148 IEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 5, MAY2003 A Fast Blind SIMO Channel Identification Algorithm for Sparse Sources David Luengo, Student Member, IEEE, Ignacio Santamaría, Member, IEEE, Jesús Ibáñez, Luis Vielva, and Carlos Pantaleón, Member, IEEE Abstract—We address the blind identification of single-input- multiple output (SIMO) finite impulse response systems when the input signal is sparse. The problem is equivalent to under- determined blind source separation (BSS), but with temporal correlation among the sources. Exploiting the sparse character of the input signal, the algorithm solves three different problems: first, to estimate the directions of the columns of the channel ma- trix; second, to estimate the -norm of the columns; and finally, to find the correct ordering of the columns of the mixing matrix. The last step is not required for the blind source separation (BSS) problem, since any permutation of the columns is admissible for BSS. The performance and computational cost of the algorithm in a noiseless situation is compared against subspace-based techniques. Index Terms—Blind channel identification, blind source separa- tion (BSS), sparse deconvolution. I. INTRODUCTION B LIND channel identification is an important and widely studied problem that appears in many signal processing applications: equalization, seismic data deconvolution, speech coding, image deblurring, etc. When the channel impulse response has finite support and the received signal is oversam- pled, the problem can be formulated as the blind identification of a single-input multiple-output (SIMO) finite-impulse re- sponse (FIR) system. In this situation, it is known that, as long as the FIR channels have no common zeros and the channel is fully excited, the SIMO system can be identified using only second-order statistics of the output [1]. Several extensions of this idea using subspace-based methods [2] or linear prediction techniques [3] have proven useful to solve this problem. An iterative algorithm based on second-order statistics is presented in [4] where the order of the system is also estimated for channels with rational transfer functions. The main drawback of all these algorithms, however, is their high computational cost. Therefore, there is still a need for low-cost, fast algorithms that can deal with the large datasets typically available in some applications such as seismic deconvolution or nondestructive evaluation. Since, typically, the number of columns of the SIMO channel matrix is larger than the number of measurements, blind SIMO Manuscript received April 19, 2002; revised September 3, 2003. This work was supported in part by MCYT (Ministerio de Ciencia y Tecnología) under Grant TIC2001-0751-C04-03. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Petr Tichavsky. The authors are with the Department of Communications Engineering (DICOM), Universidad de Cantabria, 39005 Santander, Spain (e-mail: nacho@gtas.dicom.unican.es). Digital Object Identifier 10.1109/LSP.2003.810014 identification can be viewed as a problem of underdetermined blind source separation (i.e., more sources than sensors), but with temporal correlation among the sources. On the other hand, recent advances in BSS have shown that, when the sources are sparse, it is possible to solve the underdetermined case using simple algorithms [5]–[7]. The general idea of these techniques is that, when the sources are sparse, the measurements tend to cluster along the directions imposed by the columns of the mixing matrix, which can then be easily identified. Using similar ideas, in this letter we propose a fast blind SIMO channel identification algorithm for sparse sources with application to seismic deconvolution, nondestructive evaluation, or blurred star image deconvolution. Exploiting the sparsity of the input signal, a blind technique must solve three different problems. First, it is necessary to estimate the directions of the columns of the channel matrix; once the directions have been identified, the norm of the columns must be estimated. These two steps solve the blind source separation (BSS) problem, since any permutation of the columns of the mixing matrix is admis- sible. However, in a deconvolution problem the ordering of the columns of the channel matrix is also required. In this letter we describe a simple algorithm to solve these three steps. II. PROBLEM STATEMENT Assuming that the received signal is oversampled by a factor and that the maximum length of each of the FIR channels is ; then, in a noiseless situation, the problem can be formulated as (1) where is an matrix, formed by stacking successive observations ; is a matrix of input signals, with ; and is an channel or mixing matrix, with . Our problem consists of estimating the channel matrix using only the observations and some statistical knowledge of the input signal. Specifically, in this letter, we assume that the input signal is a sparse spike train, modeled as a statistically independent zero-mean Bernoulli–Gaussian (BG) sequence, which is commonly used in nondestructive evaluation and seismic deconvolution [8]. The BG samples are generated according to (2) 1070-9908/03$17.00 © 2003 IEEE