Analysis and Filtering Using the Optimally Smoothed Wigner Distribution

We consider the analysis and filtering of a deterministic signal with slowly time-varying spectra (or, simply, a slowly time-varying signal) using the optimally smoothed Wigner distribution (OSWD). We compare this mixed time-frequency representation (MTFR) to other MTFR's such as the spectrogram, the short-time Fourier transform (STFT), and the Wigner and pseudo-Wigner distributions. We show that the normalized average frequency of the OSWD of a slowly time-varying signal whose phase can be locally expanded in a quadratic function of time yields the exact instantaneous frequency even if the signal is amplitude modulated. Using concepts from modulation theory, we show that the bandwidth (BW) of the OSWD is almost independent of the length of the analysis window, unlike the BW of the spectrogram. We also consider the decomposition of a slowly time-varying signal into its components in the presence of noise by masking the optimal STFT, which is the STFT computed with the window of the OSWD. We propose a new approach to designing linear time-varying filters for slowly time-varying signals which is based on the concept of local nonstationarity cancellation, and we show that it is equivalent to masking the optimal STFT. The performance of the filter in suppressing white noise, and in decomposing a slowly time-varying signal into its components is studied and compared to the performance of the techniques based on the STFT.