A theory of polyspectra for nonstationary stochastic processes

Harmonizable processes constitute an important class of nonstationary stochastic processes. In this paper, we present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic quantities: the nth-order moment function, the nth-order time-frequency polyspectrum, the nth-order ambiguity function, and the nth-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous nth-order moment and the conventional nth-order stationary polyspectrum, respectively. All quantities except the nth-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order n, which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels.