Research On Complex-valued Non-orthogonal Joint Diagonalization Algorithms For Blind Source Separation

Blind source separation (BSS) is a traditional yet challenging topic in signal processing, in which the separation is performed only depends on the observed signals, without knowing source signals and the mixing channels. BSS has important theoretical significance and practical value, which is widely used in the field of speech signal processing, array signal processing, biomedical engineering, etc.As a class of algebraic algorithms for BSS, joint diagonalization estimates the mixing matrix of mixture model with the specific joint diagonalization structure of target matrices, according to the statistical properties of source signals. According to whether there is an orthogonal constraint imposed on the mixing matrix, these algorithms can be classified into orthogonal and non-orthogonal ones. In order to meet the orthogonal constraint imposed on the mixing matrix, the orthogonal joint diagonalization algorithms must prewhiten the observed signals, however, this preprocess would introduce some errors. For this reason, non-orthogonal joint diaogonalization without prewhitening receives increasing attention. So far, most non-orthogonal joint diagonalization algorithms are only suitable to process real-valued signals. However, some complex-valued signals usually need to be processed in practical applications, such as speech signals in frequency-domain, the electromagnetic vector sensor array signals, etc. For this purpose, this paper focuses on complex-valued non-orthogonal joint diagonalization algorithms for complex-valued BSS. The main contributions are summarized as follows:●Firstly, a complex-valued non-orthogonal joint diagonalization algorithm based on Givens and Hyperbolic rotations is proposed. The solution for Givens rotation corresponds to orthogonal joint diagonalization. The solution for Hyperbolic rotation turns into a generalized eigen-decomposition of matrices with Lagrange multiplier method. Simulation results show that the proposed algorithm provides a more accurate separation performance than its competitors, when target matrices are four-order cumulant.●Secondly, two complex-valued non-orthogonal joint diagonalization algorithms based on LU/LQ factorization are proposed. The high-dimensional mixing matrix are factorized into triangular matrix and unitary matrix, which further turns into a series of low-dimensional sub-problems with one or two unknown parameters. Simulation results show that the two proposed algorithms can provide a more accurate separation performance than their competitors, in the difficult case that fewer target matrices are available. Moreover, in the presence of high noise level, the algorithm based on LQ factorization still maintains fast convergence.●Based on the above, aiming at the symmetric complex-valued target matrices, two non-orthogonal joint diagonalization algorithms based on LU/LQ factorization are proposed. So far, the algorithms suitable to this structure are very few. Compared with ACDC algorithm, simulation results show that two proposed algorithms have advantage in both accuracy and speed of convergence.●Finally, the above proposed non-orthogonal joint diagonalization algorithms are applied to speech separation. After solving the inherent amplitude and permutation ambiguity of BSS in frequency-domain, the artificial and the real mixture speech are separated, respectively. Simulation results demonstrate that five proposed algorithms are able to separate convolutive speech mixtures blindly.