Blind Identification of Convolutive MIMO Systems with 3 Sources and 2 Sensors
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Blind Identification of Convolutive MIMO Systems with 3 Sources and 2 Sensors Binning Chen Department of Electrical and Computer Engineering, Drexel University, Philadelphia, PA 19104, USA Email: [email protected]
Athina P. Petropulu Department of Electrical and Computer Engineering, Drexel University, Philadelphia, PA 19104, USA Email: [email protected]
Lieven De Lathauwer ´ ´ ´ Equipe Traitement des Images et du Signal, Ecole Nationale Sup´erieure d’Electronique et de ses Applications, Universit´e de Cergy-Pontoise, Cergy-Pontoise, France Email: [email protected] Received 4 July 2001 and in revised form 6 March 2002 We address the problem of blind identification of a convolutive Multiple-Input Multiple-Output (MIMO) system with more inputs than outputs, and in particular, the 3-input 2-output case. We assume that the inputs are temporally white, non-Gaussian distributed, and spatially independent. Solutions for the scalar MIMO case, within scaling and permutation ambiguities, have been proposed in the past, based on the canonical decomposition of tensors constructed from higher-order cross-cumulants of the system output. In this paper, we look at the problem in the frequency domain, where, for each frequency we construct a number of tensors based on cross-polyspectra of the output. These tensors lead to the system frequency response within frequency dependent scaling and permutation ambiguities. We propose ways to resolve these ambiguities, and show that it is possible to obtain the system response within a scalar and a linear phase. Keywords and phrases: MIMO system identification, tensor decomposition, higher-order statistics.
1.
INTRODUCTION
The goal of blind r-input n-output (n × r) system identification is to identify an unknown system H(z), driven by r unobservable inputs, based on the n system outputs. Blind identification of a Multiple-Input Multiple-Output (MIMO) system is of great importance in many applications, such as speech enhancement in the presence of competing speakers, digital multiuser/multiaccess communications systems, biomedical engineering [1, 2, 3, 4, 5]. Most of the literature on n × r MIMO problems refers to the case of n ≥ r. In that case, system identification can lead to recovery of the inputs via deconvolution. Here we consider the case of more inputs than outputs, that is, n < r. In such a scenario, recovery of the input is generally not possible, except in cases where some a priori information about the inputs is available, such as the finite alphabet property [6, 7]. Very few results exist for the convolutive MIMO problem with more inputs than outputs. In [8], the identifiability of a Moving Average (MA) system with possibly more inputs
than outputs has been studied. A special case of a blind 2 × 3 convolutive system, where the cross-channels are simple delay elements, has been studied in [9]. The delays were estimated via a polyspectra based method. The scalar 2 × 3 MIMO case has been approached in [6, 10], based on the canonical decomposition of tensors, w
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