A balanced approach to multichannel blind deconvolution

In some general state-space approaches to the multichannel blind deconvolution problem, e.g., the information backpropagation approach (Zhang and Cichocki 2000), an implicit assumption is usually involved therein, viz., the dimension of the state vector of the mixer is known a priori. In general, if the number of states in the state space is not known a priori, Zhang and Cichocki (2000) suggested using a maximum possible number of states; this procedure will introduce additional delays in the recovered source signals. In this paper, our aim is to relax this assumption. The objective is achieved by using balanced parameterization of the underlying discrete-time dynamical system. Since there are no known balanced parameterization algorithms for discrete-time systems, we need to go through a "circuitous" route, by first transforming the discrete-time system into a continuous-time system using a bilinear transformation, perform the balanced parameterization on the resulting continuous-time system, and then transform the resulting system back to discrete-time balanced parameterized system using an inverse bilinear transformation. The number of states can be determined by the number of significant singular values in the ensuing singular value decomposition step in the balanced parameterization.