Gaussian Maximum Likelihood Blind Multichannel Multiuser Identification

We consider a Spatial Division Multiple Access (S.D.M.A.) situation in which p users operate on the same carrier frequency and use the same linear digital modulation format. We consider m > p antennas receiving mixtures of these signals through multi-path propagation (equivalently, oversampling of the received signals of a smaller number of antenna signals could be used). Current approaches to multiuser blind channel identification include subspace-fitting techniques [7], deterministic Maximum-Likelihood (DML) techniques [13] and linear prediction methods [13]. The two first techniques are rather closely related and give the channel apart from a triangular dynamical multiplicative factor (see [7]), moreover, they are not robust to channel length overestimation. The latter approach is robust to channel length overestimation and yields the channel estimate apart from a unitary static multiplicative factor, which can be determined by resorting to higher order statistics. On the other hand, Gaussian Maximum Likelihood (GML) methods have been introduced in [5] for the single user case and have given better performances than DML. Extending GML to the multiuser case, we can expect good performances, and, as will be shown in the identifiability section, we will get the channel apart from a unitary static multiplicative factor. I Problem Formulation Consider linear digital modulation over a linear channel with additive Gaussian noise. Assume that we have p transmitters at a certain carrier frequency and m antennas receiving mixtures of the signals. We shall assume that m > p. The received signals can be written in the baseband as yi(t) = p Xj=1Xk aj(k)hji (t kT ) + vi(t) (1) where the aj(k) are the transmitted symbols from source j, T is the common symbol period, hji (t) is the (overall) channel impulse response from transmitter j to receiver antenna i. Assuming the aj(k) and fvi(t)g to be jointly (wide-sense) stationary, the processes fyi(t)g are (wide-sense) cyclostationary with period T . If fyi(t)g are sampled with period T , the sampled processes are (wide-sense) stationary. Sampling in this way leads to an equivalent discrete-time representation. We could also obtain multiple channels in the discrete-time domain by oversampling the continuous-time received signals, see [11],[14]. We assume the channels to be FIR. In particular, after sampling we assume the (vector) impulse response from source j to be of length N j. Without loss of generality, we assume the first non-zero vector impulse response sample to occur at discretetime zero. Let N = Ppj=1N j and N1 = maxj(N j) . The discrete-time received signal can be represented in vector form as y(k) = p Xj=1 Nj 1 Xi=0 hj(i)aj(k i) + v(k) = N1 1 Xi=0 H(i)a(k i) + v(k) = p Xj=1HjAjNj (k) + v(k) = HAN (k) + v(k) (2) y(k) = yH 1 (k) yH m (k) H ;v(k) = vH 1 (k) vH m(k) H ; hj(k) = hhjH 1 (k) hjH m (k)iH ; Hj = hj(N j 1) hj(0) ;H = H1 Hp ; H(k) = h1(k) hp(k) ;a(k) = a1H(k) apH(k) H ; Ajn(k) = ajH(k n+1) ajH(k) H ; AN (k) = hA1H N1 (k) ApH Np (k)iH : (3) where superscript H denotes Hermitian transpose. We consider additive temporally and spatially white Gaussian circular noise v(k) with Rvv(k i) = E v(k)vH (i) = 2 vIm ki. Assume we receive M samples : Y M (k) = T p M (H) AN+p(M 1)(k) + V M (k) (4) where Y M (k) = hY H (k M + 1) Y H(k)iH and V M (k) is defined similarly whereas T p M (H) is the multichannel multiuser convolution matrix of H, with M block lines. Therefore, the structure of the covariance matrix of the received signal Y (k) is RYY = T p M (H)RAAT pH M (H) + 2 vImM (5) where RAA = EnAN+p(M 1)(k)AHN+p(M 1)(k)o. From here on, we will assume mutually i.i.d. white sources with power 2 a (RAA = 2 aI). 1The work of Luc Deneire is supported by the EC by a Marie-Curie Fellowship (TMR program) under contract No ERBFMBICT950155 II GML : Gaussian Maximum Likelihood. In stochastic ML, the input symbols are modeled as Gaussian quantities. ML estimation with a Gaussian prior for the symbols has been introduced in [15] and [5] for the single user case and its robustness properties have been shown. Rewriting equation (4) in shorthand : Y = T (H)A + V , with Gaussian hypotheses on the noise and the symbols (V N (0; RVV ) and A N (0; RAA)). We want to maximize f(Y jH). Hence, Y N (0; RYY ) and the corresponding log-likelihood function to be minimized is : min H nln(detRYY ) + Y HR 1 YY Y o : (6) A Identifiability conditions Parameters are considered identifiable when they are determined uniquely by the probability distribution of the data (i.e. 8Y , f(Y j ) = f(Y j 0) ) = 0). In the models we will consider, data have a Gaussian distribution, so identifiability in this case means identifiability from the mean and the covariance of Y , hence from the covariance of Y , since its mean is zero. Another indicator of identifiability is regularity of the Fisher Information Matrix (FIM). This point of view is not equivalent however [10]. In particular, discrete valued ambiguities cause unidentifiability but don’t lead to singularity of the FIM. On the basis of the number of parameters (excluding a unitary mixture) compared to the number of equations, we arrive at : Necessary condition M > N m + 12 Sufficient condition In the Gaussian model, the m-channel H is identifiable blindly up to a unitary static mixture factor if 1. (i) The channel is irreducible and column reduced. 2. (ii) M L (L = lN p m pm). Identifiability means identifiability from RYY , which is equivalent to having enough data such that T (H) is tall, which leads to (ii), and full column rank, which leads to (i). We have 2 v = min(RYY ). H can be identified from the de-noised RYY by linear prediction [13] or Schur triangularization (see here under). The unitary static mixture is in fact block diagonal, where the different blocks correspond to channels of the same length. Indeed an arbitrary unitary mixture can be undone by forcing the proper channel lengths for the different users with different channel lengths. Sufficient condition Any minimum-phase channel (i.e. H(z) = H(z)R(z) with H(z) irreducible and column reduced, and R(z) minimum-phase) can be identified up to a static mixture for a largerM . Indeed, linear prediction allows to identify H(z) and the correlation sequence of R(z) [13, 14] from which R(z) can be identified up to a unitary mixture by spectral factorization. B Robustness to channel length overestimation A nice feature of Blind GML methods is their robustness to channel length overestimation. Indeed, let Syy(z) = 2 aH(z)Hy(z) + 2 vIm (7) where H is of length N1 (with some users having possibly shorter channels). Now, take H of length N1 N1, then, asymptotically, GML givesH such that : H(z)Hy(z) = H(z)Hy(z) ) H(z) = H(z) (z) ; (z) y(z) = Ip (8) and (z) is a p p FIR filterof length N1 Np (assuming N1 N2 Np). Now, a lossless FIR square transfer function is necessarily of the form (z) = diagfz n1 ; : : : ; z npg (9) 0 nj N1 N j 1; j = 1; : : : ; p where H = Ip. This means that GML based methods (as well as Linear Prediction methods and the Schur method developped here under) yield consistent channel estimates, even when the channel lengths have been overestimated. C Prediction Based GML Let P(z) = PLi=0 p(i)z i with p(0) = Im be the MMSE multivariate prediction error filter of order L for the noisefree received signal Y (k). If L L = lN p m pm, then it can be shown [13] that T (P )T (H) = T (h(0)), or equivalently P(z)H(z) = h(0). From this expression, it is clear that H(z) and fP(z);h(0)g are equivalent parameterizations. Expressing T (P )Y N (0; 2 aT (h(0))T H (h(0)) + 2 vT (P )T H (P )), we can apply the GML procedure described here above, and minimize w.r.t. fP ;h(0)g [5]. A consistent estimate to initialize the IQML procedure can be obtained by a linear prediction algorithm (e.g. the multichannel Levinson algorithm).