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 Loéve spectrum are complete theoretical descriptions of second-order behavior. That is, each may be used to synthesize the random process itself, according to the Cramér–Loève 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 cases, they are true time variations on the Blackman–Tukey–Rosenblatt–Grenander (BTGR) spectrogram. The latter are variations on the Stankovic class of WV estimators.