Research On Algorithms Based On The Utilizations Of Parametric Structures For Blind Source Separation

Blind source separation (BSS) aims to estimate the source signals, i.e. to demix the mixtures captured by a number of sensors, without knowledge about the actual sources or the mixing procedure. In the past two decades, BSS has received much attention in various fields, such as speech and audio processing, wireless communication, image recognition and enhancement, data analysis and biomedical signal processing, etc. This dissertation attempts to do some research on BSS algorithms based on the utilizations of parametric structures. The main contributions of this dissertation are summarized as follows:1. The joint diagonalization structure in instantaneous BSS case is studied. The identifiability theorems of both the orthogonal joint diagonalization based BSS and non-orthogonal joint diagonalization based BSS are introduced, providing a theoretical basis for joint diagonalization based BSS algorithms. Based on these, a multi-step algorithm (MSA) for orthogonal joint diagonalization is proposed to solve BSS problem. This algorithm manages not only to recover all source signals simultaneously but also to extract source signals one by one. Compared with SOBI, which is one classic algorithm for orthogonal joint diagonalization, the MSA has better estimation performance.2. Detailed analysis on merits and drawbacks of existed cost functions for joint diagonalization is done. Thereafter, a fast algorithm, named CVFFDIAG (Complex-Valued Fast Frobenius DIAGonalization), for seeking the non-unitary approximate joint diagonalizer of a given set of complex-valued target matrices is proposed. The proposed algorithm adopts a multiplicative update to minimize the Frobenius-norm formulation of the approximate joint diagonalization problem. In each of multiplicative iterations, a strictly diagonally-dominant updated matrix is obtained. This scheme ensures the invertibility of the diagonalizer and thus guarantees the avoidance of the trivial solution to Frobenius-norm cost function. The special approximation of the cost function, the ingenious utilization of some structures and the skilful denotation of concerning variables result in the highly computational efficiency of the algorithm. Furthermore, the CVFFDIAG relaxes several constraints on the target matrices, e.g. unitarity and positive-definiteness assumptions or the real-valued or Hermitian assumption, and thus has more general utilizations. Detailed computational load analysis also shows the low computational complexity of the algorithm. Extensive numerical simulations are performed to illustrate the fast convergence and good performance of the CVFFDIAG. Due to these obvious merits, the CVFFDIAG could be used to solve many problems. We take the algorithm being used to estimate the DOA estimation and harmonica retrieval in array signal processing area as an example to show its various utilizations.3. The joint block diagonalization problem of convolutive BSS is paid much attention. The multi-step algorithm for orthogonal joint diagonalization to solve the instantaneous BSS problem is parallel extended to orthogonal joint block diagonalization to solve the convolutive BSS problem.4. An non-orthogonal joint block-inner diagonalization (JBID) algorithm, getting rid of the pre-whitening operation, is proposed. The discarding of pre-whitening operation ensures the avoidance of whitening errors and preserves the block Toeplitz structure of the mixing matrix. By fully considering the block-inner diagonalization structure of source signals correlation matrices at different time lags and the block Toeplitz structure of the mixing matrix, the JBID attains to recover all source signals in only one step.5. After careful analysis, we discover that there are abundant structure traits contained in parameters of convolutive BSS when the source signals are assumed to be real and stationary. These traits which could be used in our new algorithm are as follows. The mixing matrix possesses Toeplitz structure. The correlation matrices of source signals at successive time lags are block Toeplitz and block-inner diagonalization. Furthermore, these correlation matrices have many common entries. And the coordinates of these entries could be known ahead as a prior knowledge. 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 to develop 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). Since the corresponding variables are highly structured, the derivation procedures in three sub-steps of TIA appear complex and need to be dealt with skill. But we still clearly explain the derivation procedures. We also take a close look at the computational complexity of our algorithm. This algorithm also discards the pre-whitening operation, manages to recover all source signals in only one step, and possesses good performance. Moreover, the asymptotical convergence of JBTBID is shown.6. We furthermore take a close look at the parametric structure traits of convolutive BSS and introduce three kinds of structures. We now introduce the representative one among them. Following all assumptions of JBTBID, when the received data are arrayed in different way, different parametric structures can be derived. Concretely speaking, the mixing matrix is block-inner Toeplitz. The correlation matrices of source signals at successive time lags possess block diagonalization and block-inner Toeplitz structures. Also, these correlation matrices have many common entries and the coordinates of these entries could be known ahead. It is obvious that, all these structure traits could be utilized when considering a new algorithm for the convolutive BSS problem.