Research On Several Key Issues In Blind Source Separation

Blind source separation (BSS) has been found wide applications in many scientific areas, especially in the areas of speech source separation and recognition, biological signal processing, and wireless communication, etc. In fact, due to its high value in theory and promising applications in practice, BSS has been one of the hottest topics in signal processing. Currently, many substantial contributions to BSS have been made. However, some important theoretical and practical issues are still unsolved and they will be focused in this dissertation. The main contributions of this dissertation include:1. an improvement to the famous minimum-range (MR) geometric BSS method (GBSS). GBSS provides visible separation procedure and geometric interpretation of separation. The MR method is proved reliable theoretically and has no restrictions on the number of observations. However, the optimization algorithm of the method has disadvantages in reliability and efficiency. By virtue of some favorable properties of a convex hull, an improved MR algorithm (iMR) is proposed in this thesis. It can be seen that iMR is more reliable and efficient. Moreover, the maximum-range criterion is also derived to deal with sparse signals, which leads to an extended range of applications of the MR method.2. a first and in-depth study on the condition number of diagonalizers. Joint diagonalization is one of the most powerful tools to solve BSS problems. However, existing algorithms often can not avoid ill-conditioned solutions strictly. To overcome this disadvantage, the approximate joint diagonalization problem is reviewed as a multi-objective optimization problem for the first time. Then, a new algorithm for nonorthogonal joint diagonalization is developed. The new algorithm yields diagonalizers which have as small condition numbers as possible while minimizing the diagonalization error. Meanwhile, degenerate solutions, trivial solutions and unbalanced solutions are totally avoided strictly. Besides, the new algorithm imposes few restrictions on the target set of matrices to be diagonalized, which makes it widely applicable. Primary results on convergence are presented and we also show that, for exactly jointly diagonalizable sets, no local minima exist and the solutions are essentially unique under mild conditions. The practical use of our algorithm is shown in blind source separation problems, especially when ill-conditioned mixing matrices are involved. Finally, the guided joint diagonalization algorithm (GDiag) is proposed, which is the first applicable joint diagonalization algorithm in online BSS scenario. Indeed, GDiag itself is also a regular joint diagonalization algorithm, although it has some mild restrictions on the set of matrices to be diagonalized.3. an in-depth separability analysis for the covariance rate method. Our results show that the sources are separable via the covariance rate method if and only if they have different temporal structures (i.e., autocorrelations). Consequently, the applicability and limitations of the covariance rate method are clarified. Also, the relation between the covariance rate method and slow feature analysis, and second-order statistics methods is revealed. In addition, instead of using generalized eigendecomposition, joint approximate diagonalization algorithms are introduced to improve the robustness of the method. A new criterion is presented to evaluate the separation results. Numerical simulations are performed to demonstrate the validity of the theoretical results. With these efforts, the reliability and validity of the covariance rate method is considerably enhanced.4. a new mixing matrix estimation technique with unknown source number. Although many approaches have been proposed to estimate the mixing matrix, however, they often need to know either the exact value or the upper bound of the source number (i.e. the number of columns of the mixing matrix) as a priori. In this paper, a new method, called NPCM (nonlinear projection and column masking), is proposed to estimate the mixing matrix. In NPCM, the objective function is based on a nonlinear projection such that its maxima just correspond to the columns of the mixing matrix. Then, the columns of the mixing matrix are estimated and deflated sequentially by locating each maximum followed by a masking procedure. As a result, NPCM does not need any prior information on the source number. Because the masking procedure may cause many small and useless local maxima, particle swarm optimization (PSO) is introduced to optimize the objective function. Feasibility and efficiency of PSO are discussed. Comparative experimental results show that NPCM is reasonably competent in the estimation of the mixing matrix, especially in the case that the number of sources is unknown and the sources are less sparse.5. a new nonnegative matrix factorization (NMF) method to separate highly dependent sources based on minimum-volume constraint (MVC_NMF). NMF is one of the most promising tools to separate statistically dependent sources. The major advantage of our methods is that they can give desirable results even if in very weak sparseness situations. Close relation between NMF and sparseness NMF, MVC_NMF is theoretically analyzed and the reasonability of the new model is justified. Two algorithms are proposed to optimize the objective function. The first one is quite efficient for the relatively small scale problems and the second one is more suitable for larger ones. Simulations show that the new algorithms can not only separate some highly statistically dependent sources but also improve the ability of learning parts of NMF significantly, therefore can be found wide applications in related areas.To summary, this thesis is focused on the theoretical analysis and algorithms development for the linear instantaneous mixing of BSS. Particularly, we perform in-depth study on the covariance rate method and widely applicable joint diagonalization techniques.