Convolutive blind source separation based on joint block Toeplitzation and block-inner diagonalization

This paper takes a close look at the block Toeplitz structure and block-inner diagonal structure of auto correlation matrices of source signals in convolutive blind source separation (BSS) problems. The aim is to propose a one-stage time-domain algorithm for convolutive BSS by explicitly exploiting the structure in autocorrelation matrices of source signals at different time delays and inherent relations among these matrices. The main idea behind the proposed algorithm is to implement the joint block Toeplitzation and block-inner diagonalization (JBTBID) of a set of correlation matrices of the observed vector sequence such that the mixture matrix can be extracted. For this purpose, a novel tri-quadratic cost function is introduced. The important feature of this tri-quadratic contrast function enables the development of an efficient algebraic method based on triple iterations for searching the minimum point of the cost function, which is called the triply iterative algorithm (TIA). Through the cyclic minimization process in the proposed TIA, it is expected that the JBTBID is achieved. The source signals can be retrieved. Moreover, the asymptotic convergence of the proposed TIA is analyzed. Convergence performance of the TIA and the separation results are also demonstrated by simulations in comparison with some other prominent two-stage time-domain methods.