Polynomial time–frequency distributions and time-varying higher order spectra: Application to the analysis of multicomponent FM signals and to the treatment of multiplicative noise

Abstract

One of the fundamental problems of time–frequency analysis that remained unsolved until recently is the time–frequency representation of signals with an arbitrary non-linear frequency variation in time. A certain type of higher-order time–frequency distributions (TFDs), referred to as polynomial Wigner-Ville distributions (PWVDs), can achieve delta function concentration in the time–frequency plane for mono-component polynomial FM signals (Boashash and O’Shea, 1994). This paper is a sequel to Boashash and O’Shea (1994) dealing with a broader class of signals and presenting an application to the treatment of multiplicative noise. The first part of the paper presents a general design procedure for PWVDs and the main properties of PWVDs. A specific class of polynomial time–frequency distributions that deals effectively with multicomponent signals is then described. In the second part of the paper we deal with random signals and introduce time-varying higher-order spectra (TV-HOS) an ensemble averaged PWVDs. TV-HOS are shown to be natural tools for analysis of non-stationary random signals, and we demonstrate this in the context of FM signals affected by multiplicative noise. Both moment and cumulant fourth-order TV-HOS, referred to as the Wigner-Ville trispectra are shown to be superior to the second-order methods for this application.