Space-time block coding (STBC) and a number of derivative techniques have been developed to maximize the diversity gain of a multi-input multi-output (MIMO) channel. This was later generalized for multi-user DS-CDMA systems through space-time spreading (STS), [3] and [4], which assumed flat fading as well as the availability of full channel state information (CSI) at the receiver. This paper focuses on the development of a blind chip-rate method for the equalization of STS over channels with inter-symbol interference (ISI). Simulation results are presented to demonstrate the convergence and noise resilience of the derived algorithm

Bendoukha, Samir

Weiss, S.

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryBendoukha, Samir and Weiss, S. (2008) A CM based equalizer for space-time spreading overchannels with inter-Symbol interference. In: 16th European Signal Processing Conference, 2008-08-25 - 2008-08-29, Lausanne, Switzerland.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/A CM BASED EQUALIZER FOR SPACE-TIME SPREADING OVER CHANNELSWITH INTER-SYMBOL INTERFERENCESamir Bendoukha and Stephan WeissInstitute of Communications and Signal Processing, University of StrathclydeGlasgow, UKphone: + (44) 141 548 2808, email: samir@eee.strath.ac.ukABSTRACTSpace-time block coding (STBC) and a number of deriva-tive techniques have been developed to maximize the diver-sity gain of a multi-input multi-output (MIMO) channel.This was later generalized for multi-user DS-CDMA sys-tems through space-time spreading (STS), [3] and [4],which assumed flat fading as well as the availability of fullchannel state information (CSI) at the receiver. This paperfocuses on the development of a blind chip-rate method forthe equalization of STS over channels with inter-symbol in-terference (ISI). Simulation results are presented to demon-strate the convergence and noise resilience of the derivedalgorithm.Keywords: Space-time spreading, multi-user, constantmodulus algorithm, blind equalization, frequency selectivechannels.1. INTRODUCTIONMulti-input multi-output (MIMO) channels are known to in-crease the capacity of a transmission link. This can be ex-ploited to increase either the multiplexing gain or the diver-sity gain, which leads to a higher data throughput or a betterresilience of the link to fading, respectively.In the pioneering work by Alamouti [2], a transmitter di-versity scheme, named Space-Time Block Coding (STBC),was derived which maximizes the level of diversity obtainedin flat fading channels. Inspired by STBC, a transmit diver-sity scheme referred to as space-time spreading (STS) wasderived in [4] for the downlink of DS-CDMA systems. Un-like most other algorithms, STS only requires one orthogonalcode per user, which makes it an attractive technique. How-ever, in [4] it is assumed that the MIMO channel matrix isnon-dispersive and perfectly known at the receiver. For dis-persive channels, a few schemes have been proposed in theliterature. These are mainly block based and introduce re-dundancy as well as assuming block stationarity of the chan-nel. Knowledge of the channel can be obtained through theinjection of training sequences but the channel is assumedstaionary during the transmission of data. This assumptionis not always valid, thus a blind equalizer appears advanta-geous.The most widely used blind equalizer is the constantmodulus algorithm (CMA), first introduced in [6], whichadapts the weight vectors based on the assumption that thetransmitted constellation has a constant modulus. In a multi-user scenario, this property is not preserved. In [1], a Con-stant Modulus (CM) algorithm was derived for the blindequalization of DS-CDMA systems, named FIRMER-CMA.The algorithm operates at the chip rate and forces the var-ious user signals onto a constant modulus without any ad-ditional requirements such as minimizing the mutual cross-correlation between the multiple users.In this paper, a similar CM algorithm for the blind equal-ization of STS systems over frequency selective channels isderived. The new algorithm forces the despread signals tohave a constant modulus.In the following, we introduce the data model for STSand the proposed equalisation algorithm. A derrivation ofthe algorithm is detailed in section 3. In order to assess theperformance of the novel algorithm, the zero-forcing solutionhas been chosen as a benchmark. The inverse of the polyno-mial channel matrix is calculated in frequency domain, as in[5]. Simulation results are presented in section 5 to demon-strate the convergence and error performance of the STS-CMalgorithm.2. DATA MODELInspired by STBC coding, STS was introduced in [4] asa transmit diversity scheme for wideband-CDMA systems.Similar to STBC, the source data is split into odd and evensymbol sequences, b1[n] and b2[n], respectively. The index nis referred to as symbol time, whereas the transmission willbe at a chip rate K times higher than the symbol rate, withchip index m. The chip rate signals transmitted from the twoantennas, s1[m] and s2[m], are defined ass1[m] =1√2∞∑n=−∞(c1[m−nK]b1[n]+ c2[m−nK]b∗2[n])s2[m] =1√2∞∑n=−∞(c1[m−nK]b2[n]− c2[m−nK]b∗1[n]) ,where (·)∗ denotes complex conjugation and c1[m] andc2[m] are orthonormal codes of length K, whose coefficientsin the following are arranged in vectors ci ∈ RK , i = {1,2}.The code length defines the CDMA spreading gain as K/2.If only one code, c, is assigned to every user, then the twocodes can be defined asc1 =[cc]and c2 =[c−c]. (1)Note that the orthonormality of the user codes is preserved,i.e. cH1 c2 = 0, where superscripts (·)T and (·)H denote thetranspose and Hermitian transpose operations, respectively.The received signals over a window of L samples is given16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIPWi-iiiﬀﬀﬀﬀ6ﬀﬀﬀﬀﬀ6ﬀﬀ6?---- --a1[m]a2[m]cH1cH2cH1cH2y2[n]y1[n] d1,1[n]d1,2[n]d2,1[n]d2,2[n](.)∗(.)∗v1[m]v1[m]s2[m]s1[m]HFigure 1: Space-Time equalization for STS.byRm =[rT1rT2]=N−1∑ν=0HνSm−ν +Vm. (2)where Sm is the matrix of transmitted dataSm =[s1[m] · · · s1[m−L+1]s2[m] · · · s2[m−L+1]]. (3)Matrix Hν is the ν th time slice of the channel transferfunction and is defined asHν =[h1,1[ν ] h1,2[ν ]h2,1[ν ] h2,2[ν ]], (4)where hi, j is the frequency selective channel from the jthtransmit antenna to the ith receive antenna. The channelshave a maximum length of N. The transfer function of thedispersive MIMO channel can be written asH(z) =N−1∑ν=0Hνz−ν . (5)Additive white Gaussian noise (AWGN), Vm, with zeromean and E{VVH}= LσrI2×2 is assumed to corrupt thesignals at the receiver.As shown in Fig. 1, a MIMO equalizer is used, wherebyeach of the four subfilters has length L. It was proven in [7]that perfect Zero-Forcing (ZF) equalization of the p-by-uMIMO channel can be achieved if the length of the sube-qualizers, L satisfiesL≥ L=⌈N(p−1)u− p⌉, (6)where d·e denotes rounding operation to the next larger inte-ger.3. THE STS-CMA ALGORITHMDue to its simplicity and low requirements, the constant mod-ulus algorithm (CMA) is the most commonly used blindequalizer. The algorithm, first introduced in [6], assumesa constant modulus constellation set at the transmitter andworks by forcing the received signal to have the same mod-ulus. A MIMO-CMA algorithm was derived in [8] which inaddition explicitly forces the cross-correlation between theoutputs to zero in order to avoid multiple extractions of thesame source.As shown in Fig. 1, the equalizer outputs a1[m] anda2[m] will ideally identify the transmitted sequences s1[m]and s2[m], respectively. Due to the fact that the transmit-ted signals are not constant modulus and the fact that multi-ple users are present in the system, the conventional MIMO-CMA cannot be used. In [1], a CM algorithm was derivedfor the equalization of a single antenna DS-CDMA systembased on the orthogonality of the match filtered outputs. Toexploit the STS scenario similar to [1], we consider the costfunctionξ = E{2∑i=12∑l=1(|di,l [n]|2− γ2)2}, (7)where γ is the modulus of the source constellation and di,l [n]are the despread signals as shown in Fig. 1. Due to the ex-plicit orthogonality of the despread outputs, the proposedcost function is simply the CM criterion calculated over thefour signals. The despread outputs are defined asdi,l [n] = cHl · ai[nK]...ai[nK−K+1] , (8)where the output of the ith space-time equalizer is given byai[nK] =wHi ·r1[nK]...r1[nK−L+1]r2[nK]...r2[nK−L+1]with wi =[wi,1wi,2].(9)Expanding (8) yields:di,l [n] = cHl ·Wi ·r1[nK]...r1[nK−K−L+2]r2[nK]...r2[nK−K−L+2], (10)where Wi is the convolutional matrix for the ith equalizerdefined byWi = wHi,1 0 wHi,2 0. . . . . .0 wHi,1 0 wHi,2 . (11)16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIPTable 1: Summary of the STS-CMA equalization algorithm.1: Update xl [n] =Cl ·rnK , for l = 1,2.2: di,l [n] =wHi [n] ·xl [n], for i, l = 1,2.3: ei,l [n] = di,l [n] ·(|di,l [n]|2− γ2), i, l = 1,2.4: wi[n+1] =wi[n]−µ ·∑2l=1 e∗i,l [n]xl [n], i= 1,2.The terms of Equation (10) can be rearranged to produce:di,l [n] =wHi ·Cl ·rnk, (12)whereCl =cHl 0. . . 00 cHlcHl 00. . .0 cHl, (13)andrnK =r1[nK]...r1[nK−K−L+2]r2[nK]...r2[nK−K−L+2]. (14)Estimating the instantaneous gradient of ξ through omit-ting the expectation operator in (7), the stochastic gradientdescent for the new algorithm is defined aswi[n+1] =wi[n]−µ ·5w∗i ξˆ1, for i= 1,2. (15)The instantaneous gradient of ξ for the ith space-time equal-izer is calculated similar to the conventional CMA as follows:5w∗i ξˆ1 =2∑l=1(e∗i,l [n]ClrnK), (16)whereei,l [n] = di,l [n] ·(|di,l [n]|2− γ2) , for i, l = 1,2. (17)A summary of the iterative STS-CM algorithm is detailedin Tab. 1.4. ZERO-FORCING EQUALIZATIONThe Zero-Forcing (ZF) equalizer has been chosen as a bench-mark for the derived STS-CMA algorithm. The ZF receiverpre-multiplies the received data matrix with the inverse ofthe channel matrix. The transfer function of the equalizer isgiven byW(z) =[w1,1(z) w1,2(z)w2,1(z) w2,2(z)](18)Table 2: Frequency domain inversion of the polynomial ma-trix C.1: D is a 2×2×N matrix with Dν = I for all ν .2: G(e jΩi) = DFT (Gi),where Ωi = 2piiL for i= 0, · · · ,NDFT −13: D(e jΩi) = DFT (Di)4: for i= 1,2, ...G−1(e jΩi) = D(e jΩi)∗G†(e jΩi)5: G−1 = IDFT (W(e jΩi))With a 2-transmit and p receive antenna configuration,the polynomial channel matrix H(z) may be rectangular,which makes it non-invertible. In order to overcome thisproblem, the pseudo inverse is used as follows:W(z) = H†(z)=(H˜(z)H(z))−1 H˜(z)= G−1H˜(z), (19)where ˜(·) denotes the para-Hermitian operator.In [5], a frequency domain approach was proposed for theinversion of a polynomial matrix. After taking the DiscreteFourier Transform (DFT) of the channel matrix, the algo-rithm performs a bin-wise pseudo-inverse operation. A reg-ularization matrixD is used to ensure that the optimal filtersdecay away quickly enough to minimize the circular convo-lution effect. A summary of the inversion algorithm is givenin Tab. 2.5. SIMULATION RESULTSComputer simulation results are presented in this section toprovide insight into the propose scheme and to demonstratethe convergence of the derived STS-FIRMER-CMA algo-rithm. A 2× 2 MIMO model as indicated in Figure 1 isused for simulations with k = 4 users. QPSK modulationis used at the transmitter with a modulus equal to√2. Atthe receiver, signals are corrupted by AWGN at a Signal-to-Noise Ratio (SNR) of 20 dB. The length of the subequalizersis set to L = 7. The step size is initialized to µ = 3 · 10−3,and the coefficient vectors for the four subequalizers are setto all zeros except the first entries ofw11 andw21, which areset to unity. Figure 2 shows the pole-zero diagrams of thedispersive channels used in the simulations.Fig. 3 shows the symbol values at the output of theadapted MIMO equalizer, indicating that the equalizer hascorrectly extracted the transmitted sequences of the user ofinterest. A residual rotation is present at the output due tothe CMA’s phase invariance.To demonstrate the robustness in noise, the channelmodel shown in Fig. 2 has been randomized by using the16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIPFigure 2: Pole-zero diagrams for the dispersive channels.Figure 3: Equalizer outputs and their cross-correlation withthe source signals, SNR = 20dB.Figure 4: BER curve for the derived STS-CM Algorithm.subchannel impulse responses to define a power profile, fromwhich 50 Rayleigh distributed ensemble probes are drawn.The BER performance of the proposed algorithm and theZero-Forcing receiver averaged over the 50 runs is shown inFig. 4 over a range of SNR values. It can be clearly observedthat the new algorithm performs better over low SNR values.This is due to the noise amplification inherent in the matrixinversion operation of the Zero-Forcing algorithm. When theSNR increases beyond 8dB, the performance of the derivedalgorithm drops in comparison to that of the Zero-Forcingreceiver, which can be expected since the former assumes noknowledge of the channel.6. CONCLUSIONIn this paper, a novel algorithm based on the Constant Modu-lus (CM) criterion has been derived for the blind equalizationof STS systems over frequency selective channels. Similar toFIRMER-CMA derived in [1], the new algorithm forces theconstant modulus property of the matched filtered signals ofthe user of interest rather than the immediate equalizer out-puts, and therefore implicitly achieves orthogonality of theextracted signals.Extensive computer simulations have been performed tohighlight the stability of the algorithm, whereby some rep-resentative results have been presented here. The noise per-formance of the derived algorithm has been compared to thatof the Zero-Forcing equalizer. Over low SNRs, the derivedalgorithm achieves better performance without the need forany Channel State Information (CSI).REFERENCES[1] M. Hadef, S. Weiss, and M. Rupp, “Adaptive blind mul-tiuser DS-CDMA downlink equaliser,” Electronics Let-ters, 41(21): 1184–1186, 2005.[2] S.M. Alamouti, “A Simple transmit Diversity Techniquefor Wireless Communications,” IEEE Journal on Se-lected Areas in Commus, 16(8): 1451–1458, 1998.[3] R.M. Buehrer, R. Soni, and J., -A. Tsai, “Downlink Di-versity in CDMA Using Walsh Codes”, Engineer’s NoteJW9 111 000-990 218-01EN, : Feb 1999.[4] B. Hochwald, T.L. Marzetta, and C.B. Papadias, “ATransmitter Diversity Scheme for Wideband CDMASystems Based on Space Time Spreading,” IEEE Jour-nal on Selected Areas in Comms., 19(1): 48–60, 2001.[5] O. Kirkeby, P.A. Nelson, H. Hamada, and F. Orduna-Bustamante, “Fast Deconvolution of Multichannel Sys-tems Using Regularization,” IEEE Trans. on Speech andAudio Processing, 6(2): 189–195,1998.[6] Dominique N. Godard, “Self-Recovering Equalizationand Carrier Tracking in Two-Dimensional Data Commu-nication Systems,” IEEE Trans. on Comms. [legacy, pre-1988], 28(11): 1867–1875, 1980.[7] D.T.M. Slock, “Blind joint equalization of multiple syn-chronous mobile users using oversampling and/or mul-tiple antennas,” Proc. 28th Asilomar Conf. on Signals,Systems and Computers,, 2: 1154–1158, 1994.[8] C.B. Papadias and A. Paulraj, “A space-time constantmodulus algorithm for SDMA systems,” IEEE VTCConf., 1(28): 86–90, 1996.16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 2008, copyright by EURASIP

A Simple transmit Diversity Technique for Wireless Communications,”

A space-time constant modulus algorithm for SDMA systems,”

A Transmitter Diversity Scheme for Wideband CDMA Systems Based on Space Time Spreading,”

Adaptive blind multiuser DS-CDMA downlink equaliser,”

Blind joint equalization of multiple synchronous mobile users using oversampling and/or multiple antennas,”

Downlink Diversity in CDMA Using Walsh Codes”,

Fast Deconvolution of Multichannel Systems Using Regularization,”

Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems,”

A CM based equalizer for space-time spreading over channels with inter-Symbol interference

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A CM based equalizer for space-time spreading over channels with inter-Symbol interference

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