1999 Direct blind equalizers of multiple FIR channels a deterministic approach

Direct Blind MMSE Channel Equalization Based onSecond-Order StatisticsJunqiang Shen and Zhi Ding,Senior Member,IEEEAbstract—A family of new MMSE blind channel equalization algorithms based on second-order statistics are proposed.Instead of estimating the channel impulse response,we directly estimate the cross-correlation function needed in Wiener–Hopf filters.We develop several different schemes to estimate the cross-correlation vector,with which different Wiener filters are derived according to minimum mean square error(MMSE).Unlike many known sub-space methods,these equalization algorithms do not rely on signal and noise subspace separation and are consequently more robust to channel order estimation errors.Their implementation requires no adjustment for either single-or multiple-user systems.They can effectively equalize single-input multiple-output(SIMO)sys-tems and can reduce the multiple-input multiple-output(MIMO) systems into a memoryless signal mixing system for source separa-tion.The implementations of these algorithms on SIMO system are given,and simulation examples are provided to demonstrate their superior performance over some existing algorithms.Index Terms—Blind channel estimation,channel equalization, communications,multiuser analysis.I.I NTRODUCTIONB LIND channel equalization has become a very popularresearch topic in recent years.This problem is often encountered in digital communication systems in which un-known signals are transmitted through unknown multipath channels.The intersymbol interference(ISI)caused by multi-path channels can severely degrade the communication system performance.Moreover,for multiuser systems,there are also co-channel interferences(CCI)from other users.Both ISI and CCI can be detrimental to the correct reception of the trans-mitted channel input sequence.In order to overcome the effect of ISI and CCI,a number of algorithms have been presented for blind channel identification and equalization of single-input multiple-output(SIMO)and multiple-input multiple-output (MIMO)systems[1]–[15].In blind channel identification/equalization,the channel is identified/equalized without using a training sequence.Gener-ally,at the receiver,only the output sequence and some a priori statistical information on the input sequence are utilized.Such methods offer potential improvement in system capacity by eliminating the training overhead.Traditionally,blind channel identification and equalization can be achieved by exploitingManuscript received July31,1998;revised September15,1999.This work was supported by the National Science Foundation under Grant CCR-99-96206. The associate editor coordinating the review of this paper and approving it for publication was Prof.Kon Max Wong.J.Shen is with the Engineering Resource Center,Delphi Delco Electronics Systems,Kokomo,IN46904-9005USA(e-mail:jshen@). Z.Ding is with the Department of Electrical and Computer Engineering,Uni-versity of Iowa,Iowa City,IA52242USA(e-mail:zding@). Publisher Item Identifier S1053-587X(00)02351-5.the higher order statistics of baud-rate sampled channel output signals[12]–[15].More recently,a class of second-order statistical(SOS)algorithms have been presented that rely on the SIMO system model for additional information.These SOS algorithms can achieve satisfactory performance using fewer output data samples.The algorithm by Tong et al.[1],which is known as the TXK algorithm,is one of the first subspace-based methods exploiting only SOS of SIMO systems resulting from fractionally spaced sampling.Utilizing the SIMO model,a number of SOS algorithms[2]–[11]have been proposed. However,many existing SOS methods suffer from one crit-ical weakness in that they are sensitive to channel order esti-mation errors.When channel order is known,channel estima-tion is satisfactory.However,when channel order is unknown, accurate channel order estimation under noisy conditions be-comes difficult,and performances of SOS algorithms tend to be poor.In this paper,we develop a type of blind channel equaliza-tion algorithm that is less sensitive to the inaccuracy of channel order estimation.We will present SOS methods that do not need to identify the unknown channel impulse response but instead generate equalizer parameters based on the MMSE criterion for SIMO and MIMO systems.These algorithms will estimate the cross-correlation between the measurable channel output signal and desired channel input signal.This cross-correlation vector can then be used to construct the MMSE Wiener equalizer.More specifically,for a systemwith inputs,based on our algo-rithms,we can form a matrix that equals the outer product of only one preselected block column(vector based on SOS are proposed in Section III,and in Section IV ,the blind channel equalization algorithms for SIMO system are considered.Simulation results for the SIMO system are presented in Section V to demonstrate the performance of the proposed algorithms.II.P ROBLEM F ORMULATIONConsider a baseband QAM linear systemwithuser channels are all linear and causal withimpulseresponseis the symbol baud period,andis stationary,white,and independent of channelinputsequencesis a “composite”channel impulse responsethat includes transmitter and receiver filters as well as the physical propagation channel response.In a typical multiuser system,multiple channels of observations will be available from multiple sensors.If,,and areall vectors.However,for simplicity of presentation,weassumethroughout this paper.The objective of blind channel equalization is todemodulate the inputsequencefrom the outputsequence ,given only some a priori statistical knowledgeof without knowing the channel response.We proceed to develop a discrete-time MIMO signal model,which is key to developing the blind equalization algorithms in the subsequent sections.Suppose that the channel is oversam-pled by (aninteger)is the number of output samples derived from oversam-pling (assuming that there is excess bandwidth).Supposethat(2.2)it is evidentthatis called the smoothing lag orthe equalizer memory .We formanblock Toeplitzmatrix...has full column rank andis thus identifiable [1],[2].Assume that both the channel inputsignal(3.1)(3.2)with the specialcase(3.3)noisevariance;identitymatrix.with outputsignalSHEN AND DING:DIRECT BLIND MMSE CHANNEL EQUALIZATION 1017The minimum MSE filter toestimate is the solution to theWiener–Hopfequation(3.6)whereas the cross-correlation vectorequals(3.7)wheredenotesthe,whichis.In subsequent sections,we will show how thisvector can be estimated blindly without training data.It should be noted that in [17],a blind MMSE equalizer was determined from partial knowledge of the desired user channel in the form of its CDMA spreading code.In this work,we present a dif-ferent method that does not require such partial eful Matrix PropertiesFor notational convenience,wedefine(3.8)(3.9)We also definethatis an identity matrix,except for its first zero di-agonal entries,and is all zero except for unit entries onitsto th diagonal elements.It can be directly shownthathas full columnrank,the following equation is true [19],[20]:(3.14)wheremust be estimated and subtracted from the covariance ma-trices.A.Method AObserve that from the expressionof(4.7)It is readily seen thatmatrixhasrankunitary matrix.As a result,the blindequalizer output in the absence of channel noise issimply(4.9)which is a memoryless mixtureof1018IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.48,NO.4,APRIL 2000B.Method B Another way togetis toutilizeand,as in the previous subsection.Noticethat(4.12)where the second equality follows from (4.11).Theoretically,obtained from this method should be equalto ob-tained in the previous subsection in (4.7).Because more matrix multiplications with matrix inverse are involved in this method,method B is more sensitive to the numerical errors due to the matrix inversion,and more computational work is involved here than in method A.However,noise may affect the actual perfor-mance of method A and method B in practice.In fact,because method A and method B involve different utilizationofand,and only involves two ma-trix multiplications,which is the most computational effectiveamong these three methods.However,and only contains thefirst.Recall the definitionofin (3.11).Forthecase...............istheis givenby.For single-usersystems,,which means that can be estimatedwith a (nonzero)scalar ambiguity via eigendecomposition,QR factorization,or singular value decomposition (SVD)ofmatricesor ,theMMSE equalizers can be designed based on our estimate of the cross-correlation vector with this ambiguity.B.Optimum Delay SelectionIt should be noted that the MMSE equalizer is designed forsignalestimateat a specific delay for the equalizers.Let the estimated input symbol withdelayThe MSE for the MMSE equalizer isMSESHEN AND DING:DIRECT BLIND MMSE CHANNEL EQUALIZATION1019 Hence,the optimum delay can be foundbybaud samples of the channel outputdata,form the auto-correlationsubmatricesand form the estimated auto-covariancematricesaccordingto..................fromMDLand estimate the noise variance as the average ofthesmallesteigenvalues3)According to(3.6)and(4.1),obtain the noise-adjustedauto-covariance matrixasfor asuitableaccording to(5.4)or(5.6).6)Use the cross-correlation vector for the optimum delayand determine the optimum equalizer parametervector.For mod-erate to high SNR systems,this will only result in a smallperformance degradation.•The performance of MMSE equalizers relies criticallyon the accuracy of the cross-correlation vector estimate.Different methods to estimate the cross-correlation vectormay be advantageous in terms of performance or compu-tational complexity.•Our simulations show that the performance of the equal-izers will not differ very much once the delayto beequalto.Generally speaking,selectingmeans more computational ually,we cannotaffordan is notexactly known and is estimated from the collected data.In our simulations,wechooselimitedin with roll-off factor0.10and a two-ray multipathchannelwhich results in the overall channel impulse responseofis shown in Fig.1.The data input signal is i.i.d.QPSK,and the oversampling factoris.The noise is zero mean,white,and Gaussian.Fromthe channel impulse response,it can be seen that the maximumsubchannel orderis1020IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.48,NO.4,APRIL2000Fig.1.Overall channel impulse response h (t )used in the simulations.For the purpose of comparison,we also implement the sub-space method (SSM)[6]for channel equalization.We first iden-tify the channel using SSM and then apply the MMSE algorithm for channel equalization.We label this method MMSE-SSM.To make SSM work properly for our examples,we choose thesmoothing lag for SSM tobe ,whereas for our algo-rithms,wechoose=6(true channel order),and thelength of the data samples is K =350.Fig.3.Bit error rate at the outputs of the different equalizers versus the estimated channel order.hboxSNR =18dB,and the length of the data samples is K =350.the BER of MMSE-SSM drops to an unacceptable level if the estimated channel order deviates from the true channel order.All three proposed algorithms are far less sensitive to channel order estimate mismatches.C.Experiment 3:Data Length EffectAlgorithm performance is affected by output data length.In this experiment.The SNR is fixed at 18dB,and the true channelorderThe equalizer output MSE is affected by a different selection of delaySHEN AND DING:DIRECT BLIND MMSE CHANNEL EQUALIZATION1021Fig.4.Bit error rate at the outputs of the different equalizers versus the length of data samples.SNR=18dB.Estimated channel order^m=7;,and K=350.for differentdelaywhere,to save computation,selecting=7.E.Scatter PlotsFinally,we show in Fig.6the signal scatter plot before andafter blind equalization.In this simulation,SNR1022IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.48,NO.4,APRIL2000[7]K.Abed-Meriam et al.,“Prediction error methods for time-domainblind identification of multichannel FIR filters,”in Proc.IEEE Int.Conf.Acoust.,Speech,Signal Process.,Detroit,MI,May1995,pp.1968–1971.[8]G.Xu,H.Liu,L.Tong,and T.Kailath,“A least-squares approach toblind channel identification,”IEEE Trans.Signal Processing,vol.43, pp.2982–2993,Dec.1995.[9]K.Abed-Meraim and Y.Hua,“New linear prediction algorithm for blindequalization in spatially colored noise,”presented at the IEEE Workshop Digital Signal Process.,Bryce Canyon,UT,Aug.1998.[10]H.Liu and G.Xu,“Closed-form blind symbol estimation in digital com-munications,”IEEE Trans.Signal Processing,vol.43,pp.2714–2723, Nov.1995.[11]J.Zhu,X.Cao,Z.Ding,and J.Shen,“A blind intersymbol interferencecancellation method for multi-user systems with channel diversity,”pre-sented at the31st Asilomar Conf.Signals,Syst.,Comput.,Pacific Grove, CA,Nov.1997.[12] A.Benveniste,M.Goursat,and G.Ruget,“Robust identification of anonminimum phase system:Blind adjustment of a linear equalizer in data communications,”IEEE Trans.Automat.Contr.,pp.385–399,June 1980.[13]Z.Ding,R.A.Kennedy,B.D.O.Anderson,and C.R.Johnson,“Ill-con-vergence of Godard blind equalizers in data communication systems,”IEEE mun.,vol.39,pp.1313–1327,Sept.1991.[14] D.N.Godard,“Self-recovering equalization and carrier tracking in two-dimensional data communications systems,”IEEE mun.,vol.COMM-28,pp.1867–1875,Nov.1980.[15] D.Hatzinakos and C.Nikias,“Estimation of multipath channel responsein frequency selective channels,”IEEE J.Select.Areas Commun.,vol.7,pp.12–19,Jan.1989.[16]X.Cao and R.Liu,“General approach to blind source separation,”IEEETrans.Signal Processing,vol.44,pp.562–571,Mar.1996.[17] D.J.Gesbert,J.Sorelius,and A.J.Paulraj,“Blind multiuser MMSEdetection of CDMA signals,”in Proc.IEEE Int.Conf.Acoust.,Speech, Signal Process.,Seattle,W A,May1998,pp.3161–3164.[18]M.Wax and T.Kailath,“Detection of signals by information theoreticcriterion,”IEEE Trans.Acoust.,Speech,Signal Processing,vol.ASSP-33,pp.387–392,Apr.1985.[19]G.H.Golub and C. F.Van Loan,Matrix Computations,3rded.Baltimore,MD:Johns Hopkins Univ.Press,1996.[20]T.L.Boullion and P.L.Odell,Generalized Inverse Matrices.NewYork:Wiley-Interscience,1971.[21]S.Barnett,Matrices Methods and Applications.Oxford,U.K.:Clarendon,1990.Junqiang Shen was born in Shaoxing,China.Hereceived the B.S.and M.S.degrees in computerscience and engineering from Shanghai JiaotongUniversity,Shanghai,China,in July1989and March1992,respectively,and the Ph.D.degree in electricalengineering from Auburn University,Auburn,AL,in March1999.From March1992to June1995,he worked asan electrical engineer in Eastcomm Corporation,Hangzhou,China,developing digital switchingsystems and GSM switching systems.Since June 1999,he has been with Delphi Delco Electronics Systems,Kokomo,IN, working on the smart automotive airbag sensing system.His research interests include automotive airbag sensing algorithms,multiuser detection in CDMA systems,digital communications,digital signal processing,statistical signal processing,and blind channelindentification/equalization.Zhi Ding(SM’95)received the Ph.D.degree from theSchool of Electrical Engineering,Cornell University,Ithaca,NY,in August1990.He is an Associate Professor with the Departmentof Electrical and Computer Engineering,Universityof Iowa,Iowa City.From1990to1998,he was aFaculty Member with the Department of ElectricalEngineering,Auburn University,Auburn,AL,first asan Assistant Professor and later as an Associate Pro-fessor.He has had visiting positions with the Aus-tralian National University,Canberra,the Hong Kong University of Science and Technology,the NASA Lewis Research Center,and the USAF Wright Laboratory.His research covers a number of research issues involving statistical signal processing that include communications system de-sign,signal detection and classification,and blind signal separation.Dr.Ding has been an Associate Editor of the IEEE T RANSACTIONS ON S IGNAL P ROCESSING.He currently serves as a Member of the IEEE Signal Processing for Communications Technical Committee.。