Polynomial Wigner-Ville distributions

Abstract

We propose a representation of the derivitive (phi) ' of any general polynomial (phi) of degree N in terms of Q equals N + 1 given parameters: t1,...,tQ. This representation allows us to express the derivative as a linear comination of q arbitrary ratios of [(phi) (t + (tau) (kappa )) - (phi) (t - (tau) (kappa ))]/(tau) (kappa ) calculated at q arbitrary points (tau) 1,...,(tau) q, where q denotes the integer part of (N + 1)/2. The coefficients appearing in this decomposition are independent of the polynomial coefficients. As an application, we give a formula that allows us to compute (phi) '(t) without using the coefficients of the polynomial (phi) (t) and establish a property of the polynomial Wigner-Ville distribution.