Polynomial phase signal analysis based on the polynomial derivatives decompositions

An exact decomposition of the derivatives of any order of a polynomial phi(t) is proposed in terms of phi(t-t(0))..., phi(t-t(n)). This result allows us to introduce generalized time-frequency distributions for studying signals having a polynomial phase and a constant amplitude in order to determine the degree and the coefficients of the corresponding phase. The relationships between these distributions and the already known polynomial distributions, i,e,, the polynomial phase transform and the polynomial Wigner-Ville distribution, are discussed. Illustrations by example are proposed.