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

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) as 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. (C) 1998 Elsevier Science B.V. All rights reserved