The Hilbert space geometry of the stochastic Rihaczek distribution

Beginning with the Cramer-Loeve spectral representation for a nonstationary discrete-time random process, one may derive the stochastic Rihaczek distribution as a natural time-frequency distribution. This distribution is within one Fourier transform of the time-varying correlation and the frequency-varying correlogram, and within two of the ambiguity function. But, more importantly, it is a complex Hilbert space inner product, or cross-correlation, between the time series and its one-term Fourier expansion. To this inner product we may attach an illuminating geometry. Moreover, the Rihaczek distribution determines a time-varying Wiener filter for estimating the time series from its local spectrum, the error covariance of the estimator, and the related time-varying coherence. The squared coherence is the magnitude-squared of the complex Rihaczek distribution, normalized by its time and frequency marginals. It is this squared coherence that determines the time-varying localization of the time series in frequency. Most of these insights extend to the characterization of time-varying and random channels, in which case the stochastic Rihaczek distribution is a fine-grained characterization of the channel that complements the coarse-grained characterization given by the ambiguity function.