Constantinides, AG

Sansrimahachai, P

Ward, DB

English

Sansrimahachai,P.

Ward,D.B.

Constantinides,A.G.

Spiral - Imperial College Digital Repository

Multiple-Input Multiple-Output Least-SquaresConstant Modulus AlgorithmsP. Sansrimahachai, D. B. Ward and A. G. ConstantinidesImperial College LondonDepartment of Electrical and Electronic EngineeringExhibition Road, London SW7 2BT, UKAbstract— In this paper, we propose novel blind sepa-ration techniques for multiple-input multiple-output sys-tems based on least-squares constant modulus algorithm(LSCMA). To ensure that each output signal is extractedfrom different input signals, the proposed algorithms havebeen derived by using successive interference cancellationand Gram-Schmidt orthogonalization procedure. The per-formance is observed through computer simulations andcompared with the multitarget LSCMA (MT-LSCMA)and LS multi-user CMA (LS-MU-CMA). Simulation re-sults have shown that the proposed algorithms exhibitbetter performance in terms of both bit error rate (BER)and convergence speed.I. INTRODUCTIONBlind source separation (BSS) algorithms for in-stantaneous mixtures have been popularly investigated.These systems of instantaneous mixtures include theBell Labs layered space-time (BLAST) architecture [1],[2], [3] where a number of antenna elements are usedat both the transmitter and receiver. BLAST has beenproved theoretically and experimentally that it has ahigh transmission capacity. In practical situations wheretraining can be expensive or impossible, blind separationtechniques are therefore considered to detect transmittedsignals. In this paper, blind algorithms based on least-squares constant modulus algorithms are of interest.Least-squares constant modulus algorithm (LSCMA)was first proposed in [4] for a two-sensor array. Thebasic concept of the LSCMA is to combine the well-known least-squares estimator (LSE) and constant mod-ulus (CM) properties to blindly extract communicationsignals. This algorithm is rapidly convergent and glob-ally stable for any linearly-independent set of inputsignals. Therefore, the algorithm was later modified in[5] and [6] for using in a more general system, i.e. amulti-user system.In order to prevent two or more output signalsextracted from the same input signal, multitargetLSCMA (MT-LSCMA) proposed in [5] uses a soft-orthogonalized technique which includes the Gram-Schmidt orthogonalization method followed by a soften-ing procedure. To satisfy the same purpose, least-squaresmulti-user CMA (LS-MU-CMA) [6] exploits the crosscorrelation between output signals. In addition, this termtogether with the LS concept has been used in [7] forthe similar purpose, but the algorithm has been derivedin a different way.Due to the slow convergence speed of the above men-tioned algorithms, we propose in this paper new blindseparation techniques based on the LSCMA. The firstalgorithm is formulated from the concept of successiveinterference cancellation, whereas the second algorithmis derived from the orthogonality constraint which canbe done by using the Gram-Schmidt orthogonalizationprocedure. These novel techniques exhibit faster conver-gence speed and have better performance in terms of biterror rate (BER).The paper is organised as follows. Sections II andIII explain some notations used in this paper and thevertical-BLAST (V-BLAST) system. A revision of least-squares techniques based on the constant modulus al-gorithm is presented in section IV. We formulate theproposed algorithms, namely the successive interferencecancellation LSCMA (SIC-LSCMA) and the Gram-Schmidt LSCMA (GS-LSCMA), in section V and VI.Simulation results are given in section VII and the paperis finally concluded in section VIII.II. NOTATIONS• k is the sample index.• m is the block iterative index.• ξ is a small stopping parameter, i.e. 10−4.• (·)T is the transpose operation.• (·)∗ is the conjugate operation.GLOBECOM 2003 - 2084 - 0-7803-7974-8/03/$17.00 © 2003 IEEEVectorEncodera1aMa2Txdata BLASTAlgorithmr1r2rNRxdataFig. 1. The BLAST system consisting of N receive antennas andM transmit antennas.• (·)H is the conjugate transpose operation.•〈|x|2〉L =1L∑Lk=1 x(k)x(k)∗.III. SYSTEM MODELWe consider the Bell Labs Layered Space-Time(BLAST) architecture [1], [2], [3] as a system model inthis paper. The architecture, as shown in Fig. 1, consistsof N receive antennas and M transmit antennas. Denotinga(k), v(k) and H the M × 1 transmitted signal vector,the N × 1 noise vector and the N × M channel matrix,respectively, the corresponding N × 1 received signalvector is given asr(k) = Ha(k) + v(k). (1)It is assumed that each component of a(k) has thesame statistical properties and is drawn from the sameconstellation, i.e. QAM [2]. The channel matrix H canbe modelled by a matrix having independent identicallydistributed (iid), complex, zero-mean, unit-variance en-tries. We assume that H is unitary. If it is not unitary,the received signals are then prewhitened. The noise atthe receiver is assumed to be a complex additive whiteGaussian noise. Each component of the noise vector isstatistically independent and of identical power at theantenna outputs.Here we consider the data of block NB and the N ×NB received signal matrix can be formulated asX = [r(1) · · · r(NB)]= HA + V (2)where A and V are formulated in the similar fashion asX. This problem is regarded as a problem of blind sourceseparation (BSS). It is therefore possible to recover thetransmitted signals A using only the received signals X.Denoting W = [w1 · · ·wM ] the N × M equalizermatrix, the estimate of A is obtained byY = [y(1) · · · y(NB)]= WHX. (3)It is necessary to note that the matrix W is feasible toseparate the sources, except for a possible permutationof Y and an arbitrary scaling of each source signal.In the non-blind case, W can be estimated from theclassical least-squares estimator (LSE) via some trainingsymbols. The equalizer matrix W is therefore obtainedby minimizing a least-squares criterionWLS = arg minWN−1TNT∑k=1|a(k) − WHr(k)|2= R−1T PT (4)where NT denotes the length of training, RT = XTXHTand PT = XTAHT .IV. LEAST-SQUARES TECHNIQUESA. Least-Squares Constant Modulus AlgorithmThe least-squares constant modulus algorithm(LSCMA) [4] was developed based on the Gauss’smethod of complex-argument cost functions. Considerthe constant modulus algorithm whose cost function isrepresented in a formJ(W) =NB∑k=1φ2k(W) = ‖Φ(W)‖22 (5)where Φ(W) = |WHX| − 1. Its partial Taylor-seriesexpansion with sum-of-squares component can be ap-proximately given asJ(W + ∆) ≈ ‖Φ(W) + Q(W)H∆‖22. (6)Gauss’s method updates W by the offset ∆ whichminimizes J(W + ∆) resulting inWm+1 = Wm − ∆= Wm − (RQ)−1Q(Wm)ΦH(Wm) (7)where Q(W) = ∇WJ(W) and RQ =Q(Wm)Q(Wm)H . By assuming that the transmittedsignals are linearly independent, the update equationfor LSCMA is therefore obtained asWm+1 = Wm − (XXH)−1X(Ym − Ψm)H= (XXH)−1XΨHm (8)where ψi(k) = yi(k)/|yi(k)|, i = 1, ..,M . The blockupdate algorithm can be summarised as follows. FindGLOBECOM 2003 - 2085 - 0-7803-7974-8/03/$17.00 © 2003 IEEER = XXH and its inverse Rinv = R−1. Then initializeW0 = I.for m = 0, 1, ..Ym = WHmXψm,i(k) = ym,i(k)/|ym,i(k)|, i = 1, ..,MPm = XΨHmWm+1 = RinvPmuntil ‖Wm+1 − Wm‖/‖Wm‖ < ξB. Multitarget Least-Squares Constant Modulus Algo-rithmThe multitarget least-squares constant modulus algo-rithm (MT-LSCMA) was proposed in [5] as an extensionof the LSCMA to blindly adapt a narrowband beam-former. The algorithm separates and captures multiplecommunication signals by exploiting only their modulusvariation. The original MT-LSCMA consists of threeprinciple components:• a soft-orthogonalized LSCMA• a set of sorting and classification algorithms• a fast acquisition algorithm.Only the first component will be considered here andwe will observe the iterative update version of thisalgorithm. The summary of the algorithm can be givenas follows. At the end of the LSCMA, the soft orthog-onalization is proceeded through the following steps.• Compute the hard-orthogonalized equalizer matrixŴ by using a standard Gram-Schmidt Orthogonal-ization procedure and find its corresponding outputsignalsŶ = ŴHX (9)• Compute a softening parameter λi, i = 1, ...,M byinitializing ỹ0,i = yi and calculatingεi =〈|yi(k) − ŷi(k)|2〉NB〈|yi(k)|2〉NB. (10)for p = 0, 1, ...ε̃p,i =〈|ỹp,i(k) − ŷi(k)|2〉NB〈|ỹp,i(k)|2〉NB> 0 (11)λp,i =√ε̃p,i(1 − εi)εi(1 − ε̃p,i)(12)ỹp,i(k) = λp,i yi(k) + (1 − λp,i) ŷi(k) (13)until εp,i, ε̃p,i < εmax.C. Least-Squares Multi-User Constant Modulus Algo-rithmAnother technique based on the LSCMA was pro-posed in [6], namely least-squares multi-user con-stant modulus algorithm (LS-MU-CMA). This algorithmcombines the LSCMA and MU-CMA [8], and servesas a block CM type algorithm that efficiently performsblind spatial co-channel interference mitigation. The LS-MU-CMA was derived from the MU-CMA cost functiondefined asJi = E{(|yi| − d)2} + 2i−1∑l=1|y∗i yl|2. (14)where d is a constant modulus, i.e. d = 1. Afterseparating and expanding the expectation in Ji, theresulting weight update equation is obtained in a formidentical to the least-squares estimator. Similar to theprevious algorithm, we will consider the iterative updateversion of this algorithm which can be given as follows.Calculate R and then initialize W0 = I. For m = 0, 1, ..until ‖Wm+1 − Wm‖/‖Wm‖ < ξ, we perform thefollowing steps at each m.for i = 1, ..,MRi = Rφii = wHm,iRwm,ifor j = 1 : i − 1uj = wHm,jRφji = ujwm,iif |φji| > γ|φii| do Ri = Ri + uHj ujPm,i = XΨHm,iwm+1,i = R−1i Pm,iV. SUCCESSIVE INTERFERENCE CANCELLATIONLEAST-SQUARES CONSTANT MODULUS ALGORITHMIn this section, we propose the first new technique thatcould serve as an alternative way for blind signal sep-aration. The proposed algorithm combines the LSCMAand the separation technique namely the successiveinterference cancellation to blindly separate the receivedsignals. This algorithm is therefore named successiveinterference cancellation least-squares constant modulusalgorithm (SIC-LSCMA).Consider the cost function of the SIC-LSCMA givenasJ(wi) =NB∑k=1(|wHi ri(k)| − 1)2 = ‖Φ(wi)‖22 (15)GLOBECOM 2003 - 2086 - 0-7803-7974-8/03/$17.00 © 2003 IEEEwhere Φ = |wHi Xi| − 1 andXi = Xi−1 − h̃i−1Ψi−1 (16)represents the interference cancellation procedure. Thechannel h̃i−1 used in this procedure can be estimatedfrom [9]h̃i−1 =E{r(k)ψ∗i−1(k)}E{|ψi−1|2}. (17)The proposed algorithm can be summarised as follows.Find R = XXH and its inverse Rinv = R−1. Theninitialize W0 = I. For m = 0, 1, .. until ‖Wm+1 −Wm‖/‖Wm‖ < ξ, we perform the following successiveinterference cancellation at each m by first setting X̃1 =X.for i = 1, ...,M• ym,i = wHi X̃i = ith row of Ym• ψm,i(k) = ym,i(k)/|ym,i(k)|• h̃m,i = X̃iΨHm,i(Ψm,iΨHm,i)−1• X̃i+1 = X̃i − h̃m,iΨm,iand then updateWm+1 = RinvXΨHm.The SIC-LSCMA requires O(NMNB) flops forthe successive interference cancellation and anotherO(NMNB) flops to update W.VI. GRAM-SCHMIDT LEAST-SQUARES CONSTANTMODULUS ALGORITHMBy adding a constraint to the cost function of theLSCMA given in equation (5), the problem of the Gram-Schmidt LSCMA (GS-LSCMA) can be formulated asmin J(W) = ‖Φ(W)‖22 (18)subject to WHW = Iwhere I is the identity matrix. This orthogonality con-straint ensures that the output signals are recoveredfrom different input signals and can be implemented byusing the well-known Gram-Schmidt orthogonalizationtechnique.The LSCMA is therefore simply modified by perform-ing the Gram-Schmidt orthogonalization procedure afterWm+1 is calculated. The Gram-Schmidt orthogonaliza-tion procedure is implemented aswnew1 = w1/‖w1‖P =i−1∑l=1((wnewl )Hwi)wnewl (19)wnewi =wi − P‖wi − P‖, i = 2, . . . ,M.This procedure is the hard orthogonalization step inthe MT-LSCMA which has been claimed in the samepaper that the excessive misadjustment can occur if thistechnique is used to separate the output signals. In thismentioned algorithm, the Gram-Schmidt procedure isperformed after W of the LSCMA has converged. Itis therefore possible that this procedure may degradethe performance of the MT-LSCMA.In the GS-LSCMA, however, the Gram-Schmidt or-thogonalization procedure is performed immediately af-ter the update of W. The output of this procedureconverges simultaneously as the LSCMA converges. Thealgorithm could therefore be more robust and exhibitbetter performance than the MT-LSCMA. In addition,when the channel is assumed to be unitary, this pro-cedure has been adopted in an adaptive blind sourceseparation algorithm, namely the multi-user kurtosis(MUK) algorithm proposed in [10], and provides apromising separation performance.Unfortunately, the Gram-Schmidt orthogonalizationprocedure is computationally expensive. The modifiedversion of this procedure, that yields less computationalcomplexity of 2NM2 flops [11], is then used. As a con-sequence, the GS-LSCMA requires O(NMNB) flops toobtain Y and another O(NMNB) flops to update W.VII. SIMULATION RESULTSPerformance comparison is observed through simula-tions. A V-BLAST system consisting of N = M = 2was considered and a random instantaneous channelH was assumed to be unitary. We compared the biterror rate (BER) performance of MT-LSCMA, LS-MU-CMA and SIC-LSCMA by varying the signal-to-noiseratio (SNR). In addition, we also observed the BERperformance versus the number of block iterations whenSNR was fixed at 12 dB.Parameters were set as follows. The stopping param-eter ξ was chosen to be 10−4. NB was chosen to be20. The parameters in the algorithms were defined asfollows: εmax = 0.1, γ = 1 and δ = 2 × 10−3. InGLOBECOM 2003 - 2087 - 0-7803-7974-8/03/$17.00 © 2003 IEEE0 2 4 6 8 10 12 14 16 18 2010−510−410−310−210−1100SNR (dB)Average BER (dB)Performance comparisonMT−LSCMA LS−MU−CMASIC−LSCMAGS−LSCMA Fig. 2. Bit error rate (BER) comparison (MT-LSCMA, LS-MU-CMA, SIC-LSCMA, GS-LSCMA) for NB = 20.0 10 20 30 40 50 60 70 80 90 10010−510−410−310−210−1100block iteration (m)Average BER (dB)Performance comparisonMT−LSCMA LS−MU−CMASIC−LSCMAGS−LSCMA Fig. 3. Bit error rate (BER) comparison (MT-LSCMA, LS-MU-CMA, SIC-LSCMA, GS-LSCMA) for SNR = 12 dB.addition to these settings, the number of block iterationswas limited to 100.Figures 2 and 3 show the comparison of the blindalgorithms (MT-LSCMA, LS-MU-CMA, SIC-LSCMAand GS-LSCMA). It can be observed that the proposedalgorithms exhibit better performance and these algo-rithms need 20 block iterations to achieve its minimumBER at SNR = 12 dB. In addition, the GS-LSCMAoutperforms the other algorithms by having the bestBER performance.VIII. CONCLUSIONSWe have presented two novel algorithms based onthe least-squares constant modulus algorithm (LSCMA).The first algorithm can be referred to as multi-stageblind signal detection since the interference cancellationis performed after each output is detected. The secondalgorithm has been derived from the orthogonality con-straint which ensures that each output is obtained fromdifferent inputs. This algorithm is simply performedby implementing the Gram-Schmidt orthogonalizationprocedure after the LSCMA equalizer matrix is updated.The performance has been observed through simula-tions and compared with the multitarget LSCMA (MT-LSCMA) and LS multi-user CMA (LS-MU-CMA). Theproposed algorithms have exhibited better performancein terms of both bit error rate (BER) and convergencespeed with reasonable computational complexity.REFERENCES[1] G. J. Foschini, “Layered space-time architecture for wirelesscommunication in a fading environment when using multi-element antennas”, Bell Labs Technical Journal, Vol. 1, No.2, pp.41-59, 1996.[2] P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valen-zuela, “V-BLAST: An architecture for realizing very high datarates over the Rich-scattering wireless channel”, invited paper,Proc. ISSSE-98, Pisa, Italy, Sept. 29, 1998.[3] G. D. Golden, C. J. Foschini, R. A. Valenzuela, P. W. Wolnian-sky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture”, ElectronicsLetters, vol. 35, no. 1, January 1999.[4] B. G. Agee, “The least-squares CMA: a new technique forrapid correction of constant modulus signals,” Proc. ICASSP,vol. 11, pp. 953 - 956, 1986.[5] B. G. Agee, “Blind separation and capture of communicationsignals using a multitarget constant modulus beamformer,”Proc. IEEE Military Communications Conference, vol. 2, pp.340 - 346, 1989.[6] J. Leary, “Least-squares multi-user CMArray: a new algorithmfor blind adaptive beamforming,” Proc. Thirty-First AsilomarConference on Signals, Systems and Computers, vol. 1, pp. 902- 905, 1998.[7] F. R. P. Cavalcanti, J. M. T. Romano, A. L. Brandão, “Least-squares CMA decorrelation for fast blind multiuser signalseparation,” Proc. ICASSP, vol. 5, pp. 2527 - 2530, 1999.[8] C. B. Papadias, A. J. Paulraj, “A constant modulus algorithmfor multiuser signal separation in presence of delay spreadusing antenna arrays,” IEEE Signal Processing Letters, vol. 4,issue 6, pp. 178 - 181, 1997.[9] C.-Y. Chi, C.-Y. Chen, C.-H. Chen, C.-C. Feng, “Batch pro-cessing algorithm for blind equalization using higher-orderstatistics,” IEEE Signal Processing Magazine, pp. 25 - 49,2003.[10] C. B. Papadias, “Globally convergent blind source separationbased on a multiuser kurtosis maximization criterion”, IEEETransactions on Signal Processing, vol. 48, no. 12, pp. 3508 -3519, 2000.[11] G. H. Golub, C. F. Van Loan, Matrix Computations. Baltimore,MD: John Hopkins University Press, 1990.GLOBECOM 2003 - 2088 - 0-7803-7974-8/03/$17.00 © 2003 IEEE

Multiple-input multiple-output least-squares constant modulus algorithms

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Multiple-input multiple-output least-squares constant modulus algorithms

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