基于EM算法的差分酉空时调制检测方案

Journal of Southeast University(English Edition) Vol 23, No 4, pp. 484-488 Dec. 2007 ISSN 1003--7985EM-based detection schemefor differential unitary space-time modulationDu Zhengfeng Chen Jie Pan Wen Gao XiNational Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China)Abstract: The performance loss of an approximately 3 dB signal-to-noise ratio is always paid with conventionaldifferential detection compared to the related coherent detection, A new detection scheme consisting of two stepsis proposed for the differential unitary space-time modulation DUSTM) system. In the first step, the datasequence is estimated by conventional unitary space-time demodulation( DUSTD)and differentially encodedagain to produce an initial estimate of the transmitted symbol stream. In the second step, the initial estimate ofthe symbol stream is utilized to initialize an expectation maximization( EM)-based iterative detector. In eachiteration, the most recent detected symbol stream is employed to estimate the channel, which is then used toimplement coherent sequence detection to refine the symbol stream. Simulation results show that the proposeddetection scheme performs much better than the conventional dUSTD after several iterationsKey words: unitary space-time modulation; differential detection; expectation maximization(EM) algorithmRecent information theory results show that the transmits a phase-shift keying( PSK) symbol in turncapacity of wireless channels can be substantially in- Like other differential space-time codes, differentialcreased by employing multiple transmit and receive anadulationTM) enables us totenna, especially when the channel state information is decode the received signals without the knowledge ofknown at the receiver.However, in many situa- the Csr 6. 8), but both suffer a loss of an approximatelytions, the channel state information( CSI) may not be 3 dB signal to noise ratio when compared to their reavailable to the receiver. Even if the receiver does not spective ideal coherent receivers when the CSI is per-know the fading coefficients, a substantial increase in fectly known, so it is of great importance to devise newchannel capacity is still possible. So it is of interest detection scheme for the DUSTM to improve systemto develop techniques for modulation and coding that performance. In this paper, we present an expectationdo not require CSI. Tarokh et al. first came up with maximization EM)-based detection scheme for thea differential STBC scheme for a slow Rayleigh fading DUSTM systemchannel with two transmitter antennas. the scheme em-The EM algorithm is a general approach forploys block-by-block detection, in which neither trans-mitter nor receiver knows the CSI. the same authorscomputing maximum likelihood ML) estimates Forgeneralized the differential detection for STBC to moresingle-antenna channels, receivers based on the EM althan two transmit antennas. In Ref [6], a new class gonthm have been shown to perform well under fastof signals called unitary space-time signals was pro-fading and multipath fading conditions. In partiposed, which is well tailored for Rayleigh flat-fadingcular, EM-based detectors for space-time block codeschannels where neither the transmitter nor the receiverhave been derived in Refs. 12-13]. Ref. 14]pro-knows the fading coefficients. Ref [7] presented a sys- posed an iterative receiver for differential STBC usingtematic approach to designing unitary space-time sig. the EM algorithm; however, since the differential STBCnals. In particular, Ref [8] introduced so-called diago. employed was based on the Alamouti transmit diversnal codes that are simple to generate: every antennaschemes), only two transmit antennas were assumeIn this paper, we employ the EM algorithm to deReceived 2007-01-26rive an iterative detection scheme for DUSTM sys-Foundation items: The National Natural Science Foundation of China tems. We consider the detection of a differential unitaryNo. 60572072, 60496311) the National High Technology Research andDevelopment Program of China(863 Program)( No. 2006AA01Z264)space-time constellation proposed in Ref. 8]. Thehe National Basic Research Program of China(973 Program)scheme which we propose employs two steps. In the2007CB310603), the Ph. D. Programs Foundation of Ministry of Educa- first step of our scheme, a data sequence is estimatedtion of China(No. 20060286016)Biographies: Du Zhengfeng (1976--), male, graduate: Gao Xiqi(coby conventional differential unitary space-time demodulation( DUSTD)and is differentially encoded againEM-based detection scheme for differential unitary space-time modulationto produce an initial estimate of the transmitted symbcomesstream obtained from the first step is utilized to initial- where W=m. x=osH+Wstream. In the second step, the estimation of the symbolP]. For simplicity, we assume Tize an EM-based iterative detector. In each iteration, =M in this paper. Suppose that 4, z,, . with z, e 10,the most recent detected symbol stream is used to esti- 1,.,L-1) is a data sequence to be transmitted,eachmate the fading coefficients, which are then utilized to z, corresponding to a constellation matrix from the con-implement coherent detection to renew the detected stellation set (v1, 1=0, 1,., L-1, where L=2RM forsymbol stream. The process of iterations improves the a data rate of R bit per channel use. The transmitteroverall performance of the DUSTM system Simulation sends the symbol stream S,, S,,which is determinedresults show that the detection scheme we propose per. by the following fundamental differential transmissionforms better than the conventional dustdequation1 DUSTM and System ModelS,=vS1t=1,2,where So=ly and IM denotes the M x M identity ma-Consider a communication link comprising M trix. At the receiver, the maximum-likelihood (ML)transmitter antennas and n receiver antennas and operdemodulator for DUSTM is given by)ating in a Rayleigh flat-fading environment. Let sm de-2=ang.min‖x,-vX1-1‖(5)note the transmitted signal by transmitter antenna m1,2,……, M at time slot t=1,2,…, T and satisfy thewhere li is the Frobenius norm which is defined aslA=vtr(A'A)and tr()denotes the trace oper1. The received2 New Detection Scheme for DUSTMsignal xm by the receiver antenna n=1, 2,...,N at timslot t=l, 2,... T is given byDue to the performance loss with the conventionaled to the related coherent detection txn=vp∑ h s+wm(1) devise a new detection scheme for dUSTM in this sec-where hm is the complex-valued fading coefficient be- tion In the first step of the proposed scheme, an initialtween the m-th transmitter antenna and the n-th receiv. estimate of the transmitted data sequence is derived byer antenna. The fading coefficients are assumed to be the conventional DUSTD, and then differentially enco-independent with respect to both m and n, and are cn ded again to generate the estimate of the transmitted(0, 1)-distributed( complex normal zero-mean unit-va- symbol stream, which is to be used to initialize an EM-riance distribution where the real and imaginary com-based iterative detector in the next step In the secondponents of each random variable are independent and step, we employ the EM algorithm for ML estimationto derive an iterative detector for the DUSTM systembe constant over P symbol intervals, where P is the to improve overall performance. Fig. I provides anlength of the received frame. In other words, a quasi- overview of the proposed detection scheme forstatic fading case is considered in this paper. wm is theDUSTMadditive noise at time t and at receiver antenna nwhich is independently identically distributed CN(O,C DUSIM1), with respect to both t and n p is the average signalto-noise ratio(SNR) per receiver antenna. Alternative-lily we can write this in matrix form asestimationDUSTMX=√oSH+Wwhere X, is the TxN symbol matrix of received signalEM-based iterative receiverxm,S, is the TxM symbol matrix of transmitted signalsFig. I Diagram of the proposed detection schemeS,m, and the subscript i denotes the i-th symbol within aframe; H is the M x N matrix of the Rayleigh fading 2.1 Initial estimate of the transmitted symbol streamcoefficients h: and w. is the txN matrix of additiveAs shown in Fig. 1, 20=410, 2noise w. Now let matrices S=(SI,., Sp)and x= tial estimate of the data sequence, which can beX;,.,Xp i' denote the transmitted and received sym- tained from the ML detector for DUSTM, i.ebols of one entire frame, respectively, where"+"de-(5), and is then differentially encoded to generate thnotes complex conjugate transpose, then the system be- initial estimate of the transmitted symbol stream So)486Du Zhengfeng, Chen Jie, Pan Wen, and Gao XiqiS1,... This estimate of the symbol stream is used to whereinitialize the iterative receiver in the next step2.2 Iterative detection via EM algorithmB=Ex,s"=-2、e1Xa detailed description of the EM algorithm can(12)be found in Ref [9]. Here we give a brief overview andof the algorithm. Suppose that we wish to estimate aparameter SE. based on an observation X. The EM=EHHX,S1=-,1+m“(m)algorithm, which is an iterative procedure, considerssome larger sets of data Y. Then we wish to find S to where A and n are respectively the conditionalmaximize log f(Y I S), but we do not have the data y mean and the second moment of the fading coeffito compute the log-likelihood. So instead, we maxi- cients, given the received symbol X,,..., Xp and themize the expectation of log f(y I S)given the obser- most recent estimate of the transmitted symbols S!)vation y and the current estimate of s. the data X andY are referred to incomplete data and complete dataThe M-step of the iteration then performs coherrespectively. Each iteration of the EM algorithm con- ent data detection assuming that the channel estima-sists of two steps: an expectation step(E-step)and a tion at the E-step is correct; i. e, finds S+)thatmaximization step(M-step. Let S)indicate the esti- maximize g(S Se))mate of S after the k-th iteration, k= 1, 2, ... The ES+)= arg maxQ(SS())steD computee(SIS))= logp( IS)IX, S1(6arg max tr Re(ShX)-siS'9where the expectation is with the conditional probabil-ity density function (pdf) p(Y X, S )). The M-stepgmxr(Ra(∑Sx)}(14)then computes the next estimate bywhere S is the collection of all possible transmittedS*+D= arg maxQ(S|§)(7) symbol streams and Re( )is the real operator. In othEach iteration is guaranteed to increase the likeli- er words, the M-step updates the decision on the sym-hood and the algorithm is guaranteed to converge to a bol stream according to the most recent estimate of thelocal maximum of the likelihood function! 16)fading coefficients. From the M-step of Eq. (14), it isFor the problem at hand, let the complete data y known that the metric for optimization to find S"+l)isbe the incomplete data X,,x2,... along with the fadependent of n2, so the E-step can be implementedding coefficients H, and let the unknown parameter to only by computing Eq(12). Since the received signalbe estimated be the transmitted symbol stream S,, S,is known, the fading gain estimate is assumed to bece., where S, and X, are the i-th transmitted and re- correct, and the transmitted symbols are effectivelyeived symbol matrices respectively within one framememoryless, the M-step can be solved on a sym-At the E-step of the iteration, the detector estimates bol-by-symbol basisthe fading coefficients, conditioned on the receivedS+)= arg max tr{Re(Sm“x)}i=1,2,…,Psignals and the most recent estimate of the transmitted(15)symbols. The E-step of the EM algorithm then yields where/is the collection of all possible transmittedQ(S!S)=ELlogp(X, H S)X, S](8) symbols. The EM-based iterative receiver consists ofSince H and S are independent matrices, the joint pdf choosing an initial value, then performing the E-stepp(x, H S)in Eq (8)can be factored asand the M-step successively. The iterations ceaseP(X, H S)=P(X H, S)P(H)(9) when the estimate of the symbol stream does notThe pdf p(X H, s)is given bychange during two subsequent iterations, or after ap(x\r veApt-trI(X-PSH)(X-pSH)'D specified number of iterations. Finally, the estimate ofthe data sequence can be obtained by differentially de-At the same time, it should be pointed out thatthe proposed detection scheme introduces the latencye(S s)=t Re(SA( x')-IstS'and additional processing complexity at the receiverdue to the em-based iterative detection in the secondEM-based detection scheme for differential unitary space-time modulation487algorithm for computing Eq (15)is O(M)when MN. However, compared with the conventional coher-ent detection scheme with training, the proposedscheme avoids the overhead used for channel soundin10during transmitting. Since the bandwidth is a scarceresource and the training overhead may be excessiveespecially for multiple-antenna communication, andwith more and more powerful processing units emer10-41ging, obtaining higher bandwidth efficiency at the costof some computing complexity is sometimes appealIng3 Performance SimulationsWe now apply the proposed detection scheme toConventionalthe DUStM system where the channel has unknownEM.4Rayleigh flat fading coefficients. For simplicity, in thisH knownpaper we use a simple unitary space-time constellationproposed in Ref [8]y,=diagl=0,1,……,L-1(16)where j denotes the imaginary unit and k,, k,,..., kMare optimized integer parameters to achieve the maximum diversity product. The frame length P= 1 024is used in this paper. The symbol error rate( SEr)H knownperformances as a function of the signal-to-noise ratio(SNR)when M=2 or M =3 are shown in Fig. 2. TheSNR/dBinteger q in the notation EM- q refers to the number ofEM iterations. For simplicity, we assume that therate per channel use R=l, which means L=2Mcomparison, we also plot the performance curves ofthe conventional DUSTD and give the results of theML coherent detector when the channel h is known第10Conventional- EM-4perfectly at the receiver as a lower bound10→EM8As shown in Fig. 2, the conventional DUSTD10-5suffers approximately a 3 dB performance loss com-1o6pared with its respective ideal coherent one when H isknown. The proposed detection scheme performsmuch better than the conventional DUStd after a fewiterations. Specifically, at a SER of 10 in Fig. 2(e)when M=3 and N= 3, the performance of the pro-posed detection scheme is roughly I dB better than the旁10Conventionalconventional dustd when the number of EM itera-tions is 4, and an approximately 1.9 dB gain is ob-H knowntained when the number of em iterations is 8. It is also obvious from these figures that the performance ofthe proposed detection scheme has moved a significantstep toward the lower bound given by the ML coherFig. 2 The SER performances as a function of SNR. (a)ent detectorM=2 and N=l: (b)M=2 and N= 2; (c) M=3 and N=l: (d)M=3 and N=2; (e)M=3 and N=3488Du Zhengfeng, Chen Jie, Pan Wen, and Gao Xiqifor transmit diversity [J]. IEEE J Select Areas Commun4 Conclusion2000,18(7):1169-1174In this paper, a new detection scheme based on [5] Jafarkhani H, Tarokh V Multiple transmit antenna differ-the EM algorithm is proposed for DUSTM, when neiential detection from generalized orthogonal designs [J]IEEE Trans Inform Theory, 2001, 47(9): 2626-2631ther the transmitter nor the receiver knows the channel [6] Hochwald B M, Marzetta T L Unitary space-time modulafading coefficients. A data sequence produced by theon for multiple-antenna communication in Rayleigh flat-conventional dUStD is differentially recoded to ob-fading [J]. 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[16] Wu C F On the convergence properties of the EM algo-4] Tarokh V, Jafarkhani H. a differential detection schemerithe. Ann Star,1983,1l(1):95-103基于EM算法的差分酉空时调制检测方案杜正锋陈杰潘文高西奇(东南大学移动通信国家重点实验室,南京210096)摘要:与相干检测相比传统差分检测会带来约3B的性能损失.提出一种新的差分酉空时调制检测方案,该方案分为2步:首先将传统差分检测获得的数据序列进行差分再编码,作为对发送符号序列的初始估计;然后由期望最大化(EM)算法进行迭代检测,利用上一步得到的发送符号序列的初始估计值作为EM算法的初始值.在毎一次迭代时,最新检测到的发送符号序列用来进行信道估计,随后利用估计出的信道实施相干序列检测,进一步提高对发送符号序列检测的准确性.仿真结果表明,经过几次迭代后,提出的检测方案性能大大优于传统的差分酉空时调制检测方案关键词:酉空时调制;差分检测;期望最大化(EM)算法