Optimal Kernels for Nonstationary Spectral Estimation

Journal PaperCurrent theories of a time-varying spectrum of a nonstationary process all involve, either by definition or by difficulties in estimation, an assumption that the signal statistics vary slowly over time. This restrictive quasi-stationarity assumption limits the use of existing estimation techniques to a small class of nonstationary processes. We overcome this limitation by deriving a statistically optimal kernel, within Cohen's class of time-frequency representations (TFRs), for estimating the Wigner-Ville spectrum of a nonstationary process. We also solve the related problem of minimum mean-squared error estimation of an arbitrary bilinear TFR of a realization of a process from a correlated observation. Both optimal time-frequency invariant and time-frequency varying kernels are derived. It is proven that, in the presence of any additive noise, optimal performance requires a nontrivial kernel, and that optimal estimation may require smoothing filters very different from those based on a quasi-stationarity assumption. Examples confirm that the optimal estimators often yield tremendous improvements in performance over existing methods. In particular, the ability of the optimal kernel to suppress interference is quite remarkable, thus making the proposed framework potentially useful for interference suppression via time-frequency filtering