Study On Blind Source Separation Of Linear Mixture

Blind source separation(BSS) consists of recovering mutually independent/ uncorrelated but otherwise unobserved source signals from their mixtures without any prior knowledge of the channel. BSS has attracted growing attention in statistical signal processing and unsupervised neural learning society, since it is a fundamental problem encountered in various fields, such as biomedical signal processing, audio and acoustics, multi-user communications, data analysis, and so on. In this dissertation, we focus on the method of algebra and optimizing contrast function to linear mixture BSS, and we investigate the BSS algorithms of joint diagonalization(JD), joint block diagonalization(JBD), tensor decomposition and natural gradient. In addition, some preliminary exploration on BSS of correlated sources are carried out.The primary contributions included in this dissertation are summarized blow:1. Non-orthogonal joint diagonalization could avoid the performance degradation caused by pre-whitening, but the solution is not necessarily only. We analyzed the identifiability for nonorthogonal joint diagonalization(NJD). We proposed the uniqueness condition of the solution to NJD, and pointed out that uniqueness condition for NJD is that the vectors consisting of diagonal elements in the same position of diagonal matrix are pairwise linearly independent. From this proposition,the necessary and sufficient condition for BSS is deduced. For second-order statistics based BSS, the condition is that the source signals have not the identical autocorrelation shape. For higher-order cumulant, there is not Gaussian signal in sources. The above conclusion provides a mathematical foundation for the BSS methods based on the NJD.2. We proposed an algorithm of orthogonal joint block diagonalization based on GIVENS rotation. In this method, we improved Jacobi algorithm which perfect solve the problem of orthogonal joint diagonalization. We transformed the selection of GIVENS matrix's parameter into the optimization of univariate quadruplicate trigonometric function and adjusted the cycle of rotation. So the new algorithm could solve the problem of orthogonal JBD. The new algorithm combination with pre-whitening procedure can solve Multidimensional blind source separation(MBSS) and convolutive BSS.3. In this paper, a fast algorithm of tensor canonical decomposition is proposed through tucker decomposition.Same as the estimation of underdetermined mixture matrix, there are permutation and scaling indeterminacies in factor matrix of canonical decomposition. So the estimation of underdetermined mixture matrix can be performed through tensor canonical decomposition. In order to overcome the flaw of high computational complexity and long running time of existing canconical decomposition algorithm,compress the tensor into lower order core one using tucker decomposition.The factor of tucker decomposition can be obtained by left singular value of the original tensor's mode-3 matrix. The mixture matrix can be estimated by the alternating least squares based canonical decomposition of the core tensor. The proposed algorithm has much lower computational complexity with no performance loss than existing algorithm.4. The accuracy of activation function is an important factor that influences the convergent speed and stability of the natural gradient algorithm. A new estimating activation function approach based on the method of function approximation is proposed. In this approach, activation function was approximated by the linear combination of a set of orthogonal polynomials. The accuracy of approximating is measured by mean square error(MSE). Using the property of score function ,coefficients of the linear combination can be obtained by adaptive minimizing MSE. The new BSS algorithm is developed by substituting the estimated activation function into natural gradient iterative formula. Compared with the traditional one, convergent speed of the new algorithm is highly improved.5. Dependent sources BSS is a difficult problem in this research field. We proposed an algorithm which could resolve the problem of dependent sources BSS of time domain sparseness. For BSS of sparse souces, there are some instants at which only one source existing.The vectors of observation at these instants are the estimation of the corresponding columns at mixing matrix. Using this property, mixing matrix can be estimated correctly. These instants can be obtained by comparing variances of the ratio between two sensors. By this special way, the sparse dependent BSS problem can be resolved.