A Unifying Framework for Blind Source Separation Algorithms Based on Generalized Eigen-value Decomposition
Changli Li · 2022
In this paper, we present a unifying framework for linear blind source separation (BSS) via generalized eigen-value decomposition (GEVD). The derived algorithms via GEVD for BSS turn the underlying optimization problem into a GEVD problem, thus they can be easily implemented. There are two classes of algorithms via GEVD for BSS. The first class of algorithm directly accomplishes the GEVD of a matrix pencil which is composed of observed signals from sensors in a practical system. The second class of algorithm constructs some cost function with a form of generalized Rayleigh quotient (GRQ), and its corresponding separation vector can be easily obtained by GEVD. However, algorithms of this type lack reasonable explanation. In this paper, we focus on proposing a unifying framework for BSS via GEVD. Besides, it is pointed out that their separability relies on source signals’ non-property: non-Gaussianity (statistical non-property), non-stationarity (time non-property), or non-whiteness (frequency non-property). Hence, the non-property is the essential characteristic of all GEVD-based algorithms for BSS. Under the proposed unifying framework, we prove that any generalized eigenvector of the GEVD problem is the same as the de-mixing vector which can successfully separate one of the original source signals. Finally, simulation experiments are made and theoretical explanation for our unifying framework is given.