Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 2, FEBRUARY 1999 199

Transactions Papers

Space–Time Codes for High Data Rate WirelessCommunication: Performance Criteria in the

Presence of Channel Estimation Errors,Mobility, and Multiple Paths

Vahid Tarokh,Member, IEEE, Ayman Naguib,Member, IEEE, Nambi Seshadri,Senior Member, IEEE,and A. R. Calderbank,Fellow, IEEE

Abstract—Space–time coding is a bandwidth and power effi-cient method of communication over fading channels that realizesthe benefits of multiple transmit antennas. Specific codes havebeen constructed using design criteria derived for quasi-staticflat Rayleigh or Rician fading, where channel state informationis available at the receiver. It is evident that the practicality ofspace–time codes will be greatly enhanced if the derived designcriteria remain valid in the absence of perfect channel stateinformation. It is even more desirable that the design criteria notbe unduly sensitive to frequency selectivity and to the Dopplerspread. This paper presents a theoretical study of these issuesbeginning with the effect of channel estimation error. Here itis assumed that a channel estimator extracts fade coefficientsat the receiver and for constellations with constant energy, itis proved that in the absence of ideal channel state informationthe design criteria for space–time codes is still valid. The analysisalso demonstrates that standard channel estimation techniquescan be used in conjunction with space–time codes provided thatthe number of transmit antennas is small. We also derive themaximum-likelihood detection metric in the presence of channelestimation errors. Next, the effect of multiple paths on theperformance of space–time codes is studied for a slowly changingRayleigh channel. It is proved that the presence of multiple pathsdoes not decrease the diversity order guaranteed by the designcriteria used to construct the space–time codes. Similar resultshold for rapid fading channels with or without multiple paths.The conclusion is that the diversity order promised by space–timecoding is achieved under a variety of mobility conditions andenvironmental effects.

Index Terms—Diversity, multipath channels, multiple anten-nas, space–time codes, wireless communication.

I. INTRODUCTION

BANDLIMITED wireless channels are narrow pipes thatdo not accommodate rapid flow of data. Deploying multi-

Paper approved by O. Andrisano, the Editor for Modulation for FadingChannels of the IEEE Communications Society. Manuscript received April15, 1997; revised November 15, 1997; July 15, 1998; and August 15, 1998.

The authors are with AT&T Labs-Research, Florham Park, NJ 07932 USA(e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(99)01931-5.

ple transmit antennas broadens this data pipe and informationtheory provides measures of capacity [5], [7], [8]. We believethat Telatar [8] was the first to obtain expressions for capacityand error-exponents for multiple transmit antenna systems inthe presence of fading and Gaussian noise. Here, capacityis derived under the assumption that fading is independentfrom one channel use to another. At about the same time,Foschini and Gans [5] derived the outage capacity under theassumption that fading is quasi-static; i.e., constant over a longperiod of time and then changing in an independent manner. Aparticular layered space–time architecture was shown to havethe potential to achieve a substantial fraction of capacity [4]. Amajor conclusion of these works is that the capacity of a multi-antenna system far exceeds that of a single antenna system. Inparticular, the capacity grows at least linearly with the numberof transmit antennas as long as the number of receive antennasis greater than or equal to the number of transmit antennas.A comprehensive information theoretic treatment for many ofthe transmit diversity schemes that have been studied beforeis presented by Narulaet al. [7].

In [13], we constructed space–time coding techniques fortransmission using multiple transmit and receive antennas.Simulation results were included to demonstrate that thesecodes perform as close as 2–3 dB from the theoretical limits.Space–time coding is a bandwidth and power efficient methodof communication over wireless Rayleigh or Rician fadingchannels. The spatial properties of space–time codes canguarantee that the diversity burden is placed at the transmitterwhile maintaining optional receive diversity. On the otherhand, the temporal properties guarantee that the diversityadvantage is achieved, without any sacrifice in the transmissionrate. Every space–time code has a well defined trellis structure,and therefore standard soft decision decoding techniques canbe used at the receiver. The design of space–time codesguarantees the highest possible transmission rate for a givendiversity gain. Previous work [13] has shown that in theory,space–time codes can provide the best theoretical tradeoff

0090–6778/99$10.00 1999 IEEE

200 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 2, FEBRUARY 1999

between diversity gain, transmission rate, constellation size,signal space dimension, and trellis complexity.

Space–time codes were constructed using design criteria de-rived for the flat Rayleigh fading channels where ideal channelstate information is available at the receiver. The goal of thispaper is to present a systematic study of effects of channel esti-mation error, frequency selectivity, and the speed of the changeof channel on the diversity advantage provided by space–timecoding. First, we consider the situation when exact channelstatistics are not known and a channel estimator is used toestimate the channel state information (CSI). We analyze thecase when all points in the signal constellation have equalenergy. In this case, an approximation can be made providinga tractable criteria for design. This criteria is identical withthe design criteria assuming ideal channel state informationat the receiver. Our analysis shows that space–time codingremains effective even in the presence of channel estimationerror, though as the number of transmit antennas increases, thesensitivity of the system to channel estimation error increases.However, this is not a major problem if the number of transmitantennas is small, and this is likely to be the case.

Next, we study the effect of multiple rays on the designcriteria for space–time codes. We prove that the presence ofmultiple paths does not decrease the diversity order promisedby the design criteria used to construct the space–time codes.The result is then extended to rapid fading channels as wellas to multiple path rapid fading channels. As diversity is thesingle most important factor determining the performance ofwireless systems, this result indicates that the combinationof space–time coding with novel equalization and estimationtechniques remains attractive for a variety of environmentsand mobility conditions.

The outline of this paper is as follows. Section II describesour system model for a mobile communication system withmultiple transmit and optional receive antennas. To keep thispaper self-contained, we give a brief review of space–timecoding. In Section III, we derive the optimum decoding metricin the presence of channel estimation error for the case whenthe fade coefficients are constant during one frame and varyfrom one frame to another. This metric can be used in thestandard Viterbi algorithm applied to the trellis structure ofthe space–time codes in order to provide estimates of thetransmitted signal. Section IV provides a performance analysisfor the case when the channel is quasi-static flat fading andthe constellation signals have equal energy (such as PSK)and only estimates of the channel state are available. Thisanalysis leads to the same design criteria as when completechannel state information is available. The latter assumptionwas used in the original design of space–time codes. Thisproves that space–time coding remains effective even in thepresence of channel estimation error. Based on the resultspresented, we can conclude that standard channel estimationtechniques carry over to this case as well, provided that thenumber of transmit antennas is small. In Section V, we studythe performance of space–time codes in a multiple path envi-ronment. We analyze a two-path model, but the generalizationto higher numbers of paths is straightforward. It is proved thatthe presence of multiple paths cannot decrease the diversity

Fig. 1. The block diagram of the transmitter and the receiver.

advantage promised by the design criteria used in the originalconstruction of space–time codes. Fast fading channels areconsidered in Section VI, where a similar result is proved inthe presence of ideal channel state information and multiplepaths. In Section VII, we provide some simulation results, andin Section VIII present conclusions and final comments.

II. THE SYSTEM MODEL AND REVIEW OF SPACE-TIME CODING

Given a slowly flat fading channel, we consider a com-munication system equipped with antennas at the base and

antennas at the mobile. Data is encoded by the channelencoder, the encoded data goes through a serial-to-parallelconverter and is divided into streams of data. Each streamof data is used as the input to a pulse shaper. The outputof each shaper is then modulated. At each time slot, theoutput of modulator is a signal that is transmitted usingtransmit antenna. We emphasize that thesignalsare transmitted simultaneously, each from a different transmitantenna, and that all these signals have the same transmissionperiod . The block diagram of the transmitter is shown inFig. 1.

Signals arriving at different receive antennas undergo inde-pendent fades. The signal at each receive antenna is a noisysuperposition of the transmitted signals corrupted by fading.

We assume that the elements of the signal constellation arecontracted by a factor of , chosen so that the averageenergy of the constellation is one. The signalreceived byantenna at time is given by

(1)

where the noise components, at time ,are modeled as independent samples of a zero-mean Gaussianrandom variable with variance per dimension. Thecoefficient , , is the pathgain from transmit antennato receive antenna, and thesecoefficients are modeled as independent samples of a complexGaussian random variable with mean zero and variance 0.5per dimension. The fading is assumed to be constant over aframe of length and to vary from one frame to another.

When complete channel state information is available [13],we considered the probability that the receiver decides erro-neously in favor of a signal

assuming that

was transmitted. The analysis given in [13] leads to thefollowing design criteria.

TAROKH et al.: SPACE-TIME CODES FOR HIGH DATA RATE WIRELESS COMMUNICATION 201

Fig. 2. 4-PSK and 8-PSK constellations.

Design Criteria For Rayleigh Space–Time Codes:

• The Rank Criterion:In order to achieve the maximumdiversity , the matrix

......

......

. . .. . .

...

has to be full rank for any codewordsand . Ifhas minimum rank over the set of pairs of distinctcodewords, then a diversity of is achieved.

• The Determinant Criterion:Suppose that a diversity ben-efit of is our target. The minimum of th roots ofthe sum of determinants of all principal cofactorsof taken over all pairs ofdistinct codewords and corresponds to the codinggain, where is the rank of . Special attention inthe design must be paid to this quantity for any codewords

and . The design target is making this sum as largeas possible. If a diversity of is the design target,then the minimum of the determinant of takenover all pairs of distinct codewords and must be maximized.

Furthermore, it was proved in [13] that a code designed usingthe above criteria will continue to perform well in Ricianenvironments.

In designing trellis codes using the above criteria, werequired that at the beginning and the end of each frame,the encoder must be in the zero state. Each transition in thetrellis representation is labeled with a sequenceof constellation elements, and this corresponds to sending thesymbol via transmit antenna for all .

To illustrate the construction of space–time codes, we con-sider the following examples.

Example 1: Consider signal constellations 4-PSK and 8-PSK, where the signal points are labeled as in Fig. 2. Thenthe codes of Figs. 3 and 4 can be used for transmission of2 and 3 bits/s/Hz using two transmit antennas. These trelliscodes are examples of space–time codes providing a diversityadvantage of two.

The performance of these codes and those constructed in[13] were shown to be as close as 2.5 dB to the theoreticallimit assuming complete channel state information.

Fig. 3. 4-PSK space–time code, four-states, 2 bits/s/Hz.

III. M AXIMUM LIKELIHOOD DECODING METRIC IN THE

ABSENCE OFCOMPLETE CHANNEL STATE INFORMATION

In practice, there will be errors in the channel state infor-mation available to the receiver. A channel estimator extractsfrom the received signal approximations to the fade coeffi-cients during each data frame. This can be done using pilottones. One method of channel estimation is to turn off alltransmit antennas apart from antennaat some time instantand to send a pilot signal using antenna. The fade coefficients

are then estimated for . This procedureis repeated for until all the coefficients ,

, are estimated. A secondmethod of estimation is to send orthogonal sequences ofsignals (similar to those given by Walsh functions) for pilotsignals, one from each transmit antenna.

Regardless of the method used for estimation, the channelestimator provides estimates for , ,

. It is assumed that is a zero-meancomplex Gaussian random variable only dependent onwith correlation coefficient . The correlation has a simpleexpression in terms of the variance of channel estimation error.In fact, in general we might assume

where the variable represents the channel estimation error.It will be assumed that is a complex Gaussian randomvariable independent of having mean zero and variance

. It follows that the correlation is given by

In general, is small at reasonable signal-to-noise ratios, sothat is close to one.

Since it was assumed that the fade coefficients arepairwise independent, the coefficients are also indepen-dent complex Gaussian random variables. Letdenote thevariance of the real and imaginary components of . Thecovariance of and is easily seen to be

, where is the complex conjugate of .


Fig. 4. 8-PSK space–time code, eight-states, 3 bits/s/Hz.

During each frame, the estimates , ,, and the received word are input to the

decoder. The variance can be computed as a functionof the signal-to-noise ratio, the number of pilot signals, andthe method of estimation. Thus the receiver can estimatethe signal-to-noise ratio and. We assume that the corre-lation coefficient is known to the decoder and derive themaximum-likelihood decoding rule and the appropriate branchmetrics.

Suppose that a codeword

has been transmitted, and

has been received.At the receiver, optimum decoding amounts to choosing a

codeword

for which thea posteriori probability

is maximized. Since only depends on for ,we can take logarithms of the above probability and decidein favor of at the receiver if this codeword minimizes thequantity

(2)Conditioned on , the random variable has mean

and variance per dimension. As-suming that was transmitted, it follows that conditionedon , , the random variable is complexGaussian with mean and variance

per dimension.

Thus from (2), the decision statistic to be minimized foroptimal decoding at the receiver is easily seen to be

(3)

Note that if the signals in the constellation have equal energy(as is the case in any PSK constellation), then the metric of(3) reduces to

(4)

In any case, it is clear that the metric of (3) and the Viterbi de-coding algorithm can be easily used for maximum-likelihooddecoding at the receiver if the code has a trellis representation.This is the case for space–time codes.

IV. EXAMINATION OF THE DESIGN CRITERIA IN THE

ABSENCE OFPERFECT CHANNEL STATE INFORMATION

We examine the design criteria in the case when constel-lation points have equal energy, as is the case for the PSKconstellations. In this important special case, we will recoverthe criteria for coding gain and the diversity advantage givenin Section II.

Suppose that was transmitted. If we suppose that thesignals in the constellation have equal energy, then consideringthe mean and variance of conditioned on ,

, it follows from (4) that the probability of decidingin favor of is given by

(5)


where

(6)

This is just the standard approximation to the Gaussian tailfunction. Notice that the process is Gaussianwith zero mean and variance 0.5 per dimension, and we maywrite

(7)

For any complex vectors andin C , let denote the inner product

of and where denotes the complex conjugate of.For any matrix , let denote the Hermitian (transposeconjugate) of . Setting , it is easy tosee that

(8)

where and , , ,for , . Thus

(9)

Since is Hermitian, there exists a unitary matrixand a real diagonal matrix such that . Therows of are a complete orthonormal basis ofC given byeigenvectors of . Furthermore, the diagonal elementsof are the eigenvalues , of countingmultiplicities. Since , where thematrix is defined in Section II, it follows (see [10])that the eigenvalues of are nonnegative real numbers.We express in terms of the eigenvalues of the matrix

. Let and , then

(10)

Combining the above with (9), and knowing that ,completely determine , and

vice-versa, we conclude that

(11)

Since is unitary, are independent complex Gaussian ran-dom variables with zero-mean and variance 0.5 per dimension.Thus , , are independentrandom variables with Rayleigh distribution

for . Thus to compute a bound on the probability oferror, we average the right side of (11) with respect to theseprobability density functions to arrive at

(12)

Let denote the rank of the matrix . Then the kernelof has dimension and exactly eigenvaluesof are zero. If we assume that the nonzero eigenvalues of

are , then it follows from inequality(12) that

(13)It follows from the above formula that the design criteria forspace–time codes presented in Section II are still valid in thepresence of channel estimation error.

It is instructive to compare this result with similar resultsfor trellis codes subject to the rapid fading model analyzed byCavers and Ho [1], where there is one transmit and one receiveantenna. Individual fade coefficients vary from one symbolto another and are not constant over a frame. Under theseassumptions, Cavers and Ho [1] establish a formula which issimilar in spirit to (13). Comparing this analogous formulato (13) provides further evidence that standard estimationmethods used for the fading channel carry over to our modelof transmit diversity provided that is small. This is likely tobe the case in practical applications.

V. EXAMINATION OF THE DESIGN CRITERIA

IN A MULTIPLE RAY FADING ENVIRONMENT

Although space–time codes were designed using a criteriadeveloped for quasi-static flat fading channels, it would be verydesirable if this design criteria not be too sensitive to multiplepaths. This motivates our studies in this section, where weanalyze the performance of space–time codes in a multiplepath environment. The analysis is presented for a two-raymodel, but the generalization to a higher number of paths isstraightforward.

We model the channel by an ensemble of slowlyfading multi-ray subchannels corresponding to theth transmitand th receive antennas for and . Theimpulse response for each of these subchannels is modeled by


where and are independent complexGaussian random variables with respective variance 0.5 and

per dimension and is the Dirac delta function. Itis assumed that the delay parameteris a random variablewith a probability density function . In some cases,it may make sense to model by an exponential randomvariable [6].

If the codeword is output from the encoder (shown inFig. 1), then the signal transmitted from antennacan berepresented by the complex function

where is the number of symbols in a frame and isa complex function, time-limited to , with total energy

.We assume that the channel is slowly varying and that the

delay parameters and the fade coefficients are constant over aframe and vary from one frame to another. If a frame consistsof signaling periods, then the time length of each frame is

. Assuming linear superposition in the propagation media,the received signal at receive antennaat time is

(14)

where , , and are, respectively, independent sam-ples of , , and . Also is a complex Gaussian stochasticprocess with two-sided power spectral density .

In computing the performance of the communication systememploying the space–time codes, we assume that the decoderhas perfect knowledge of , , and although it ispossible to extend this analysis to the case when only imperfectestimates for these quantities are available at the receiver.

Let C denote the space of complex dimensionalsignals of finite energy. The -distance between two signals

and is defined to be

Assuming that the codeword was transmitted, the decodermakes an erroneous decision in favor of, if the receivedsignal vector is closer in -distance to

than where

and

(15)

(16)

The standard approximation to the tail of Gaussian distributioncan be applied, and it follows that the probability of error given

, , and is upper bounded by

(17)

where

and is some matrix whose elements only dependon , , and . The matrix was defined in the previoussection. The diagonalization method employed in the previoussection of this paper can be applied to the quadratic form

in a straightforward manner to arrive at

(18)

where , are the eigenvalues ofcounting multiplicities. We can now prove the following

theorem.Theorem 5.1:Let denote a space–time code constructed

to provide a diversity order in a quasi-static flat fadingRayleigh environment using transmit and receive anten-nas. The diversity order provided by the codein a multiplepath channel is at least equal to .

Proof: Suppose that , ,are given. Let , denote the nonzeroeigenvalues of the matrix where . It followsfrom (18) that

We can conclude from the above that conditioned on,, , a diversity order of

is achieved. Next, we observe that independent rows of thematrix correspond to independent rows of . Hence

and


We conclude that conditioned on , ,, a minimum diversity order of is

achieved. The result follows by averaging with respect to,, since the rank of is

independent of these variables.Because the diversity order of a code is the single most

important contributor to reliable wireless communications, theabove theorem promises that a space–time code designed fora slow flat fading channel will continue to perform well in amultiple path slowly varying Rayleigh environments.

VI. EXAMINATION OF THE DESIGN CRITERIA

OVER RAPID FADING CHANNELS

When the fading is rapid, we assume that the received signalis given by

(19)

where the coefficients , ,, are modeled as independent

samples of a complex Gaussian random variable with meanzero and variance 0.5 per dimension. These assumptionscorrespond to very fast Rayleigh fading, but the generalizationto Rician fading is straightforward. The noise terms aresamples of independent zero-mean complex Gaussian randomvariables with variance per dimension.

Prior work [13] considered the probability that a maximum-likelihood decoder decides in favor of a codeword

assuming that the codeword

was transmitted. The following diversity criterion was estab-lished.

The Distance Criterion:In order to achieve the diversityin a rapid fading environment, for any two distinct

codewords and , the strings andmust differ in at least time instants as ranges from 1 to.

We can now establish the following theorem.Theorem 6.1:Let denote a space–time code constructed

to provide a diversity order in a quasi-static flat fadingRayleigh environment using transmit and receive anten-nas. The diversity order provided by the codein a flat fastfading Rayleigh channel is at least equal to .

Proof: Suppose this is not the case. Then there existcodewords and in such that

in at most time instants . It follows thatthe rank of the matrix is at most and thusthe same is true for violating the assumption that thedesigned diversity order in a quasi-static flat fading Rayleighenvironment is .

The following theorem can now be established for rapidmultiple path Rayleigh fading environments.

Theorem 6.2:Let denote a space–time code constructedto provide a diversity order in a quasi-static flat fadingRayleigh environment using transmit and receive anten-nas. The diversity order provided by the codein a multiplepath fast fading Rayleigh channel is at least equal to.

Proof: The proof is similar to that of Theorem 5.1.Since the diversity of a code is the single most important

contributor to reliable wireless communications, the abovetheorems demonstrate that a space–time code designed fora slow flat fading channel will continue to perform well invarious multiple path and flat environments under a variety ofmobility conditions.

VII. SIMULATION RESULTS

We proved that the absence of perfect channel state infor-mation at the receiver, multiple paths, and fast fading do notdecrease the diversity order guaranteed by the design criteriaused to construct the space–time codes. We now support thistheory by demonstrating some of these results via simulation.

In this section, we provide simulation results for the per-formance of the space–time codes [13] in the absence ofperfect channel state information. For additional simulationscovering the effect of fast fading and multiple paths andchannel estimation errors, we refer the reader to [11], wherea space–time coded modem is engineered.

We first discuss a channel estimation method used in oursimulations. At the beginning of each frame of symbols to betransmitted from transmit antenna, a sequence of length

pilot symbols

is appended. The sequences are designedto be orthogonal to each other

whenever .Let be the observed sequence of

received signals at antennaduring the training period. Then

where the coefficients are independent samples of a com-plex Gaussian random variable with mean zero and variance0.5 per dimension, and are independent samples of a zero-mean complex Gaussian random variable with varianceper dimension. Let . Our goal isto estimate , , using thestatistic .

The unbiased estimator having the least variance isgiven by the ratio of inner products .Indeed, since , it is easy to see that

thus


Fig. 5. Performance of four and 16 states 4-PSK codes in the presence ofchannel estimation error, 2 bits/s/Hz, two receive and two transmit antennas.

In other words,

The random variable has zero-mean. The variance ofthe estimation error is per dimension which is theminimum given by the Cramer–Rao bound.

In our simulations , , and the frame lengthis 130 symbols. In Fig. 5, we present the performance of thefour-state and 16-state codes designed in [13], using accuratechannel state information as well as under mismatch scenario.Simulation results indicate that under the mismatch scenario,there is a loss of almost 1 dB for both codes compared to thecase of ideal channel state information. Some of this SNR lossaccounts for the training energy loss. We note that even in thepresence of channel estimation error we are within 4.5 dB ofthe theoretical limit as computed by Foschini and Gans [5].

VIII. C ONCLUSION

We provided a theoretical demonstration that space–timecodes designed to provide a certain diversity order withperfect channel state information on slowly varying flat fadingRayleigh channels provide the same diversity order in variousmultipath and flat fading channels under a variety of mobilityconditions. Furthermore, this diversity order is preserved whenonly imperfect estimates for the channel state information areavailable to the receiver.

ACKNOWLEDGMENT

The authors thank the three anonymous reviewers whosecomments have greatly enhanced the quality of this paper.

REFERENCES

[1] J. K. Cavers and P. Ho, “Analysis of the error performance of trelliscoded modulations in Rayleigh fading channels,”IEEE Trans. Commun.,vol. 40, pp. 74–83, Jan. 1992.

[2] D. Divsalar and M. K. Simon, “The design of trellis coded MPSK forfading channel: Performance criteria,”IEEE Trans. Commun.,vol. 36,pp. 1004–1012, Sept. 1988.

[3] , “The design of trellis coded MPSK for fading channel: Setpartitioning for optimum code design,”IEEE Trans. Commun.,vol. 36,pp. 1013–1021, Sept. 1988.

[4] G. J. Foschini, Jr., “Layered space-time architecture for wireless commu-nication in a fading environment when using multi-element antennas,”Bell Labs Tech. J.,pp. 41–59, Autumn 1996.

[5] G. J. Foschini, Jr. and M. J. Gans, “On limits of wireless communicationin a fading environment when using multiple antennas,”WirelessPersonal Commun.,vol. 6, no. 3, pp. 311–335, Mar. 1998.

[6] W. C. Jakes,Microwave Mobile Communications.Piscataway, NJ:IEEE Press, 1993.

[7] A. Narula, M. Trott, and G. Wornell, “Information theoretic analysis ofmultiple-antenna transmission diversity,”IEEE Trans. Inform. Theory,to be published.

[8] E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T BellLaboratories Internal Tech. Memo., June 1995.

[9] R. D. Gitlin, J. Salz, and J. H. Winters, “The capacity of wirelesscommunication systems can be substantially increased by the use ofantenna diversity,”IEEE J. Select. Areas Commun.,to be published.

[10] R. A. Horn and C. R. Johnson,Matrix Analysis. New York: CambridgeUniv. Press, 1988.

[11] A. Naguib, V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-timecoding modem for high data rate wireless communications,”IEEE J.Select. Areas Commun.,Nov. 1998.

[12] N. Seshadri and J. H. Winters, “Two signaling schemes for improvingthe error performance of frequency-division-duplex (FDD) transmissionsystems using transmitter antenna diversity,”Int. J. Wireless Inform.Networks,vol. 1, no. 1, 1994.

[13] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes forhigh data rate wireless communication: Performance criterion and codeconstruction,”IEEE Trans. Inform. Theory,vol. 44, Mar. 1998.

Vahid Tarokh (M’97) received the Ph.D. degreein electrical engineering from the University ofWaterloo, Ontario, Canada, in 1995.

He is currently employed by AT&T Laboratories,Florham Park, NJ, as a Senior Member of the Tech-nical Staff, where he is doing research in wirelessand wireline communications, source and channelcoding, and multimedia information processing.

Dr. Tarokh was awarded the Gold Medal ofthe Governor General of Canada for his Ph.D.dissertation.

Ayman Naguib (S’91–M’96) received the B.Sc. de-gree (with honors) and the M.S. degree in electricalengineering from Cairo University, Cairo, Egypt, in1987 and 1990, respectively, the M.S. degree in sta-tistics and the Ph.D. degree in electrical engineeringfrom Stanford University, Stanford, CA, in 1993 and1996, respectively.

From 1987 to 1989, he spent his military serviceat the Signal Processing Laboratory, the MilitaryTechnical College, Cairo, Egypt. From 1989 to1990, he was employed with Cairo University as

a Research and Teaching Assistant in the Communication Theory Group,Department of Electrical Engineering. From 1990 to 1995, he was a Researchand Teaching Assistant in the Information Systems Laboratories, StanfordUniversity, Stanford, CA. In 1996, he joined AT&T Labs, Florham Park, NJ,as a Senior Member of the Technical Staff. His current research interestsinclude signal processing and coding for high data rate wireless and digitalcommunications and modem design for broadband systems.


Nambi Seshadri(S’81–M’86–SM’95) received theBachelor’s degree in electronics and communica-tions engineering from the University of Madras,India, in 1982 and the M.S. and Ph.D. degrees inelectrical and computer engineering from RensselaerPolytechnic Institute, Troy, NY, in 1984 and 1986,respectively.

He is Head of the Communications ResearchDepartment at AT&T Laboratories, Florham Park,NJ. Prior to this, he was a Distinguished Memberof the Technical Staff at AT&T Bell Laboratories,

Murray Hill, NJ. Research activities in the department cover a broad rangeof signal processing and communications concepts. These include signalanalysis for compression and transmission, error resilient signal compressiontechniques, new transmission techniques for wireless such as space–timecoding, radio link adaptation algorithms and protocol design, and design of aflexible wireless simulator. He was the Associate Editor of coding techniquesfor IEEE TRANSACTIONS ON INFORMATION THEORY from 1996–1998.

A. R. Calderbank (M’89–SM’97–F’98) receivedthe B.S. degree in 1975 from Warwick University,U.K., the M.S. degree in 1976 from Oxford Univer-sity, U.K., and the Ph.D. degree in 1980 from theCalifornia Institute of Technology, Pasadena, all inmathematics.

He joined AT&T Bell Laboratories in 1980, andprior to the split of AT&T and LucentTechnologies,he was a Department Head in the MathematicalSciences Research Center at Murray Hill, NJ. He isnow Director of the Information Sciences Research

Center at AT&T Labs, Florham Park, NJ. His research interests range fromalgebraic coding theory to wireless communications to quantum computing.He has developed and taught an innovative course on bandwidth-efficientcommunication at the University of Michigan and at Princeton University.

Dr. Calderbank received the 1995 Prize Paper Award from the InformationTheory Society for his work on the Z4 linearity of the Kerdock and Preparatacodes (jointly with R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane, and P.Sole).

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Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths