Polyspectra for harmonizable stochastic processes

Harmonizable processes form an important class of nonstationary stochastic processes. In this paper we present a theory of higher-order moment spectra (polyspectra) for the harmonizable class. We show that there are four basic quantities, the n-th order moment function, the n-th order time-varying polyspectrum, the n-th order ambiguity function, and the n-th order spectral correlation function that need to be considered. We offer interpretations of the polyspectra in terms of Hilbert space inner products. Based on the inner product interpretations, we derive two novel definitions of n-th order polycoherence. The polycoherences are objective measures of stationarity to order n, which can be used to construct various statistical tests.