Polynomial Wigner-Ville distributions and their relationship to time-varying higher order spectra

The Wigner-Ville distribution (WVD) has optimal energy concentration for linear frequency modulated (FM) signals. This paper presents a generalization of the WVD in order to effectively process nonlinear polynomial FM signals. A class of polynomial WVD´s (PWVD´s) that give optimal concentration in the time-frequency plane for FM signals with a modulation law of arbitrary polynomial form are defined. A class of polynomial time-frequency distributions (PTFD´s) are also defined, based on the class of PWVD´s. The optimal energy concentration of the PWVD enables it to be used for estimation of the instantaneous frequency (IF) of polynomial FM signals. Finally, a link between PWVD´s and time-varying higher order spectra (TVHOS) is established. Just as the expected value of the WVD of a nonstationary random signal is the time-varying power spectrum, the expected values of the PWVD´s have interpretations as reduced TVHOS