Relationships between instantaneous blind source separation and multichannel blind deconvolution

We present a general algebraic approach to an extended dynamic independent component analysis (EDICA) for multichannel blind signal separation/deconvolution. Precise algebraic equivalence and direct analogies between instantaneous blind source separation (BSS) and dispersive (dynamic) blind signal separation/deconvolution (referred to also as multichannel blind deconvolution, MBD) problems are shown, as well as, the equivalence of the problem in the time domain and the Z-transform domain. For circular convolution the equivalence (analogy) is precise for finite length time series, while for linear convolution such analogy is valid only in the asymptotic sense for infinite length series. Elegant and concise derivation of learning algorithms in the time domain is presented using the algebraic properties of the convolution operator and relationships between convolution and cross-correlation. Using this general concept, unsupervised learning algorithms (both batch and online algorithms) are developed for multichannel blind deconvolution/separation problems. Computer simulation experiments confirm validity and high performance of the proposed algorithms. The proposed approach and some automatic rules can be applied not only to the known, already existing algorithms for blind separation and extraction of sources but we hope could be also used for extension and generalisation of learning rules developed in future