Sylvester matrix

Abstract: In this note, we apply a hierarchical identification principle to study solving the Sylvester and Lyapunov matrix equations. In our approach, we regard the unknown matrix to be solved as system parameters to be identified, and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions. We prove that the iterative solution consistently converges to the true solution for any initial value, and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor (or step-size) appropriately. Furthermore, the iterative method is extended to solve general linear matrix equations. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.

Abstract: This paper develops a fast maximum likelihood method for estimating the impulse responses of multiple FIR channels driven by an arbitrary unknown input. The resulting method consists of two iterative steps, where each step minimizes a quadratic function. The two-step maximum likelihood (TSML) method is shown to be high-SNR efficient, i.e., attaining the Cramer-Rao lower bound (CRB) at high SNR. The TSML method exploits a novel orthogonal complement matrix of the generalized Sylvester matrix. Simulations show that the TSML, method significantly outperforms the cross-relation (CR) method and the subspace (SS) method and attains the CRB over a wide range of SNR. This paper also studies a Fisher information (FI) matrix to reveal the identifiability of the M-channel system. A strong connection between the FI-based identifiability and the CR-based identifiability is established.