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