Multichannel Blind Deconvolution: Natural Gradient Approach
Andrzej S Cichocki, Шун-ичи Амари · 2002
The main objective of this chapter is to review and extend existing adaptive natural gradient algorithms for various multichannel blind deconvolution models. Blind separation/deconvolution of source signals has been a subject under consideration for more than two decades. There are significant potential applications of blind separation/deconvolution in various fields, for example, wireless telecommunication systems, sonar and radar systems, audio and acoustics, image enhancement and biomedical signal processing (EEG/MEG signals). In these applications, single or multiple unknown but independent temporal signals propagate through a mixing and filtering medium. The blind source separation/deconvolution problem is concerned with recovering independent sources from sensor outputs without assuming any a priori knowledge of the original signals, except certain statistical features. In this chapter, we present using various models and assumptions, relatively simple and efficient, adaptive and batch algorithms for blind deconvolution and equalization for single-input/multiple-output (SIMO) and multiple-input/multiple-output (MIMO) dynamical minimum phase and non-minimum phase systems. The basic relationships between standard ICA/BSS (Independent Component Analysis and Blind Source Separation) and multichannel blind deconvolution are discussed in detail. They enable us to extend algorithms derived in the previous chapters, in particular, the natural gradient approaches for instantaneous mixture to convolutive dynamical models. We also derive a family of equivariant algorithms and analyze their stability and convergence properties. Furthermore, a Lie group and Riemannian metric are introduced on the manifold of FIR filters and using the isometry of the Riemannian metric, the natural gradient on the FIR manifold is described. Based on the minimization of mutual information, we present a natural gradient algorithm for the causal minimum phase, the finite impulse response (FIR) multichannel filter. Using information back-propagation, we also discuss an efficient implementation of the learning algorithm for the non-causal FIR filters. Computer simulations are also presented to illustrate the validity and good learning performance of the described algorithms.