Algorithms For Non-orthogonal Joint Diagonalization And Its Application In Blind Source Separation

Blind source separation is a traditional and challenging problem in the signal processing. The purpose of source separation is to extract the best estimate of the source signal just corroding to the statistics of observed signals and the independence(uncorrelation) assumption of source signals without knowing source signals and mixing process. As a kind of very effective algebraic algorithm to solve the separation of blind source signals, the joint of the matrix has been paid much attention in recent years. Joint diagonalization is divided into two types of orthogonal and non orthogonal,due to the orthogonal joint diagonalization to the observed signal is pre whitened, this will cause some errors, thus the separation effect have nothing can compensate for the influence and non orthogonal joint diagonalization can does not satisfy the orthogonality constraint conditions, so there is no need to do pre whitened, nearly 10 years to non orthogonal joint diagonalization algorithm by extensive attention, has become a research hotspot. In this paper, we also focus on this topic and do the following work:1.We will discuss the specific target matrix obtained by observed signal with diagonalization structure and joint diagonalization method is through recovery this diagonalization structure to estimate the mixing matrix or the de-mixing matrix. Thus will confirm that the non-orthogonal joint diagonalization is effective when solving the actual problem of blind source separation.2. By using an original elementary matrix, we established a Jacobi like algorithm with no approximation, called GERALD1, and two declinations, called GERALD2 a and GERALD2 b. The idea of these two algorithms is based on the so-called Jacobi algorithm for solving the eigenvalues problem of Hermitian matrix. They are based on the search of two complex parameters by the minimization of a quadratic criterion corresponding to a measure of diagonality. These algorithms take the square sum of non diagonal elements as a criterion, to search of two complex parameters. Lastly,simulation 1 show that the GERALD2 b algorithm has the best convergence speed than the GERALD1 algorithm and the GERALD2 a algorithm. Hence, we will only consider the GERALD2 a algorithm and the GERALD2 b algorithm, compare them with other algorithms. Simulation 2-5 show that the GERALD2 b algorithm has the best results interms of the convergence speed and the joint diagonal quality. Simulation 6 show that the result of the real case is the same with the results of the simulations.3. Based on the LU decomposition, we propose three non-orthogonal Jacobi-like alternating iterative algorithms with two strategies for solving the joint diagonalization problem of a set of Hermitian matrices. In this kind of algorithm, each transformation includes one upper triangular iterative step and one lower triangular iterative step, and each step involves one parameter. The optimal parameter of each step is derived analytically. The convergence of our proposed algorithms is proven. According to our convergent analysis, we also correct the decrement of the GNJD algorithm. Lastly, some numerical simulations are conducted to illustrate the effective performance of our developed algorithms and the result of the real case is the same with the results of the simulations.