Multi-set Joint Blind Source Separation Based On Tensor Diagonalization

In modern signal processing, driven by many practical problems, the joint blind source separation of multi-set has become one of the frontier research direction. Joint blind source separation can use the statistical property between latent variables of observation data, such as the intra-set independence and the inter-set dependence of the signal. And tensor can characterize the structural features of the high-dimensional data intuitively. If the tensor decomposition and joint blind source separation are combined, it will promote the development of cutting-edge technology in the theory and methods like signal processing by tensor and joint blind source separation. Multi-set joint blind source separation based on tensor diagonalization first construct the target tensors, to fit these tensors by algebra, then identify the mixing mechanism of signal, eventually restore the source signals.The joint blind source separation based on tensor diagonalization is still at the early stage. This paper mainly research the joint blind source separation methods of multi-set signals based on tensor diagonalization:third-order and fourth-order tensor diagonalization by Givens rotations, LU decomposition and successive rotations strategy. And these algorithms are successfully applied to the joint blind source separation of the actual data. The concrete contributions of this paper are summarized as follows:(1) For the problem of third-order orthogonal joint blind source separation, a third-order orthogonal tensor diagonalization by Givens rotations is proposed. The proposed method alternates among all the mixing matrices and updates each of them by a sequence of Givens rotations solved in closed-form. Numerical results are provided to illustrate the strong convergence advantage as well as the separation accuracy performance of the proposed algorithm in comparison with existing methods of similar type. And the application of the fetal electrocardiography (ECG) separation and speech separation expound that the performance of the proposed algorithm.(2) For the problem of fourth-order orthogonal joint blind source separation, an orthogonal tensor diagonalization based on Givens rotations is proposed. Through matrix factorization, the multi-parameter optimization problem is turned into a series if simple eigenvalue decomposition. The simulation experiments prove the strong convergence advantage as well as the separation accuracy. For the experiment of ECG separation, simulation results prove the separating performance of the proposed algorithm.(3) For the problem of fourth-order non-orthogonal joint blind source separation, an algorithm for non-orthogonal tensor diagonalization based on LU decomposition and successive rotations is proposed. The algorithm uses LU decomposition to convert the overall optimization into L and U stages, and then the factor matrices in these stages can be appropriately parameterized by a sequence of simple elementary triangular matrices, which can be solved analytically. Simulations show that the convergence advantage and the separation accuracy of the proposed algorithm. And the algorithm can be applied to solve the fetal ECG separation.