Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping

Abstract

By deriving a factorization of Toeplitz matrices into the product of Vandermonde matrices, we demonstrate that the Euclidean norm of a filtered signal is equivalent with the Euclidean norm of the appropriately frequency-warped and scaled signal. In effect, we obtain an equivalence between the energy of frequency-warped and filtered signals. While the result does not provide tools for warping per se, it does show that the energy of the warped signal can be evaluated efficiently, without explicit and complex computation of the warped transform. The main result is closely related to the Vandermonde factorization of Hankel matrices.