Theory of higher-order Rihaczek spectra

The Rihaczek distribution was originally proposed as a second-order complex valued time-frequency correlation function for energy signals. In this paper, we show that the generalization to higher orders and to stochastic processes arises naturally from considerations of the nonstationary moment functions. These so-called higher-order Rihaczek spectra (HORS) are shown to exhibit many interesting, important, and desirable properties as measures of the time and frequency behavior of stochastic processes. We show that the n-th order HORS is to be interpreted as the distribution of complex correlation between a stochastic process and an (n − 1) dimensional (multifrequency) random field. Under stationarity assumptions, we found that the HORS collapses to the conventional stationary moment spectrum. The frequency and time marginals of the n-th order are the n-th order instantaneous moment, and the n-th order stationary moment spectrum, respectively. Finally, we evaluated the n-th order HORS for the output process from a nonstationary channel excited by a white, stationary, and non-Gaussian source, and we found that all orders of the HORS contain useful information about the time-frequency behavior of the channel.