMAL SCHEME GIVEN DELAYED CHANNEL STATES

The results in Section IV assume that the CSI at the trans-mitter is perfect. However, in practical systems, some nonide-alities may exist. For instance, both FDD and TDD provide de-layed versions of the channel state at the transmitter. In addition,for FDD systems, CSI is quantized and suffers from feedbackbit error. Delay is by far the most prominent nonideality sincesufficiently fine quantization and low error rate feedback chan-nels can be used. In this section, we discuss optimal signalinggiven a delayed version of the channel state. A comparison be-tween multicode and beamforming schemes in the presence ofdelay is given in [7] using average BER. Codebook optimiza-tion for single path, single receive antenna systems based uponthe average received SNR gain and mutual information givennonideal CSI is reported in [4].

We use the Markov (AR-1) model used in [7] for channel stateevolution. Define as the delayed version of and we as-sume that evolve independently. Then, for ,

, , we have

E E

(25)

where , denotes the zeroth-order Besselfunction of the first kind, is the channel Doppler spread,and is the amount of delay. The parameterrepresents thechannel fading rate. As the fading rate increases,decreasesand the correlation between and decreases.

To obtain an optimal signaling scheme, we use two criteria:maximize the expected SNR gain, E with definedin (9) and minimize the expected BER, EBER . Due tothe effect of delay, maximizing E is not in generalequivalent to minimizing EBER .

The solution to maximization of expected SNR gain is similarto the ideal case in Section IV-A. Substituting forinto (12), it can be shown that the optimal solution is

E

Using the techniques shown in Appendix A, this solution can beexpressed in the form given in Theorem 2 by defining

E

...

... (26)

This result demonstrates that the rank-1 beamformer structuremaximizes E given the delayed channel state. How-ever, the rank-1 beamformer structure does not generally mini-mize EBER .

Exact minimization of EBER over is nottractable. Hence, we develop some insight by considering a

system and minimize the Chernoff upper boundBER [11]. Hereand we assume i.i.d. fading across all transmit antennas:E , where is the channel coefficient vari-ance. This implies that given the delayed channel state,

. Minimization of the Chernoff upper boundunder these conditions is discussed in detail in Appendix C. Inparticular, we derive a closed-form solution for . It isshown in Appendix C that the optimal signature code matrixfor is

(27)

(28)

where and are orthonormal vectors, , and. The coefficient is given in (38) in Ap-

pendix C. When , represents the beamforming solu-


Fig. 3. The optimal� for P = 2, Q = 1, andL = 1 as a function of� fordifferent values ofE=� with khk = � = 1.

tion depicted in Fig. 2. On the other hand, indicatesa multicode solution as is of rank 2. Notice that the rank 2

is a perturbation from the beamforming solution .This perturbation is needed to achieve optimality due to the un-certainty in the most dominant channel mode derived from thedelayed channel state. The value of in (28) is depicted inFig. 3 as a function of for various values of while fixing

and . Observe that the rank-one beamformingsolution becomes optimal at smaller values of(faster fading)as is decreased.

VII. COMPARISON AND EXAMPLES

It is shown in Section IV-B that the suboptimal beamforming(space-only optimized) scheme is inferior to the optimal(space–time optimized) scheme when CSI is available at thetransmitter. Also, for single path channel, it is shown that theoptimal scheme coincides with the suboptimal scheme.

In this section we first demonstrate that the suboptimal beam-forming scheme in Section IV-B based on CSI is superior to themulticode scheme. This is intuitively satisfying since channelstates convey more information about the channel than channelstatistics. For simplicity, assume that and

for the suboptimal beamforming scheme (29)

for the multicode scheme (30)

These conditions result in the same transmitted signal energy.For the multicode scheme, (30) results in minimum asit satistifies the optimality condition given in (23). The condi-tion (29) for the signature codein the suboptimal beamformingscheme indicates thathas the same autocorrelation character-istics as the codes in the multicode scheme. By the convolutiontheorem, it also indicates thathas an all-pass frequency re-sponse, in contrast to the optimal solution implied by Theorem

1. Let and be SNR gain as defined in (9) for (29)and (30), respectively. Then, from (20), we have

trace

This demonstrates that the best multicode scheme cannot outper-form the suboptimumbeamforming scheme. Hence, usingCSI tooptimizeonlythespatialbeamformerisstillbeneficial.Moreover,for a system with one receive antenna in a single-path channel( ), and .This indicates a (10 ) dB SNR gain for the suboptimalbeamforming over the multicode scheme.

We next compare the optimal beamforming, suboptimalbeamforming, and multicode schemes for a system withdifferent values of and and . Binary Gold codesare used for one of the suboptimal beamforming and multicodeschemes. The correlation properties of binary Gold codes aresuch that (29) and (30) are closely approximated [11]. Thesimulated channel is Rayleigh fading with .The average BER curves are depicted in Fig. 4(a)–(d). Observethat, for [Fig. 4(a) and (b)], the suboptimal beamformingscheme coincides with the optimal one. Also, the gain ofthe optimal beamforming scheme relative to the suboptimalscheme is more pronounced as the number of pathsincreases.This gain is obtained due to the spectral focusing of energy inthe frequency selective channel as discussed in Section IV-A.The suboptimal scheme only focuses spatially. Forand 4, we choose and , respectively.The suboptimal beamforming schemes utilize the Gold code,the second and the fourth dominant frequencies. Fig. 4(c)–(d)show that the schemes which use subdominant frequencies stilloutperform the one using the Gold code. The loss associatedwith use of subdominant frequencies seems to be negligible,yet more pronounced for a larger number of paths. This isbecause frequency selectivity increases with the number ofpaths .

As an example of signaling design given the channel statis-tics, we choose a , system in a singlepath channel( ) with channel statistics

(31)

Here, is the correlation between 2 channelcoefficients and and represents the en-


(a) (b)

(c) (d)

Fig. 4. Average BER comparison for optimal beamforming, sub-optimal beamforming and multicode forP = 2 and various values ofQ andL. Gold codes areused for the multicode and one of the sub-optimal beamforming schemes. (a)Q = 1,L = 1. (b)Q = 2,L = 1. (c)Q = 1,L = 2. (d)Q = 1,L = 4.

ergy difference between and . Decomposing as theproduct of an upper and lower triangular matrix and applying(24) gives

where form an orthonormal set. Fig. 5 showsBER of the optimal multicode and Walsh–Hadamard basedmulticode schemes ( ) with . In this case,the performance of the optimal scheme does not depend uponand since (23) is satisfied exactly independent ofand . Asevident, the performance gain of the optimal scheme becomesmore significant as and/or increase.

To illustrate signature code design given the channel statisticsfor a multipath channel, we consider with ,. We model the channel covariance matrix as

Fig. 5. Average BER comparison between the optimal and orthogonalmulticode schemes forP = 2, Q = L = 1. Various values of� andc areused.

where is given in (31). The parameters and rep-resent the energy difference and correlation between two paths.The code length is set to be 31. We choose


and such that the optimality condition (23) cannotbe satisfied. The above model forassumes that two differentpaths share the same spatial channel characteristic. Since the op-timal multicode scheme satisfying (23) is unknown, we employa solution that approximately satisfies (23) by solving the fol-lowing equation numerically:

(32)

where and denotesthe Frobenius norm [10]. The average BER of the optimal andGold-sequence based multicode schemes are shown in Fig. 6.Observe that the optimal multicode outperforms the Gold-basedmulticode by 1.5-dB at BER . The results with

represents an unattainable BER lower bound, whichis shown only for comparison.

To illustrate the results obtained in Section VI when delayedchannel states are used, we compare the performance of the op-timal solution in (27) to the beamforming (rank-1) and mul-ticode schemes with , . The multicodescheme consists of the two length-32 Walsh-Hadamard codesfor two transmit antennas. The simulated average BER for

are depicted in Fig. 7(a)–(c)2 . Observe that for allfading rates, the performance of the optimal solution tends tocoincide with that of beamforming for small and withmulticode for higher . At moderate , beamformingis near-optimal for low Doppler spread and multicode is nearoptimal for high Doppler spread.

These results indicate that the beamforming solution is op-timal only when the delay is small relative to the channel fadingrate or the fading is sufficiently slow, i.e., . As thefading rate increases, the optimality of the multicode schemeis attributed to the increase of uncertainty of the most domi-nant channel mode. However, the rank-1 scheme is always betterwhen is small. It is reasonable to expect that this claimalso holds for and . Furthermore, for othertypes of nonidealities, beamforming solution is perceived to beoptimal when the nonidealities are sufficiently mild. This con-clusion is similar to the results in [4].

VIII. C ONCLUSION

The design of diversity signaling schemes that use CSI atthe transmitter is investigated. The channel side informationis either the channel state or the channel statistics. It is shownthat when perfect channel state information is available atthe transmitter, the BER-minimizing scheme consists of acommon signature code for all transmit antennas, followedby a beamformer which focuses the transmission to the mostdominant spatio-temporal channel mode at any particularinstant (the space-time optimized scheme). The optimalspace-time beamformer scheme can be efficiently implemented

2For 0.667-ms delay (1 slot in WCDMA [1]),� = 0:95; 0:8; 0:6 correspondto Doppler spreads of 107, 219 319 Hz (� 58, 118, 172 kmph at 2 GHz centerfrequency), respectively.

Fig. 6. Average BER comparison between the optimal and orthogonalmulticode schemes forP = L = 2, Q = 1. We use� = � = 0:5 andc = c = 0:5. The results with� = I =4 represent an unattainable BERlower bound, which is shown only for comparison.

using IDFT-DFT bank as in multicarrier systems, followedby an adaptive array. The beamformer computation involves aDFT and finding the most dominant eigenvector of amatrix. When the temporal signature code is fixed, the CSIcan still be used to perform space-only optimization. Whenonly the channel statistics are available at the transmitter, theBER-minimizing solution suggests that the signal should betransmitted throughout all the channel modes to combat theeffects of fading. This is done by utilizing a set of linearlyindependent signature codes for the transmitter array that matchthe channel statistics so as to provide independent and identicalfading subchannels (the multicode scheme). Finally, we discusssignaling design based upon delayed channel state information.It is demonstrated that the BER-minimizing scheme is a rank-1space-time beamformer for lower fading rates and a multicodescheme for high fading rates.

When CSI is available at the transmitter, the solution pre-sented in this paper (space–time or space-only optimization) re-quires the computation of the dominant eigenvector ofmatrices. When , a closed-form solution is possible. For arelatively small , numerical techniques such as power methodand its variants [14], which compute only the dominant eigen-vector can be utilized to provide the solution in a reasonableamount of time. When is large, more sophisticated algorithmsmay be needed.

APPENDIX APROOF OFTHEOREM 2

Since is circulant, , whereis the -DFT matrix and [14]

since is the first column of . Define the matrix


(a) (b)

(c)

Fig. 7. Average BER comparison for optimal diversity scheme, beamforming and multicode schemes forP = 2, Q = L = 1 when the delayed channel stateis used at the transmitter. (a)� = 0:95. (b) � = 0:8. (c) � = 0:6.

. Then

(33)

where the last equality in (33) follows from the identity. Hence,

......

...

The matrix can be rearranged into a block diag-onal matrix as follows:

...

where , is the ( )unitary permutation matrix defined in [9] and is given in(17). Similar block diagonalization technique has been used in[12]. Thus,


To complete the proof, we need the following two basic lemmasfrom linear algebra.

Lemma 1: Let , be matrices and be unitary.Let . Then .

Lemma 2: Let ... be a block diag-

onal matrix where . Then,, where .

Applying Lemma 1, 2 and the fact that isunitary, we have

where the second and third equalities follow from the followingidentities [9].

1) For any , , ,, .

2)

APPENDIX BPROOF OFTHEOREM 3

Let , and

. The constant trace con-straint trace is equivalent to . We seekto solve the following optimization problem:

s.t.

Note that . If the opti-mizer does not depend on , it follows that isminimized at . Notice that is trivially concave in

.This problem can be solved using Lagrange multiplier tech-

niques. We form the Lagrangian and its partial derivatives

(34)

Setting all the partial derivates in (34) to 0, we have

(35)

This implies is independent of and thus. Enforcing the constraint we have

for . It is easy to checkthat this solution satisfies Kuhn–Tucker optimality condition.

APPENDIX CMINIMIZING E

Applying the eigendecompositionwhere and defining

, it can be shown that

E

E

where denotes a noncentral chi-squared distributedrandom variable with degrees of freedom and noncentralityparameter . Note that for any unitary matrix , thereexists parameters , , andangles such that

(36)

Using the above parametrization and the generating function ofthe noncentral chi-squared random variable [11], we have

E

Notice that the phase angles are immaterial. Hence,the optimization problem is reduced to finding nonnega-tive parameters such that ,

and is minimized. Note that oncethe optimal are obtained, the optimumsignature code matrix is , where is anysemi-unitary matrix, and ischosen to satisfy (36).

For , let and . Let .Then, we have

(37)


(38)

By symmetry, we only need to consider ,. The denominator in (37) is independent ofand the

numerator can be written as shown at the top of the page. Ob-serve that for . This shows that the min-imum of occurs at . Then, it follows from (36) that

where .Now, let with . That is,

Then, the optimization problem is reduced to finding. The first derivative condition implies

that the critical points of satisfy ,where

It is easy to show that the larger root of this equation is definedas in (38), shown at the top of the page. It can be shown (by thesecond derivative test) that is concave and is a globalmaximum. Since

it follows that . The solution in (28) followsfrom enforcing the constraint and using the factthat is strictly increasing within .

REFERENCES

[1] [Online]. Available: http://www.3gpp.org[2] J. H. Winters, “Smart antennas for wireless systems,”IEEE Personal

Commun., pp. 23–27, Feb. 1998.[3] A. Scaglione, S. Barbarossa, and G. Giannakis, “Filterbank transceiver

optimizing information rate in block transmissions over dispersive chan-nels,” IEEE Trans. Inform. Theory, vol. 45, pp. 1019–1032, Apr. 1999.

[4] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient useof side information in multiple antenna data transmission over fadingchannels,”IEEE J Select. Areas Commun., vol. 16, pp. 1423–1436, Oct.1998.

[5] G. Ganesan and P. Stoica, “Space-time diversity using orthogonal andamicable orthogonal designs,” inProc. ICASSP 2000, vol. 5, 2000, pp.2561–2564.

[6] G. Giannakis, Y. Hua, and P. Stoica,Signal Processing Advances inWireless Communications. Englewood Cliffs, NJ: Prentice-Hall, 2000,vol. 2.

[7] E. N. Onggosanusi, A. Gatherer, A. G. Dabak, and S. Hosur, “Perfor-mance analysis of closed-loop transmit diversity in the presence of feed-back delay,”IEEE Trans. Commun., vol. 49, pp. 1618–1630, Sept. 2001.

[8] M. S. Alouini, A. Scaglione, and G. B. Giannakis, “PCC—principalcomponents combining for dense correlated multipath fading environ-ments,” inProc. VTC Fall 2000, Boston, September 2000.

[9] J. Brewer, “Kronecker products and matrix calculus in system theory,”IEEE Trans. Circuits Syst., vol. CAS-25, pp. 772–781, Sept. 1978.

[10] R. Horn and C. Johnson,Topics in Matrix Analysis. Cambridge, MA:Cambridge Univ. Press, 1991.

[11] J. G. Proakis,Digital Communications, 3rd ed. New York: McGrawHill, 1995.

[12] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wirelesscommunication,”IEEE Trans. Commun., vol. 46, pp. 357–366, Mar.1998.

[13] S. Verdu, Multiuser Detection. Cambridge, MA: Cambridge Univ.Press, 1998.

[14] G. H. Golub and C. F. Van Loan,Matrix Computations. Baltimore,MD: John Hopkins Univ. Press, 1996.

[15] M. K. Simon and M.-S. Alouini, “A unified approach to the performanceanalysis of digital communication over generalized fading channels,”Proc. IEEE, vol. 86, pp. 1860–1877, September 1998.

Eko N. Onggosanusi received the B.Sc. (withhighest distinction), M.Sc., and Ph.D. degreesin 1996, 1998 and 2000, respectively, from theUniversity of Wisconsin-Madison, all in electricaland computer engineering.

During the summer of 1999 and 2000, he was avisiting student at the DSPS R & D Texas Instru-ments, Dallas, TX, working on antenna array diver-sity for wideband CDMA and adaptive modulationand coding for wireless personal area networks, re-spectively. Since January 2001, he has been with the

DSPS R & D Texas Instruments, where he is currently a member of technicalstaff in the mobile wireless branch of communication systems laboratory. Heis currently working on the third-generation wireless communication systems,focusing on high data rate techniques and applications. His research interestsinclude signal processing, information theory and channel coding for digitalcommunications.


Akbar M. Sayeed(S’89–M’97) received the B.S. de-gree from the University of Wisconsin-Madison in1991 and the M.S. and Ph.D. degrees from the Uni-versity of Illinois at Urbana-Champaign, in 1993 and1996, respectively, all in electrical and computer en-gineering.

While at the University of Illinois, he was aresearch assistant in the Coordinated ScienceLaboratory and was also the Schlumberger Fellowin signal processing from 1992to1995. During1996–1997, he was a Post-Doctoral Fellow at Rice

University, Houston, TX. Since August 1997, he has been with the Universityof Wisconsin-Madison, where he is currently an Assistant Professor inElectrical and Computer Engineering. His research interests are in wirelesscommunications, statistical signal processing, time-frequency and waveletanalysis.

Dr. Sayeed received the NSF CAREER Award in 1999 and the ONR YoungInvestigator Award in 2001. He is currently serving as an Associate Editor forthe IEEE SIGNAL PROCESSINGLETTERS.

Barry D. Van Veen (S’81–M’86) was born in GreenBay, WI. He received the B.S. degree from MichiganTechnological University in 1983 and the Ph.D. de-gree from the University of Colorado in 1986, bothin electrical engineering.

He was an ONR Fellow while working on thePh.D. degree. In the spring of 1987, he was with theDepartment of Electrical and Computer Engineeringat the University of Colorado–Boulder. Since Augustof 1987, he has been with the Department of Elec-trical and Computer Engineering at the University

of Wisconsin–Madison and currently holds the rank of Professor. His researchinterests include signal processing for sensor arrays, nonlinear systems,adaptive filtering, wireless communications and biomedical applications ofsignal processing. He coauthoredSignals and Systems(New York: Wiley,1999) with Simon Haykin.

Dr. Van Veen was a recipient of a 1989 Presidential Young Investigator Awardfrom the National Science Foundation and a 1990 IEEE Signal Processing So-ciety Paper Award. He served as an associate editor for the IEEE Transactions onSignal Processing and on the IEEE Signal Processing Society’s Technical Com-mittee on Statistical Signal and Array Processing from 1991 through 1997 andis currently a member of the Sensor Array and Multichannel Technical Com-mittee. He received the Holdridge Teaching Excellence Award from the ECEDepartment at the University of Wisconsin in 1997.

optimal antenna diversity signaling for wide-band systems utilizing

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