Space-Time Block Codes versus Space-Time Trellis Codes

Space-Time Block Codes versus Space-TimeTrellis CodesSumeet Sandhu,Robert W.Heath Jr.,Arogyaswami PaulrajThe authors are with Information Systems Laboratory,Stanford University,Stanford,CA94305.R.Heath was supported by Ericsson Inc.through the Networking Research Center at Stanford University.AbstractTransmit diversity schemes for the coherent multiple-antennaflat-fading channel range from space-time block codes(STBC)to space-time trellis codes(STTC).In this letter we compare the performance of STBC and STTC by means of frame error rate.We discover that a simple concatenation of STBC with traditional AWGN(additive white Gaussian noise)trellis codes outperforms some of the best known STTCs at SNRs(signal to noise ratios)of interest.Our result holds for small numbers of receive antennas and trellis states,and may extend to greater numbers of antennas and states with improved block codes.KeywordsBlock codes,Diversity methods,MIMO systems,Modulation coding,Fading channels.I.IntroductionThe challenge of transmit diversity design for the multiple-antenna fading channel has been met with several novel modulation and error correction techniques in the recent past.Prominent among these space-time block codes([1],[2])and space-time trellis codes([3],[4],[5],[6]). Space-time block codes operate on a block of input symbols producing a matrix output over antennas and time.Unlike traditional single-antenna AWGN block codes,full rate space-time block codes do not provide coding gain.Their key feature is the provision of full diversity with extremely low encoder/decoder complexity.The best known orthogonal block codes are provided in[2].Space-time trellis codes operate on one input symbol at a time producing a sequence of spatial vector outputs.Like traditional TCM(trellis coded modulation)for the single-antenna channel, space-time trellis codes provide coding gain.Since they also provide full diversity gain,their key advantage over space-time block codes is the provision of coding gain.However,they are extremely difficult to design and require an expensive encoder and decoder.This paper proposes a compromise between STBC(known designs,simple ML(maximum likelihood)decoding,no coding gain)and STTC(difficult to design,expensive ML decoding, coding gain).We concatenate AWGN trellis codes with STBC in order to obtain coding gain. Then we compare the performance of concatenated STBC with STTC by keeping transmit power and spectral efficiencyfixed.The results are somewhat surprising-with the same number of trellis states(4and8),concatenated STBC outperforms STTC for1and2receive antennas. Concatenation of TCM with STBC was also proposed in[7]but its performance was not com-pared with STTC.A performance comparison of STTC and concatenated STBC was providedin[8],but it was not fair in terms of date rate and number of trellis states.A fair comparison of STTC and concatenated STBC was provided in[9]for the fast fading channel but not for the quasi-static channel.II.Data and Channel ModelConsider a system with M r receive antennas and M t>1transmit antennas.The channel is flat-fading and quasi-static.It is unknown at the transmitter but is known at the receiver.At time n T,the channel output corresponding to the n th input block spanning T symbol times isY n T=HX n T+V n T(1)where the received signal Y n T is M r×T,the fading channel H is M r×M t,the encoded codeword X n T is M t×T,and receiver noise V n T is M r×T.The entries of H are i.i.d.(independent,identically distributed)circular complex Gaussian random variables with variance0.5in each dimension, i.e.h i,j∼ℵc(0,1).The entries of V n T are i.i.d.,v n T i,j∼ℵc(0,N0),and independent over n. The average power transmitted on M t antennas is E s.The codeword X n T is encoded using concatenated STBC or STTC,both of which are described in detail in the following sections.III.Space-Time Block CodesThe input to the encoder is a stream of real or complex modulated symbols.The encoder operates on a block of K symbols producing an M t×T codeword X n T whose rows correspond to transmit antennas and columns correspond to symbol times.At the receiver,ML decoding is simplified by the orthogonal structure of the codewords.The effective channel induced by space-time block coding of input symbols(before ML de-tection)is an AWGN channel with receive SNR equal to||H||2F E sIV.Space-Time Trellis CodesSpace-time trellis codes encode the input scalar symbol stream into an output vector symbol stream.Unlike space-time block codes,space-time trellis codes map one input symbol at a time to an M t×1vector output.Decoding is performed via ML sequence estimation.Code performance is quantified by the diversity advantage and the coding advantage[4].For a given number of transmit antennas,the design objective is to construct the largest possible codebook with full diversity advantage(=M t)and the maximum possible coding advantage.A number of hand-crafted codes with full diversity advantage were provided in[4].Codes with greater coding advantage than those in[4]were reported in[5]and[14]after exhaustive computer searches over a feedforward convolutional coding(FFC)generator.A structured method of code construction that ensures full diversity was provided in[15]along with new designs.Recently new codes with better distance spectrum properties and performance were reported in[6].V.Performance AnalysisNote that any space-time code can be analyzed in terms of the criteria presented for space-time trellis codes,namely diversity advantage and coding advantage.These criteria affect the performance curve in different ways.Diversity advantage affects the asymptotic slope of the FER(frame error rate)versus SNR graph-greater the diversity,the more negative the slope. Coding advantage affects the horizontal shift of the graph-greater the coding advantage,the greater the shift to the left.Since all codes considered here are full diversity,the asymptotic slopes of their FER graphs will be the same.The difference in coding advantage can be quantified as follows.At high SNRs the logarithm of the minimum PEP for the k th code is P k≈−M r M t s k−M r M t c k where M r M t is the diversity advantage,s k=log(E sA.Coding AdvantageWe will only consider full diversity codes for2transmit antennas at a bit rate of2bits per symbol time.We will focus on a small number of trellis states,equal to4and8,both for STTCs and concatenated STBCs.In order to predict the performance of different codes,we computed their coding advantages by means of a computer search and listed them in Table I.These values agree with those found in published literature[16],[5],[6].Note that the values in Table I are normalized by M t=2.The block code labeled STBC stands for the complex M t=2Alamouti code[1]and has a 4-PSK constellation.The codes labeled STBC-TCM represent the Alamouti code concatenated with outer AWGN trellis codes and have8-PSK constellations.The outer codes are both rate2/3 and are the best4-state(parallel transition)and8-state trellis codes in[16].The codes labeled STTC denote space-time trellis codes,of which the4and8state codes in[4](STTC-Tarokh), [5](STTC-Grimm),and[6](STTC-Yan)are shown,all of which have4-PSK constellations.The 4-state Grimm and Yan codes have the best possible coding advantage in the class of FFC codes [5].As demonstrated in[6],although the4and8state Yan codes have the same coding advantage as the Grimm codes,they outperform the Grimm codes because of better distance spectrum properties.Since the Grimm codes were shown to outperform the Tarokh codes[14],[6],we will only consider the Grimm and Yan codes in the sequel.VI.SimulationsIn this section Monte Carlo simulations of the codes in Table I are presented.Performance is measured in terms of FER for a burst of130symbols.The Rayleigh channel described in(1)is used with two transmit antennas.Channel realizations are i.i.d.from burst to burst. Performance is plotted for one,two and three receive antennas in Figures1,2and3respectively. The performance of STBC is plotted in all thefigures for reference.In Figure1(a)note that STBC by itself performs as well or better than all the STTCs,even though it provides no coding gain.This is explained by the multidimensional structure of STBC that spans two time symbols.With one receive antenna,concatenated STBC performs dramatically better than all the4-state STTCs.The performance gap reduces with increasing receive antennas in Figures2and3.The performance gap is also less noticeable with8-state codes for all receive antennas.With three receive antennas and8trellis states,in fact,STTC-Yan outperforms STBC-TCM. The loss in performance of STBC-TCM with increasing receive antennas is explained by the capacity loss incurred by STBC[11].Considering the capacity-optimized block codes in[12],we expect that those block codes will outperform STTC-Yan for even greater numbers of receive antennas and trellis states.Note that in all these comparisons,the encoding/decoding complexity of STTCs is much higher than that of concatenated STBCs for large numbers of trellis states and multiple receive antennas.A comparison that is fair with respect to complexity would entail concatenation of more sophisticated codes with improved block codes,potentially extending their performance advantage to multiple receive antennas.VII.ConclusionsWe have shown that for a small number of receive antennas and trellis states,a simple concate-nation of space-time block codes with traditional AWGN trellis codes can significantly outperform space-time trellis codes.The concatenated scheme separates spatial modulation from temporal error correction and is therefore attractive from the aspect of computational complexity as well.References[1]S.M.Alamouti,“A simple transmitter diversity scheme for wireless communications,”IEEE J.Select.Areasin Comm.,vol.16,pp.1451–1458,October1998.[2]V.Tarokh,H.Jafarkhani,and A.R.Calderbank,“Space-time block codes from orthogonal designs,”IEEErmation Theory,vol.45,pp.1456–1467,July1999.[3]J.C.Guey,M.P.Fitz,M.R.Bell,and W.Y.Kuo,“Signal design for transmitter diversity wireless commu-nication systems over Rayleigh fading channels,”in Proc.VTC,vol.I,pp.136–140,1996.[4]V.Tarokh,N.Seshadri,and A.R.Calderbank,“Space-time codes for high data rate wireless communicationI:performance criterion and code construction,”IEEE rmation Theory,vol.44,pp.744–65,March 1998.[5]J.Grimm,Transmitter diversity code design for achieving full diversity on Rayleigh fading channels.PhDthesis,Purdue University,1998.[6]Q.Yan and R.S.Blum,“Optimum space-time convolutional codes,”in Proc.WCNC,2000.[7]S.M.Alamouti,V.Tarokh,and P.Poon,“Trellis-coded modulation and transmit diversity:design criteriaand performance evaluation,”in Proc.ICUPC,1998.[8]J.-C.Guey,“Concatenated coding for transmit diversity systems,”in Proc.VTC,vol.5,pp.2500–2504,1999.[9] A.Yongacoglu and M.Siala,“Space-time codes for fading channels,”in Proc.VTC,vol.5,pp.2495–2499,1999.[10]G.Ganesan and P.Stoica,“Space-time diversity using orthogonal and amicable orthogonal designs,”inProc.ICASSP,2000.[11]S.Sandhu and A.Paulraj,“Space-time block codes:a capacity perspective.”Accepted IEEE Comm.Letters,June2000.[12] B.Hassibi and B.M.Hochwald,“High-rate codes that are linear in space and time,”in Proc.38th AnnualAllerton Conference on Communication,Control and Computing,2000.[13]S.Sandhu,R.Heath,and A.Paulraj,“Union bound for linear space-time codes,”in Proc.38th AnnualAllerton Conference on Communication,Control and Computing,2000.[14]S.Baro,G.Bauch,and A.Hansmann,“Improved codes for space-time trellis-coded modulation,”IEEEComm.Lett.,vol.4,pp.20–22,January2000.[15] A.R.Hammons and H.E.Gamal,“On the theory of space-time codes for PSK modulation,”IEEE Trans.Information Theory,vol.46,pp.524–542,March2000.[16] E.Biglieri,D.Divsalar,P.J.McLane,and M.K.Simon,Introduction to Trellis-Coded Modulation withApplications.Macmillan,1991.Table I:Coding Advantage:2transmit antennas,2bits/symbol time Figure1:One receive antennaFigure2:Two receive antennasFigure3:Three receive antennasTABLE ICode Coding AdvantageSTBC1STBC-TCM240√ 1.51 STTC-Grimm√ 1.51 STTC-Yan84.5858838 3.01STTC-Yan2IEEE COMMUNICATIONS LETTERS,VOL.XX,NO.Y,MONTH200011November9,2000DRAFT。