A novel method to determine the finite and infinite frequency structure of a rational matrix

In this paper a novel method is proposed that determines the finite and infinite frequency structure of any rational matrix. For a polynomial matrix, a natural relationship between the rank information of the Toeplitz matrices and the number of the corresponding irreducible elementary divisors (IREDs) in its Smith form is established. These special Toeplitz matrices correspond to the derivatives at some finite zero of that matrix. This relationship proves to be fundamental and efficient to the study of the finite frequency structure of a polynomial matrix. For a rational matrix, this technique can be employed efficiently to find its infinite frequency structure by examining the finite frequency structure of the dual of its companion polynomial matrix. It can also go through to find the finite frequency structure of a rational matrix via polynomial matrix fraction description (MFD). Some neat and numerically stable algorithms are developed. Corresponding Author: Telephone: Telefax: E-mail: Dr. A. C. Pugh Department of Mathematical Sciences Loughborough University Leicestershire, LE11 3TU, UK +44 (0)1509 223190 +44 (0)1509 211869 [email protected]