A semi-blind maximum likelihood approach for resolving linear convolutive mixtures

We introduce a new technique to separate a linear convolutive mixture of discrete-time sources, emitting uncorrelated data samples. The proposed approach works in the reduced dimension space of the (channel) whitened data samples, where the data matrix is highly structured: it is the product of an orthogonal and generalized Toeplitz matrices embedded in additive Gaussian noise. By itself, this factorization does not unambiguously determine the sources (even in the absence of noise), but uniqueness is restored if short fragments (inadequate for training) of the emitted messages are known beforehand. We present a locally convergent iterative algorithm which implements the joint maximum likelihood (ML) estimator of both the orthogonal mixing matrix and the user signals, subject to the known side information. We also discuss a simple (sub-optimal) adaptation of the proposed algorithm to the class of finitealphabet (FA) sources. Computer simulations show that the proposed technique permits to outperform an alternative subspace approach, specially in low signal-to-noise ratio (SNR) scenarios. 1. PROBLEM FORMULATION Methods for separating linear convolutive mixtures find direct application in many important signal processing areas, e.g., in space division multiple access (SDMA) wireless networks, where the base station has to discriminate a linear superposition of user signals transmitted over multipath channels see 11, 2, 3, 41 and the references therein. The conditional or unconditional ML techniques in 11, 21 address the problem of instantaneous mixtures for FA sources, whereas (3, 41 are not full ML approaches. Here, we introduce a semi-blind ML based technique to resolve convolutive mixtures. The side information is needed because the proposed technique is not restricted to FA sources, and uses only 2nd order statistics. The paper is organized as follows. Section 2 establishes the data model and states the main assumptions. Section 3 describes the joint ML estimator of the orthogonal mixing matrix and the user signals. Section 4 addresses some computational and implementation issues concerning the proposed algorithm. In section 5 , we compare our proposed technique with the subspace approach [3], through computer simulations. Notat ion. Matrices (capital) and vectors are in boldface type. RnX" is the set of n x m matrices with real entries. The notations ( . ) T , (.)+, 8, 0, t r {.}, and vec (.) stand for the transpose, Moore Penrose pseudo inverse, Kronecker product, Hadamard product, the trace, and the vectorization operator, respectively. The symbols I , and O n X , represent the n x n identity matrix and the all-zero n x m matrix, respectively (when the dimensions are clear from the context, we drop the subscripts). Within WX", we single out the subsets 9, = { Q : QTQ = I , } (group of orthogonal matrices), R, = { H : H = HT } (subspace of symmetric matrices) and its orthogonal complement K, = { K : K = -KT } (subspace of skew-symmetric matrices). For a vector 8 = [ 81 192 . . . 01 1' E RI, TnX, (e) denotes the n x m Toeplitz matrix generated by 8, i . e . , e' 1 r en e,, en+2 . . . where m + n 1 = 1. Diagonal concatenation of the matrices Ai, i = 1 , 2 , . . . , n (not necessarily sharing the same dimensions) is denoted by diag (AI, . . . , An). We write a: N (p , C ) , if the random vector a: is Gaussian distributed with mean p and covariance matrix C . 2. DATA MODEL AND ASSUMPTIONS Consider a noisy linear convolutive mixture of P discretetime sources, Here, a : ( k ) E WN denotes the vector of channel outputs, {h, ( 1 ) E RN : 1 = 0,1, . . . , L, l } is the finite-impulse response (FIR) of the multichannel filter corresponding to the pth user (L , denotes the pth user channel length), 0, ( k ) E W is the scalar signal transmitted by the pth user, and w ( k ) E WN denotes additive noise. Equation (1) may be 0-7803-6293-4/00/$10.00 02000 IEEE. 2809 compactly rewritten as system outputs, R, = E {z ( k ) 2 ( k l T } = p v u ~ + ~ 2 r N , (3)