Research On Joint Diagonalization Algorithm Of Asymmetric Matrix

Blind source separation is one of the classical and significant issues in blind signal processing,which has important applications in a great number of fields such as military radar,communications,and medicine.Blind source separation means to recover unobserved source signals from their observed source signals without any information of the mixing system.The covariance matrices or cumulant matrices of the observed signal has a structure that can be approximately diagonalized.The method of obtaining the estimation of the source signal is called the joint diagonalization algorithm of matrices,which is a very effective algebraic algorithm for solving blind source separation.It has been widely used in the problem of blind source separation.According to whether the joint diagonalization has orthogonality,the algorithm is divided into orthogonal joint diagonalization and non-orthogonal joint diagonalization of matrices;according to whether the structure is symmetric(Hermitian),it is divided into symmetric(Hermitian)joint diagonalization and non-symmetric(non-Hermitian)joint diagonalization algorithm.Because of the good structural properties of symmetric(Hermitian)matrices,many excellent algorithms have been designed.Relatively,there are fewer joint diagonalization algorithms for non-Hermitian matrices,which are difficult to design.This paper mainly studies the non-Hermitian joint diagonalization algorithm based on the idea of least squares and the idea of Givens rotation.The details are as follows:1.Based on the subspace fitting cost function of least squares,the left and right diagonalizers are decomposed according to the column vector,update a column of vectors of the left and right diagonalizers at each step.Finally,the cost function is converted to solve the left and right singular vectors problem corresponding to the largest singular value of the corresponding matrix.At the same time,we also analyzed the convergence of the algorithm.2.Inspired by the ideas of the algorithm for solving the eigenvalues of the Hermitian matrices by Givens rotation and the Jacobi-like algorithm for joint diagonalization of Hermitian matrices,a generalized Givens iterative matrices containing parameters is constructed,and the cost function based on the Frobenius norm is established.The cost function contains the higher-order terms of the desired parameters,which leads to the difficulty of solving.Therefore,the corresponding joint diagonalization algorithm of non-Hermitian matrices is obtained by omission and simplification under two reasonable assumptions.Finally,the numerical experiment shows that the algorithm is fast,stable and effective.