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

A novel method is proposed for determining 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 corresponding irreducible elementary divisors in its Smith form is established. These special Toeplitz matrices are constructed from certain derivatives evaluated at the Smith zeros of the 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. This is extended to find the finite frequency structure of a rational matrix via a polynomial matrix fraction description. Some neat and numerically stable algorithms are developed.