Space-time Block Codes From Orthogonal Designs .
1456IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999Space–Time Block Codes from Orthogonal DesignsVahid Tarokh, Member, IEEE, Hamid Jafarkhani, and A. R. Calderbank, Fellow, IEEEAbstract— We introduce space–time block coding, a new paradigm for communication over Rayleigh fading channels usingmultiple transmit antennas. Data is encoded using a space–timeblock code and the encoded data is split into n streams whichare simultaneously transmitted using n transmit antennas. Thereceived signal at each receive antenna is a linear superpositionof the n transmitted signals perturbed by noise. Maximumlikelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas ratherthan joint detection. This uses the orthogonal structure of thespace–time block code and gives a maximum-likelihood decodingalgorithm which is based only on linear processing at the receiver.Space–time block codes are designed to achieve the maximumdiversity order for a given number of transmit and receiveantennas subject to the constraint of having a simple decodingalgorithm.The classical mathematical framework of orthogonal designsis applied to construct space–time block codes. It is shownthat space–time block codes constructed in this way only existfor few sporadic values of n. Subsequently, a generalization oforthogonal designs is shown to provide space–time block codesfor both real and complex constellations for any number oftransmit antennas. These codes achieve the maximum possibletransmission rate for any number of transmit antennas usingany arbitrary real constellation such as PAM. For an arbitrarycomplex constellation such as PSK and QAM, space–time blockcodes are designed that achieve 1 2 of the maximum possibletransmission rate for any number of transmit antennas. Forthe specific cases of two, three, and four transmit antennas,space–time block codes are designed that achieve, respectively,all, 3 4, and 3 4 of maximum possible transmission rate usingarbitrary complex constellations. The best tradeoff between thedecoding delay and the number of transmit antennas is alsocomputed and it is shown that many of the codes presented hereare optimal in this sense as well.Index Terms— Codes, diversity, multipath channels, multipleantennas, wireless communication.I. INTRODUCTIONSEVERE attenuation in a multipath wireless environmentmakes it extremely difficult for the receiver to determinethe transmitted signal unless the receiver is provided withsome form of diversity, i.e., some less-attenuated replica ofthe transmitted signal is provided to the receiver.In some applications, the only practical means of achievingdiversity is deployment of antenna arrays at the transmitter and/or the receiver. However, considering the fact thatManuscript received June 15, 1998; revised February 1, 1999. The materialin this paper was presented in part at the IEEE Information Theory Workshop,Killarney, Ireland, 1998.V. Tarokh and A. R. Caldebank are with AT&T Shannon Laboratories,Florham Park, NJ 07932 USA.H. Jafarkhani is with AT&T Labs–Research, Red Bank, NJ 07701 USA.Communicated by M. Honig, Associate Editor for Communications.Publisher Item Identifier S 0018-9448(99)04360-6.receivers are typically required to be small, it may not bepractical to deploy multiple receive antennas at the remotestation. This motivates us to consider transmit diversity.Transmit diversity has been studied extensively as a methodof combatting impairments in wireless fading channels [2]–[4],[6], [9]–[15]. It is particularly appealing because of its relativesimplicity of implementation and the feasibility of multipleantennas at the base station. Moreover, in terms of economics,the cost of multiple transmit chains at the base can beamortized over numerous users.Space–time trellis coding [10] is a recent proposal thatcombines signal processing at the receiver with coding techniques appropriate to multiple transmit antennas. Specificspace–time trellis codes designed for 2–4 transmit antennasperform extremely well in slow-fading environments (typicalof indoor transmission) and come close to the outage capacitycomputed by Telatar [12] and independently by Foschini andGans [4]. However, when the number of transmit antennasis fixed, the decoding complexity of space–time trellis codes(measured by the number of trellis states in the decoder)increases exponentially with transmission rate.In addressing the issue of decoding complexity, Alamouti[1] recently discovered a remarkable scheme for transmissionusing two transmit antennas. This scheme is much less complex than space–time trellis coding for two transmit antennasbut there is a loss in performance compared to space–timetrellis codes. Despite this performance penalty, Alamouti’sscheme [1] is still appealing in terms of simplicity andperformance and it motivates a search for similar schemesusing more than two transmit antennas. It is a starting pointfor the studies in this paper, where we apply the theory oforthogonal designs to create analogs of Alamouti’s scheme,namely, space–time block codes, for more than two transmitantennas.The theory of orthogonal designs is an arcane branchof mathematics which was studied by several great numbertheorists including Radon and Hurwitz. The encyclopedic workof Geramita and Seberry [5] is an excellent reference. Aclassical result in this area is due to Radon who determined theset of dimensions for which an orthogonal design exists [8].Radon’s results are only concerned with real square orthogonaldesigns. In this work, we extend the results of Radon to bothnonsquare and complex orthogonal designs and introduce atheory of generalized orthogonal designs. Using this theory, weconstruct space–time block codes for any number of transmitantennas. Since we approach the theory of orthogonal designsfrom a communications perspective, we also study designswhich correspond to combined coding and linear processingat the transmitter.0018–9448/99 10.00 1999 IEEE
TAROKH et al.: SPACE–TIME BLOCK CODES FROM ORTHOGONAL DESIGNS1457The outline of the paper is as follows. In Section II, we describe a mathematical model for multiple-antenna transmissionover a wireless channel. We review the diversity criterion forcode design in this model as established in [10]. In Section III,we review orthogonal designs and describe their application towireless communication systems employing multiple transmitantennas. It will be proved that the scheme provides maximumpossible spatial diversity order and allows a remarkably simpledecoding strategy based only on linear processing. In SectionIV, we generalize the concept of the orthogonal designs anddevelop a theory of generalized orthogonal designs. Using thismathematical theory, we construct coding schemes for anyarbitrary number of transmit antennas. These schemes achievethe full diversity order that can be provided by the transmit andreceive antennas. Moreover, they have very simple maximumlikelihood decoding algorithms based only on linear processingat the receiver. They provide the maximum possible transmission rate using totally real constellations as established in thetheory of space–time coding [10]. In Section V, we definecomplex orthogonal designs and study their properties. We willrecover the scheme proposed by Alamouti [1] as a special case,though it will be proved that generalization to more than twotransmit antennas is not possible. We then develop a theoryof complex generalized orthogonal designs. These designsexist for any number of transmit antennas and again haveremarkably simple maximum-likelihood decoding algorithmsbased only on linear processing at the receiver. They provideof the maximum possible ratefull spatial diversity and(as established previously in the theory of space–time coding)using complex constellations. For complex constellations andfor the specific cases of two, three, and four transmit antennas,these diversity schemes are improved to provide, respectively, andof maximum possible transmission rate.all,Section VI presents our conclusions and final remarks.For the reader who is interested only in the code construction but is not concerned with the details, we provide asummary of the material at the beginning of each subsection.whereare independent samples of a zero-mean complexSNR per comGaussian random variable with varianceplex dimension. The average energy of the symbols transmitted.from each antenna is normalized to beAssuming perfect channel state information is available, thereceiver computes the decision metricII. THE CHANNEL MODEL AND THE DIVERSITY CRITERIONIn this section, we model a multiple-antenna wireless communication system under the assumption that fading is quasistatic and flat. We review the diversity criterion for code designassuming this model. This diversity criterion is crucial for ourstudies of space–time block codes.We consider a wireless communication system where thebase station is equipped with and the remote is equippedwith antennas. At each time slot , signalsare transmitted simultaneously from the transmit antennas.is the path gain from transmit antenna toThe coefficientreceive antenna . The path gains are modeled as samples ofindependent complex Gaussian random variables with varianceper real dimension. The wireless channel is assumed to bequasi-static so that the path gains are constant over a frame oflength and vary from one frame to another.received at antenna is given byAt time the signal(1)(2)over all codewordsand decides in favor of the codeword that minimizes this sum.Given perfect channel state information at the receiver,we may approximate the probability that the receiver decideserroneously in favor of a signalassuming thatwas transmitted. (For details see [6], [10].) This analysis leadsto the following diversity criterion. Diversity Criterion For Rayleigh Space–Time Code: In, the matrixorder to achieve the maximum diversity.has to be full rank for any pair of distinct codewordsand . Ifhas minimum rank over the set of pairsis achieved.of distinct codewords, then a diversity ofSubsequent analysis and simulations have shown that codesdesigned using the above criterion continue to perform wellin Rician environments in the absence of perfect channel stateinformation and under a variety of mobility conditions andenvironmental effects [11].III. ORTHOGONAL DESIGNS ASCODES FOR WIRELESS CHANNELSIn this section, we consider the application of real orthogonal designs (Section III-A) to coding for multiple-antennawireless communication systems. Unfortunately, these designsonly exist in a small number of dimensions. Encoding usingorthogonal designs is shown to be trivial in Section III-B.Maximum-likelihood decoding is shown to be achieved bydecoupling of the signals transmitted from different antennasand is proved to be based only on linear processing at thereceiver (Section III-C). The possibility of linear processingat the transmitter, leads to the concept of linear processingorthogonal designs developed in Section III-D. We then provea normalization result (Theorem 3.4.1) which allows us to
1458IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999focus on a specific class of linear processing orthogonaldesigns. To study the set of dimensions for which linearprocessing orthogonal designs exist, we need a brief review ofthe Hurwitz–Radon theory which is provided in Section III-E.Using this theory, we prove that allowing linear processingat the transmitter only increases the hardware complexity atthe transmitter and does not expand the set of dimensions forwhich a real orthogonal design exists.A reader who is only interested in code construction andapplications of space–time block codes may choose to focusattention on Sections III-A, III-B, and III-C as well as Theorem 3.5.1, Definition 3.5.2, and Lemma 3.5.1.A. Real Orthogonal DesignsorthogonalA real orthogonal design of size is an.matrix with entries the indeterminatesThe existence problem for orthogonal designs is known asthe Hurwitz–Radon problem in the mathematics literature [5],and was completely settled by Radon in another context at thebeginning of this century. In fact, an orthogonal design existsor .if and only ifGiven an orthogonal design , one can negate certaincolumns of to arrive at another orthogonal design where allthe entries of the first row have positive signs. By permutingisthe columns, we can make sure that the first row of. Thus we may assume without loss of generalityhas this property.thatdesignExamples of orthogonal designs are theorder of. Corollary 3.3.1 of [10] implies that the maximumtransmission rate is bits per second per hertz (bits/s/Hz).orthogonalWe provide this transmission rate using anbits arrive at the encoder anddesign. At time slot 1,. Settingforselect constellation signals, we arrive at a matrixwith. At each time slotentriesthe entriesare transmitted simultaneously.from transmit antennasClearly, the rate of transmission is bits/s/Hz. We nowdemonstrate that the diversity order of such a space–time block.code isTheorem 3.2.1: The diversity order of the above coding.scheme isProof: The rank criterion requires that the matrixbe nonsingular for any two. Clearly,distinct code sequenceswhereis the matrix constructed fromby replacingwithfor all. Theis easily seen to bedeterminant of the orthogonal matrixwhereis the transpose of. Hence(3)thedesignwhich(4)orderisnonzero. It follows thatis nonsingular and the maximum diversityis achieved.C. The Decoding Algorithmand thedesignNext, we consider the decoding algorithm. Clearly, theare all permutations of the first row ofwithrows ofdenote the permutationspossibly different signs. Letdenote the sign ofcorresponding to these rows and letin the th row of . Thenmeans thatis upth element of . Since the columnsto a sign change theof are pairwise-orthogonal, it turns out that minimizing themetric of (2) amounts to minimizing(6)(5)The matrices (3) and (4) can be identified, respectively,and the quaternionic numberwith complex number.whereB. The Coding SchemeIn this section, we apply orthogonal designs to constructspace–time block codes that achieve diversity. We assume thattransmission at the baseband employs a real signal constellawithelements. We focus on providing a diversitytion(7)
TAROKH et al.: SPACE–TIME BLOCK CODES FROM ORTHOGONAL DESIGNSand wheredenotes the complex conjugate of.only depends on the code symbol , theThe value of, the path coefficients, and thereceived symbolsstructure of the orthogonal design . It follows that minimizing the sum given in (6) amounts to minimizing (7) for all. Thus the maximum-likelihood detection rule is toform the decision variables1459Proof: Letorthogonal design, and letwhere the matricescoefficientsThen it follows thatbe a linear processingare diagonal and full-rank (since theare strictly positive).(9)(10)for alland decide in favor ofconstellation symbols ifamong all theis a full-rank diagonal matrix with positive diagonalanddenote the diagonal matrix having theentries. Let. We define.property thatsatisfy the following properties:Then the matrices(11)(12)(8)This is a very simple decoding strategy that provides diversity.It follows thatorthogonal array having the propertyis a linear processingD. Linear Processing Orthogonal DesignsThere are two attractions in providing transmit diversity viaorthogonal designs. There is no loss in bandwidth, in the sense that orthogonaldesigns provide the maximum possible transmission rateat full diversity. There is an extremely simple maximum-likelihood decoding algorithm which only uses linear combining at thereceiver. The simplicity of the algorithm comes from theorthogonality of the columns of the orthogonal design.The above properties are preserved even if we allow linearprocessing at the transmitter. Therefore, we relax the definition of orthogonal designs to allow linear processing at thetransmitter. Signals transmitted from different antennas willnow be linear combinations of constellation symbols.Definition 3.4.1: A linear processing orthogonal design inis anmatrix such that:variablesare real linear combinations of variables., whereis a diagonal matrix withth ,diagonal element of the formall strictly positivewith the coefficientsnumbers.In view of the above theorem, we may, without any loss ofgenerality, assume that a linear processing orthogonal designsatisfiesE. The Hurwitz–Radon TheoryIn this section, we define a Hurwitz–Radon family ofmatrices. These matrices encode the interactions betweenvariables in an orthogonal design.real matricesHurwitz–Radon familyDefinition 3.5.1: A set ofis called a sizeof matrices ifand The entries ofIt is easy to show that transmission using a linear processingorthogonal design provides full diversity and a simplifieddecoding algorithm as above. The next theorem shows thatwe may, with no loss of generality, constrain the matrix inDefinition 3.4.1 to be a scaled identity matrix.Theorem 3.4.1: A linear processing orthogonal design inexists if and only if there exists avariableslinear processing orthogonal design such thatWe next recall the following theorem of Radon [8]., where is odd andTheorem 3.5.1: Letwithand. Any Hurwitz–Radon family ofmatrices contains strictly less thanmatrices. Furthermore. A Hurwitz–Radon familymatrices exists if and only ifor .containingbe amatrix and letDefinition 3.5.2: Letbe any arbitrary matrix. The tensor productis thematrix given by.(13)
1460IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999Definition 3.5.3: A matrix is called an integer matrix if all.of its entries are in the setThe proof of the next Lemma is directly taken from [5] andwe include it for completeness.there exists a Hurwitz–RadonLemma 3.5.1: For anywhose members are integerfamily of matrices of sizematrices.Proof: The proof is by explicit construction. Letdenote the identity matrix of size . We first notice that ifwith odd, then. Moreover, given a familyHurwitz–Radon integer matricesofof size, the setis a Hurwitz–Radon family ofinteger matrices of size. In light of this observation, it suffices to prove the. To this endlemma for(14)(15)andis an integer Hurwitz–Radon family ofinteger.matrices, we already conWe proceed by induction. Forstructed an integer Hurwitz–Radon family of sizewith entries in the set. Now (17) gives the transitionto . By using (18) and letting,, wefromto . Similarly, with,get the transition fromand,, we get the transition fromtoandto .The next theorem shows that relaxing the definition oforthogonal designs to allow linear processing at the transmitterdoes not expand the set of dimensions for which there existsan orthogonal design of size .Theorem 3.5.2: A linear processing orthogonal design ofexists if and only ifand .sizeProof: Let denote a linear processing orthogonal design. Since the entries of are linear combinations of variableswe can write row ofas, whereis an appropriate real-valuedmatrix and. Orthogonality oftranslates into the following set of matrix equalities:(16)(19)(20)Letis the identity matrix. We now construct a Hurwherewitz–Radon set of matrices from the original design. Letfor. Thenand we have(21)(22)(23)andis a HurThese equations imply thatwitz–Radon family of matrices. By the Hurwitz–Radon Theoandrem (3.5.1), we can conclude thator .ThenIn particular, we have the following special case.We observe thatfamily of sizeis a Hurwitz–Radon integer family of sizeis a Hurwitz–Radon integerandis an integer Hurwitz–Radon family of size.is anThe reader may easily verify that ifmatrices, theninteger Hurwitz–Radon family of(17)integer matricesis an integer Hurwitz–Radon family of.is an integer HurIf, in addition,matrices, thenwitz–Radon family of(18)Corollary 3.5.1: An orthogonal design of sizeand .and only ifProof: Immediate from Theorem 3.5.2.exists ifTo summarize, relaxing the definition of orthogonal designs,by allo
theory of generalized orthogonal designs. Using this theory, we construct space–time block codes for any number of transmit antennas. Since we approach the theory of orthogonal designs from a communications perspective, we also study designs which correspond to combined coding and lin
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