Toeplitz and Hankel kernels for estimating time-varying spectra of discrete-time random processes

For a nonstationary random process, the dual-time correlation function and the dual frequency Loeve spectrum are complete theoretical descriptions of second-order behavior. That is, each may be used to synthesize the random process itself, according to the Cramer-Loeve spectral representation. When suitably transformed on one of its two variables, each of these descriptions produces a time-varying spectrum. This spectrum is, in fact, the expected value of the Rihaczek distribution. In this paper, we derive two large families of estimators for this spectrum: one based on a diagonal-Toeplitz-diagonal (dTd) factorization of smoothing kernels and the other based on a diagonal-Hankel-diagonal (dHd) factorization. The dTd factorization produces noncoherent averages of the time-varying spectrogram, and the dHd factorization produces coherent averages. Some of the dTd estimators may be called time-varying power spectrum estimators, and some of the dHd estimators may be called time-varying Wigner-Ville (WV) estimators. The former may always be implemented as multiwindow spectrum estimators, and in some casts, they are true time variations on the Blackman-Tukey-Rosenblatt-Grenander (BTGR) spectrogram. The latter are variations on the Stankovic class of WV estimators.