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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 151

On the Design of Algebraic Space–Time Codes forMIMO Block-Fading Channels

Hesham El Gamal, Member, IEEE,and A. Roger Hammons, Jr., Member, IEEE

Abstract—The availability of multiple transmit antennas allowsfor two-dimensional channel codes that exploit the spatial transmitdiversity. These codes were referred to as space–time codes byTarokh et al. Most prior works on space–time code design haveconsidered quasi-static fading channels. In this paper, we extendour earlier work on algebraic space–time coding to block-fadingchannels. First, we present baseband design criteria for space–timecodes in multi-input multi-output (MIMO) block-fading channelsthat encompass as special cases the quasi-static and fast fadingdesign rules. The diversity advantage baseband criterion is thentranslated into binary rank criteria for phase shift keying (PSK)modulated codes. Based on thesebinary criteria, we constructalgebraic space–time codes that exploit the spatial and temporaldiversity available in MIMO block-fading channels. We alsointroduce the notion of universal space–time codes as a general-ization of the smart–greedy design rule. As a part of this work,we establish another result that is important in its own right:we generalize the full diversity space–time code constructionsfor quasi--static channels to allow for higher rate codes at theexpense of minimal reductions in the diversity advantage. Finally,we present simulation results that demonstrate the excellentperformance of the proposed codes.

Index Terms—Block-fading channels, multiple transmit and re-ceive antennas, space–time codes, wireless communication.

I. INTRODUCTION

RECENT works have investigated the design of channelcodes that exploit the spatial transmit diversity avail-

able in coherent multi-input multi-output (MIMO) channels(e.g., [1]–[10]). These works have primarily focused on thequasi-static fading model in which the path gains remain fixedthroughout the codeword. In this paper, we extend our earlierwork on quasi-static fading channels to the more generalblock-fading model. In this model, the codeword is composedof multiple blocks. The fading coefficients are constant overone fading block but are independent from block to block.The number of fading blocks per codeword can be regardedas a measure of the interleaving delay allowed in the system,so that systems subject to a strict delay constraint are usuallycharacterized by a small number of independent blocks.

Manuscript received August 10, 2001; revised June 25, 2002. The materialin this paper was presented in part at the IEEE International Symposium onInformation Theory, Washington, DC, June 2001.

H. El Gamal is with the Electrical Engineering Department, the OhioState University, Columbus, OH 43210 USA (e-mail: [email protected]).

A. R. Hammons, Jr. is with Corvis Corporation, Columbia, MD 21046-3403USA (e-mail: [email protected]).

Communicated by G. Caire, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2002.806116

The information-theoretic capacity of MIMO block-fadingchannels has been studied by Biglieriet al.assuming the avail-ability of channel state information (CSI) at both the transmitterand receiver [11]. One of their key conclusions was that antennadiversity can be a substitute for temporal diversity. This conclu-sion does not necessarily hold for the case of interest in thispaper where CSI is only available at the receiver. The code de-sign objective in this scenario is to exploit both spatial and tem-poral diversity.

In this paper, we present baseband design criteria that deter-mine the diversity and coding advantages achieved in MIMOblock-fading channels. We show that the design criteria in [12],[13] for quasi-static and fast-fading MIMO channels are specialcases of the new criteria. Then, we follow the approach in [14] totranslate the diversity advantage baseband design criterion intobinary design criteria for phase shift keying (PSK) modulatedcodes. The binary design criteria facilitate the development ofan algebraic framework for space–time code design in this sce-nario. This algebraic framework opens the door for constructingcodes that realize the optimum tradeoff between diversity ad-vantage and transmission rate for systems with arbitrary num-bers of transmit antennas and fading blocks per codeword. Wealso introduce the notion ofuniversal space–time codingwhichaims at constructing codes that exploit the additional temporaldiversitywhenever available. This notion is a generalization ofthe smart greedy design principle [12] for the present scenario.

In addition to its importance for space–time code designin time-varying channels, the MIMO block-fading model hasproved to be beneficial in the frequency-selective scenario.In [15], we utilize the algebraic framework developed here toconstructspace–frequencycodes that exploit the frequency di-versity available in MIMO frequency-selective fading channels.

As a part of this work, we generalize the quasi-static full di-versity space–time code constructions in [14] to allow forhigherratecodes with arbitrary diversity advantages. This generaliza-tion offers more flexibility in the tradeoff between diversity ad-vantage and transmission rate, especially in systems with largenumbers of transmit antennas.

The rest of this paper is organized as follows. Section II in-troduces the system model and briefly reviews the design rulesin MIMO quasi-static and fast-fading channels. The design cri-teria for space–time codes in MIMO block-fading channels arepresented in Section III. Section IV is devoted to the develop-ment of the algebraic design framework. Simulation results forcodes constructed based on the new framework are provided inSection V for some representative scenarios. Finally, we offersome concluding remarks in Section VI.

0018-9448/03$17.00 © 2003 IEEE

152 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003

II. SPACE–TIME CODING CONCEPTS

In this section, we review the main concepts of space–timecode design and introduce our notation. The system model ispresented in Section II-A and we review briefly the design cri-teria for quasi-static and fast-fading channels in Sections II-Band II-C, respectively.

A. System Model

In the system under consideration, the source generatesinformation symbols from the discrete alphabet, which areencoded by the error control codeto produce codewords oflength over the symbol alphabet . The encodedsymbols are parsed among transmit antennas and thenmapped by the modulator into constellation points from thediscrete complex-valued signaling constellationfor trans-mission across the channel. The modulated streams for allantennas are transmitted simultaneously. At the receiver, thereare receive antennas to collect the incoming transmissions.The received baseband signals are subsequently decoded bythe space–time decoder.

Following [14], we formally define a space–time code to con-sist of an underlying error control code together with a spatialparser.

Definition 1: An space–time codeof size consistsof an error control code and a spatial parser thatmaps each codeword vector to an matrix whoseentries are a rearrangement of those of.

Except as noted to the contrary, we will assume that the stan-dard parser maps

to the matrix

......

. . ....

In this notation, it is understood that is the code symbol as-signed to transmit antennaat time .

Let be the modulator mapping function. Thenis the baseband version of the codeword as transmitted

across the channel. For this system, we have the following base-band model of the received signal:

(1)

where is the energy per transmitted symbol; is the com-plex path gain from transmit antennato receive antenna attime ; is the transmitted constellation point fromantenna at time ; is the additive white Gaussian noise

sample for receive antennaat time . The noise samples are in-dependent samples of zero-mean circularly symmetric complexGaussian random variable with variance per dimension.The different path gains are assumed to be spatially whitesamples of zero-mean complex Gaussian random variable withunit variance. The CSI is assumed to be knowna priori only atthe receiver.

The fading model of primary interest is that of a block flatRayleigh-fading process in which the codeword encompasses

fading blocks. The complex fading gains are constant overone fading block but are independent from block to block. Thequasi-static and fast-fading models are special cases of theblock-fading model in which and , respectively,[13], [12], [14]. For simplicity, it is also assumed here thatdivides .

B. Quasi-Static Fading Channel

For the quasi-static flat Rayleigh-fading channel the pairwiseerror probability that the decoder will prefer the alternate code-word to is upper-bounded by [13], [12]:

(2)

(3)

where and is thegeometric mean of the nonzero eigenvalues of

The exponent is sometimes called the “diversity advan-tage,” while the multiplicative factor is sometimes called the“coding advantage.” The parameteris the diversity providedby the multiple transmit antennas. The pairwise error probabilityleads to the following rank and equivalent product distance cri-teria for space–time code design in quasi-static channels [13],[12].

• Rank Criterion: Maximize the transmit diversity advantageover all pairs of distinct codewords

.• Product Distance Criterion: Maximize the coding advan-

tage over all pairs of distinct codewords.

The rank criterion is the more important of the two as it deter-mines the asymptotic slope of the performance curve as a func-tion of [12].

In [14], we developed a binary rank criterion that facilitatesthe design of algebraic binary space–time codes for binary phaseshift keying (BPSK) modulation. Using the binary rank crite-rion, we also proposed the following general construction forfull diversity space–time codes [14].

Theorem 2(Stacking Construction): Letbe binary matrices of dimension , , and let

EL GAMAL AND HAMMONS: ALGEBRAIC SPACE–TIME CODES FOR MIMO BLOCK-FADING CHANNELS 153

be the space–time code of dimensionconsisting ofthe codeword matrices

...

where denotes an arbitrary-tuple of information bits and. Then satisfies the binary rank criterion, and thus,

for BPSK transmission over the quasi-static fading channel,achieves full spatial transmit diversity , if and only if

have the property that

is of full rank unless .

The binary rank criterion and stacking construction lift to thedomain where they describe full diversity quaternary phase

shift keying (QPSK)-modulated space–time codes (see [14] fordetails). In this paper, we generalize the stacking constructionin two different ways. First, we present a generalization thatallows for increasing the transmission rate at the expense of aminimal reduction in the diversity advantage. Second, we ex-tend this construction to MIMO block-fading channels wherethe constructed codes exploit the spatial and temporal diversityavailable in the channel without compromising the transmissionrate.

C. Fast-Fading Channel

For the fast-fading channel, the pairwise error probability canbe upper-bounded by [12]

where is the th column of , is the th column of , isthe number of columns that are different from , and

The diversity advantage is now , and the coding advantageis .

Thus, the fundamental design criteria for space–time codesover fast-fading channels are the following [12]:

• Columnwise Hamming Distance Criterion: Maximize thenumber of column differences over all pairsof distinct codewords .

• Product Criterion: Maximize the coding advantage

over all pairs of distinct codewords .

Since real fading channels will be neither quasi-static nor fastfading but something in between, Tarokhet al. [12] suggestedthe ad hocstrategy of designing space–time codes based on acombination of the quasi-static and fast-fading design criteria.They refer to space–time codes designed according to the hybridcriteria as “smart greedy codes,” meaning that the codes seek toexploit both spatial and temporal diversity whenever available.The fact that the smart greedy approach only considers the twoextremescenarios of quasi-static and fast fading motivates thedesign ofuniversal space–time codesthat strive to achieve thesame goal in the parameterized MIMO block-fading channels.

III. SPACE–TIME DESIGN CRITERIA FOR

BLOCK FADING CHANNELS

Now, we consider the MIMO block-fading model and de-velop a systematic approach for designing space–time codesthat achieve the maximum diversity advantage for any codingrate, number of transmit antennas, and temporal interleavingdepth. We start in this section by developing the design criteriathat govern the code performance in such channels.

A. Baseband Design Criterion

Under the block-fading assumption, the path gains are con-stant over consecutive symbol durations. We will use thenotation to denote the single or two-dimensional vectorof values of parameter for the th fading block. Accord-ingly, we have1

......

. . .

Then, for

(4)

Assuming that codeword is transmitted, the conditional pair-wise error probability that the decoder will prefer an alternatecodeword to is now given by

where

1The dimensions of the arrays and matrices are reported as subscripts.


is a Gaussian random variable with mean

and variance

Thus,

Following the approach in [12], [13], the pairwise probabilityof error can be manipulated to yield the fundamental bound

(5)

where

and , are the nonzero eigenvalues of

Hence, the generalized diversity and product distance criteriafor space–time codes over MIMO block-fading channels are asfollows.

Block-Fading Sum of Ranks Criterion: Maximize the trans-mit diversity advantage

over all pairs of distinct codewords .Block-Fading Product Distance Criterion: Maximize the

coding advantage

over all pairs of distinct codewords .It is straightforward to see that the design criteria for

quasi-static and fast-fading channels can be obtained from theblock fading criteria by simply letting and ,respectively.

Similar to the quasi-static fading scenario, the fact that thediversity advantage baseband rank criterion applies to thecomplex-valued differences between baseband codewordsrepresents a major obstacle to the systematic design of maximaldiversity space–time codes. Following in the footsteps of[14], we will translate the baseband criterion into an equiva-lent binary design criteria for BPSK- and QPSK-modulated

space–time codes. These general binary criteria are sufficientto ensure that a space–time code achieves a certain level ofdiversity over block-fading channels. In our approach, a certainequivalence relation is imposed upon potential basebanddifference matrices, where each equivalence class contains aspecial representative that is seen to be a binary projection ofa codeword whose rank overthe binary fieldis a lower boundon the rank of any of thecomplexmatrices in the equivalenceclass.

B. BPSK Binary Design Criterion

For BPSK modulation, the natural discrete alphabet is thefield of integers modulo. Modulation is performedby mapping the symbol to the constellation point

according to the rule . Note that it ispossible for the modulation format to include an arbitrary phaseoffset , since a uniform rotation of the BPSK constellationwill not affect the rank of the matrices northe eigenvalues of the matrices

Notationally, the circled operator will be used to distinguishmodulo addition from real- or complex-valued oper-ations. Based on this, we have the following result for BPSK-modulated codes over MIMO block-fading (BF) channels.

Proposition 3(BPSK-BF Binary Design Criterion): Letbe a linear space–time code with used in a com-munication system with transmit antennas and operating overa block-fading channel with independent blocks per code-word. Let

where the rank is over the binary field. Then, for BPSK trans-mission, the space–time codeachieves a transmit diversitylevel at least as large as.

Proof: From [14], we know that for any two codewords

where the rank on the left-hand side is over the complex fieldand on the right-hand side is over the binary field. Hence,

Since is linear, then and

as was to be shown.

C. QPSK Binary Design Criterion

For QPSK modulation, the natural discrete alphabet is thering of integers modulo . Modulation is per-formed by mapping the symbol to the constellation point

according to the rule , where .Again, the absolute phase reference of the QPSK constellation


could have been chosen arbitrarily without affecting the diver-sity or coding advantages of a -valued space–time code.

For the -valued matrix , we define the binary-valuedrow and column indicant and , respectively,as in [14]. These indicant projections serve to indicate certainaspects of the binary structure of the matrix in which mul-tiples of two are ignored. For example, let be a -valuedmatrix with exactly rows that are not multiples of two. Aftersuitable row permutations if necessary, it has the following rowstructure:

...

...

The row-based indicant projection (-projection) is then de-fined as

...

...

where is the binary projection of (i.e., moduloreduction of the entries [14]). Similarly, the column-based in-

dicant projection ( -projection) is defined as

(6)

Using these binary indicants, we have the following resultthat translates the baseband rank criterion into a binary designcriterion for QPSK-modulated codes in a MIMO block-fadingenvironment.

Proposition 4(QPSK-BF Binary Design Criterion): Letbe a linear space–time code used in a communicationsystem with transmit antennas and operating over a block-fading channel with blocks per codeword. Let

where the rank is over the binary field. Then, for QPSK trans-mission, the space–time codeachieves a transmit diversitylevel at least as large as.

Proof: The proof follows the technique in [14] to relatethe binary rank of the indicants to the baseband rank, then it isstraightforward to obtain the result using the same argument asin the BPSK scenario.

IV. A LGEBRAIC CODE DESIGN

The MIMO block-fading channel binary design criteriapermit the construction of space–time codes that realize theoptimum tradeoff between transmission rate and diversityadvantage. In this section, we show that the design of full di-versity space–time codes is straightforward using the stackingconstruction in Theorem 2. However, we will also show thatusing full diversity codes in this scenario entails a significantloss in the transmission rate compared to the quasi-static case.This motivates the design of space–time codes that exploitthe additional temporal diversity without compromising thetransmission rate. We show that carefully constructed codes canachieve significant increases in the diversity advantage with arelatively small number of fading blocks per codeword (i.e.,a small penalty in terms of interleaving delay), and withoutany rate reduction. The main results are developed first inSection IV-A for BPSK-modulated codes and the extension toQPSK modulation is then briefly outlined in Section IV-C.

A. Algebraic Framework for BPSK Space–Time Codes

In a MIMO block-fading channel with transmit antennasand fading blocks per codeword, the maximum transmit di-versity is . Under this model, the space–time codeis de-fined to consist of codewords

......

...(7)

where , . Full diversity is realized in thisscenario through the multistacking code construction [14].

Corollary 5 (Multistacking Construction) : The space–time code will achieve full transmit diversityif for every , the set of matrices

satisfies the stacking constructionconditions in Theorem 2.

Proof: According to the binary design criterion for BPSKcodes, full diversity will be achieved if for every nonzero code-word we have

The conditions in Theorem 2 ensure thatfor every nonzero codeword, and hence, the resulting codeachieves full diversity in the block fading scenario.

Corollary 5 offers a simple construction for full diversityspace–time codes in MIMO block-fading channels with ar-bitrary numbers of transmit antennas and fading blocks percodeword. Starting with a full diversity rate space–timecode for quasi-static channels, one canrepeatthis code in everyfading block to obtain a rate code with diversityadvantage. More generally, one can multiplex any “” fulldiversity quasi-static codes in the different fading blocks toobtain a full diversity block-fading code. For example, thebinary convolutional codes in [14, Table I] offer a rich class ofquasi-static codes that can be tailored for this scenario.


TABLE IMAXIMUM ACHIEVABLE DIVERSITY ADVANTAGE FOR FULL RATE

TRANSMISSION (THAT IS, 1-BIT/TRANSMISSION INTERVAL)BPSK SPACE–TIME CODES

It is clear that the transmission rate of full diversity codes con-structed according to Corollary 5 is bits per transmission in-terval. Comparing this rate to full diversity codes in quasi-staticfading channels that achieve 1-bit/transmission interval, one ob-serves the significant loss in throughput. This observation is for-malized in the following result that establishes a fundamentallimit on the tradeoff between transmission rate and diversityadvantage in MIMO block-fading channels, independent of theused code.

Lemma 6: The maximum transmission rate for BPSK modu-lation in a communication system with transmit antenna, op-erating over a block-fading channel with blocks, and usinga space–time code that achieveslevels of transmit diversity is

bits per transmission interval.Proof: Follows directly from the Singleton bound.

The previous result argues that the loss in rate observed inthe multi-stacking construction is inherent to any full diversitycode in this scenario. In Table I, we report the maximumdiversity advantage for BPSK-modulated codes supporting1-bit/transmission interval (i.e., the same rate as full diversityquasi-static codes). This table shows that carefully constructedcodes can achieve a significant increase in the diversity advan-tage in MIMO block-fading channels without penalizing thetransmission rate. At this point, we note that the limit on thetransmission rate in Lemma 6 can be relaxed by using rotatedconstellations instead of the standard PSK constellations [16].This line of research, however, will not be pursued further inthis paper.

In the following, we investigate the design of space–timecodes that realize the optimum tradeoff between the transmis-sion rate and diversity advantage in Corollary 6 for arbitrarynumbers of transmit antennas, fading blocks per codeword, andtransmission rates. Our development is divided into two parts.First, we present a generalization of the stacking construction inquasi-static fading channels that allows for increasing the trans-mission rate at the expense of a minimal reduction in the di-versity advantage. In addition to being an important result inits own right, this generalization introduces the technical ma-chinery necessary for the second part where space–time codesare constructed to optimally exploit the diversity available inMIMO block-fading channels.

In the first step, we wish to generalize Theorem 2 to handlecodes that achieve levels of spatial transmit diversity.These codes are capable of supporting higher transmission ratesthan full diversity codes since, according to Lemma 6, a-diver-sity BPSK code can support bits per transmission in-terval in quasi-static fading channels. This result is particularly

important for systems with large numbers of transmit antennaswhere it may be advantageous to increase the transmission ratebeyond that possible with full diversity codes while maintaininga reasonable diversity advantage.

Before proceeding further, some definitions are needed tosimplify the presentation. The BPSK space–time codeis nowdefined as in the stacking construction (i.e., Theorem 2) withthe minor modification that can be smaller thanto allow fortransmission rates higher than 1-bit/transmission interval. Let

be the general linear group over the binary field [17].2

The number of distinct matrices, ignoring permutations,3 in thisgroup is [17]

(8)

For every matrix and every -tuple informationvector , we define

......

(9)

where are the binary matrices used to generatethe space–time code

(10)

and is the th element of . Now, we have the fol-lowing result that generalizes the stacking construction to allowfor higher transmission rates in quasi-static channels.

Theorem 7 (Generalized Stacking Construction): Let bea linear space–time code as defined in Theorem 2 with theexception that is allowed. Then, for BPSK transmissionover the quasi-static fading channel,achieves at least levelsof transmit diversity if is the largest integer such that

has full rank over the binary field .Proof: Applying the BPSK-BF binary design criterion

to the quasi-static scenario, one immediately sees thatwillachieve levels of diversity if the binary rank for every nonzerocodeword is larger than or equal to. Now, a codeword matrix

will have a binary rank equal to, if and only if all matricesresulting from applying any number of simple row operationsto have at least nonzero rows—i.e., the number of zero rowsis less than . Noting that is the set ofmatrices resulting from applying all possible combinations ofsimple row operations to the codeword matrix corresponding

2GL ( ) is the set ofL � L full-rank binary matrices over the binaryfield.

3All the matrices that can be obtained from one matrix through row permu-tations are counted only once. The reason is that all these matrices result in thesame test in our approach.


to the input stream , one can easily see thatwill achievelevels of diversity if

(11)

Using the fact that

we can see that the condition in (11) will be satisfied for everynonzero input stream if and only if

has full rank over the binary field for every .

It is clear that for , the condition in Theorem 7 re-duces to that in the stacking construction (i.e., Theorem 2). Thegeneralized stacking construction allows for significant gains inthroughput in certain scenarios. For example, withtransmitantennas, the rate can be increased from 1-bit/transmission in-terval to 2-bit/transmission interval at the expense of a reductionin diversity advantage from to .

The next step is to extend the generalized stacking construc-tion to MIMO block-fading channels. First, some more nota-tion is needed to handle this scenario: for every fading block

and input information sequence , wedefine and

......

(12)where

(13)

is the th element of , and ,are empty. Using this notation, we are ready to

generalize the quasi-static generalized stacking constructionto block-fading channels with arbitrary numbers of transmitantennas and fading blocks per codeword.

Theorem 8 (Block Fading Generalized Stacking Construc-tion): Let be a linear space–time code as defined in(7). Then, for BPSK transmission in a MIMO block channelwith transmit antennas and fading blocks per codeword,

achieves at leastlevels of transmit diversity if is the largestinteger such that

and

has a full rank over the binary field .

Proof: From the BPSK-BF binary design criterion, weknow that the diversity advantage of the space–time codeisgiven by

(14)

Let , then

(15)

where is the number of zero rows in the matrix . It isstraightforward to see that

(16)

if and only if is the smallest integer such that anyrows chosen from have at least one nonzerorow. This condition is satisfied if is the smallest integersuch that has full rank over thebinary field for any -tuple with

and . This means that if is thesmallest integer such that has fullrank over the binary field for any -tuplewith , , and

then

(17)

and, hence,

(18)

which was to be proven.

One notes that this proposition only establishes a lower boundon the diversity advantage due to the fact that the BPSK-BFbinary design criterion provides only sufficient conditions. Inmost cases considered in this paper, however, this lower boundwas found to correspond to thetrue diversity advantage as val-idated by simulation results. It is straightforward to see that themultistacking construction can be obtained by setting the diver-sity advantage in Theorem 8.

Theorem 8 enables the construction of space–time convolu-tional and block codes that optimally exploit the diversity avail-able in MIMO block-fading channels with arbitrary numbers oftransmit antennas and fading blocks per codeword. In the fol-lowing, we consider in some detail the design of space–timeconvolutional codes. The main advantage of such codes is theavailability of computationally efficient maximum-likelihooddecoders.

Following the approach in [18], let be a binary convolu-tional code of rate . The encoder processesbinaryinput sequences and producescoded output sequences which aremultiplexed together to form the output codeword. A sequence

is often represented by the formal series


We refer to as a -transform pair. The actionof the binary convolutional encoder is linear and is characterizedby the so-called impulse responses associ-ating output with input . Thus, the encoder action issummarized by the matrix equation

where

and

......

......

(19)

We consider the natural space–time formatting ofin whichthe output sequence corresponding to is as-signed to theth transmit antenna in the th fading block, andwish to characterize the diversity that can be achieved by thisscheme. Similar to Theorem 8, the algebraic analysis techniqueconsiders the rank of matrices formed by concatenating linearcombinations of the column vectors

...(20)

Following in the footsteps of Theorem 8, we define, ,

(21)

where is empty. Then, the following result char-acterizes the diversity advantage achieved by BPSK space–timeconvolutional codes in block-fading channels.

Proposition 9: In a communication system with transmitantennas operating over a block-fading channel withinde-pendent blocks per codeword, letdenote the space–time codeconsisting of the binary convolutional code, whosetransfer function matrix is ,and the spatial parserin which the output

is assigned to theth antenna in the thfading block. Then, for BPSK transmission,achieves at least

levels of transmit diversity if is the largest integer such that

and

TABLE IIRATE 1=2 UNIVERSAL BINARY SPACE–TIME

CODES FORTWO TRANSMIT ANTENNAS

TABLE IIIRATE 1=3 UNIVERSAL BINARY SPACE–TIME

CODES FORTHREE TRANSMIT ANTENNAS

has a rank over the space of all formal series.Proof: Straightforward by specializing the technique used

in the proof of Theorem 8 to the convolutional codes scenarioas in [18].

We note that, similar to [18], this framework encompasses asa special case rate convolutional codes with bit or symbolinterleaving across the transmit antennas and fading blocks.

Theorem 8 and Proposition 9 allow for generalizing thesmartand greedydesign rule to the present scenario. Since in manypractical applications the number of independent fading blocksper codeword is not knowna priori at the transmitter, it is desir-able to construct codes that exploit the temporal diversitywhen-everavailable. This leads to the notion ofuniversalspace–timecodes that achieve the maximum diversity advantage, allowedby their transmission rate according to Lemma 6, with differentnumbers of fading blocks per codeword (i.e., different). Morespecifically, we have the following definition.

Definition 10: A rate binary space–time codeis said bea universal BPSK space–time code for a ) MIMOblock-fading channel if it achieves the maximum transmitdiversity advantage, allowed by the transmission rate anddecoder complexity, with transmit antennas and arbitrary

fading blocks per codeword.

In this definition, it is assumed that the output symbols fromthe baseline code will be periodically distributed across thetransmit antennas and fading blocks for every. Rateuniversalspace–time codes are particularly important becausethey achievefull transmit diversity in quasi-static fading chan-nels. In Tables II and III, we present the best rate uni-versal codes found through an exhaustive search for two andthree transmit antennas with different numbers of fading blocks


per codeword.4 These codes also maximize the free distancewithin the codes with the same diversity advantage profile. Inour search, we assumed bit multiplexing across the differenttransmit antennas and fading blocks.

B. Suboptimality of the Concatenated Coding Approach

Several works have considered the design of concatenatedcoding schemes for time varying MIMO channels—e.g., [19].The basic idea is to construct an outer code to exploit thetem-poral diversity along with an inner code that exploits thespatialdiversity. In this approach, the outer code is composed of a base-line code and a multiplexer that distributes the coded symbolsacross the different fading blocks. The input assigned to eachfading block isindependentlycoded with an innerspatialcode.5

The following result establishes an upper bound on the di-versity advantage that demonstrates the suboptimality of thiscoding scheme. The bound also offers some guidelines on theoptimal rate allocation for the inner and outer codes.

Proposition 11: Let be a linear concatenated space–timecode using a rate outer code that achieves levels oftemporal transmit diversity and rate inner code that achieve

levels of spatial transmit diversity used in a communicationsystem with transmit antennas and independent blocksper codeword. Then, achieves at least levels oftransmit diversity for BPSK transmission with maximum-like-lihood decoding. Furthermore, for a fixed transmission rate

bits per transmission interval, we have

where equality can be achieved if and only if the rates are allo-

cated as , .Proof: Let denote a codeword in the

outer code . In the concatenated coding approach,the codeword is partitioned into substreams (i.e.,

), where each substream is inde-pendently encoded by the inner spatial code. Letdenote the binary weight of , and

if

otherwise (22)

Using the linearity of the code , it is easy to see that

(23)

From the BPSK-BF rank criterion, we know that the diversityadvantage is lower-bounded by

(24)

4Bold letters are used to identify the cases where the maximum diversity ad-vantage is achieved.

5In most proposed approaches, this inner code is derived from the orthogonaldesigns.

The fact that the inner code is designed to achievelevels ofspatial diversity means that

if

if (25)

Combining (23), (24), and (25), we obtain the first part of theproposition

(26)

From Lemma 6, we know that

(27)

(28)

and hence,

(29)

Subject to the constraint of a transmission rate equal tobits pertransmission interval, the right-hand side of (29) is maximized

by assigning , (this result can be obtainedusing the standard Lagrange multiplier optimization approach).This allocation results in

(30)

Component convolutional codes for this approach can be de-signed using Proposition 9 after setting , , re-spectively. Compared with the achievable diversity advantagesuggested by Lemma 6, Proposition 11 implies that the con-catenated coding approach is usually suboptimal. For example,assuming that , 1-bit/transmission interval,the achievable diversity advantage is . The possibletransmit diversity for the concatenated coding approach is, how-ever, .6 Furthermore, the optimal rate allocation strategysuggested by Theorem 11 argues that using afull spatial diver-sity inner code is not always the optimal solution since full spa-tial diversity is only possible with rate inner codes whichmay not agree with the optimal rate-allocation strategy. In theprevious example, using full spatial diversity inner code will re-sult in . The inner code uses all the available redundancy,and hence, the temporal diversity is not exploited.

C. QPSK Modulation

As a consequence of the QPSK-BF binary design criterion,one sees that the binary constructions proposed for BPSK modu-lation in Section IV-A can also be used to construct QPSK mod-ulated codes. For example, the binary connection polynomialsof Tables II and III can be used to generate linear,-valued,trellis QPSK codes that achieve the same diversity advantage asthebaselinebinary code [14]. More generally, one may use anyset of -valued connection polynomials whose modulopro-jections [14] appear in the tables. Ideally, it is desirable to findthe code that offers the best coding gain (i.e., product distance)

6Again, there is some looseness in this argument sinced may be lower thanthe true diversity advantage.


Fig. 1. Performance of 16-state BPSK space–time code withL = 2,L = 1, and different numbers of fading blocks per codeword.

within this class of codes. However, in the absence of a system-atic approach for optimizing the coding gain, one has to restoreto empirical searching techniques. Experimentally, we foundthat codes obtained by replacing each zero in the binary gen-erator matrix by a two yield the best simulated frame error rateperformance in most cases. Another way for designing codesfor QPSK modulation is to combine two binary codes, withthe same diversity advantage in a dyadic format .The resulting code achieves the same diversity advantage asthe binary codes for QPSK transmission [14]. It is worth notingthat theselifting techniques are general for arbitrary numbers oftransmit antennas and fading blocks per codeword.

V. NUMERICAL RESULTS

Fig. 1 highlights the performance gains possible withincreasing the number of fading blocks per codeword. In thefigure, we report the performance of the universal 16-stateBPSK code in a system with and . It is clearfrom the figure that the diversity advantage increases as thenumber of fading blocks increases, and hence, the performancecurve becomes steeper. It is worth noting that the gain inperformance in this scenario does not entail any additionalreceiver complexity. Fig. 2 highlights the importance of carefuldesign to optimize the diversity advantage. In the figure, wecompare the four-state optimal free distance space–timecode with the four-state code in a BPSK system with

, , and . Using Proposition 9, onecan see that the code achieves forrespectively. The code, on the other hand, achievesin both cases (note that for , is the maximumpossible diversity advantage for this throughput). As shown inthe figure, for the case, the superior diversity advantageof the code is apparent in the steeper slope of the frameerror rate curve. This results in a gain of about 1 dB atframe error rate. For the case, however, it is clear thatthe code exhibits a superior coding gain compared withthe code.

Fig. 3 compares the performance of Tarokh, Seshadri, andCalderbank (TSC) 16-state code [12], and a new linearcodeobtained by “lifting” the binary code7 with QPSK mod-ulation in the same scenario (i.e., ). By examiningthe error events of the TSC code, one can easily see that this codeonly achieves , whereas the new code achieves asargued in Section IV-C. It is clear from the figure that the supe-rior diversity advantage of the new code allows for significantgains especially at high signal-to-noise ratios. Fig. 4 comparesthe performance of the recently proposed turbo space–time code[5] with the new 64-state linear code with generator polyno-mial 8 in a systemwith and . One observes the significantperformance gain and reduction in receiver complexity offered

7The generator polynomial of the new code is(1+D+D ; 1+D+2D ).8This code is obtained by lifting the 16-state universal code in Table I.


Fig. 2. Performance comparison of two 4-state BPSK space–time codes withL = 2,L = 1, and different numbers of fading blocks per codeword.

Fig. 3. Performance comparison of the new 16-state QPSK space–time code and the TSC space–time code withL =M = 2, andL = 1.


Fig. 4. Performance comparison of the new 64-state QPSK space–time code and the new Turbo space–time code withL =M = 2, andL = 1.

Fig. 5. Performance comparison of the algebraic QPSK space–time code and the concatenated space–time code in a system withL = 2,L = 1, andM = 4.


by the new code compared to the turbo code in this scenario withlimited degrees of diversity.9

In Fig. 5, we compare the proposed algebraic QPSK space–time codes with the concatenated coding approach in a sys-tems with , , and . The QPSK codeis obtained by lifting the eight-state binary code in Table II tothe ring. Alamouti orthogonal design is used as an innercode in the concatenated coding approach. The outer code is theeight-state binary code proposed for the block-fading channelin [20]. To achieve a 2-bits/channel use, similar to the algebraicQPSK code, we used 16-QAM modulation in the concatenatedcoding approach. As shown in the figure, the algebraic QPSKcodematched to the MIMO-BF channeloffers a gain of 1.5 dB.It is also worth noting that further gain may be achieved if the16-QAM modulation along with a lower ratealgebraically con-structedcode are used. This, however, requires extending ourresults to higher order constellations which is beyond the scopeof this paper.

VI. CONCLUSION

In this paper, we considered the design of space–time codesfor MIMO block-fading channels. We derived the baseband de-sign criteria that determine the diversity and coding advantageachieved by space–time codes in MIMO block-fading channels.For BPSK and QPSK modulated codes, we presented binary de-sign criteria that allow for designing space–time codes that ex-hibit the optimum diversity-versus-throughput tradeoff. Thesebinary criteria were then utilized to develop an algebraic frame-work for space–time code design in MIMO block-fading chan-nels. Several design examples for trellis codes suitable for BPSKand QPSK transmissions were also presented together with sim-ulation results that demonstrate the efficacy of the proposedframework.

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