The Wigner distribution of noisy signals with adaptive time-frequency varying window

Time-frequency representations using the Wigner distribution (WD) may be si gnificantly obscured by the noise in observations. The analysis performed f or the WD of discrete-time noisy signals shows that this time-frequency rep resentation can be optimized by the appropriate choice of the window length . However, the practical value of this analysis is not significant because the optimization requires knowledge of the bias, which depends on the unkno wn derivatives of the WD, A simple adaptive algorithm for the efficient tim e-frequency representation of noisy signals is developed in this paper. The algorithm uses only the noisy estimate of the WD and the analytical formul a for the variance of this estimate. The quality of this adaptive algorithm is close to the one that could be achieved by the algorithm with the optim al window length, provided that the WD derivatives were known in advance, T he proposed algorithm is based on the idea that has been developed in our p revious work for the instantaneous frequency (IF) estimation, Here, a direc t addressing to the WD itself, rather than to the instantaneous frequency, resulted in a time and frequency varying window length and showed that the assumption of small noise and bias is no longer necessary. A simplified ver sion of the algorithm, using only two different window lengths, is presente d. It is shown that the procedure developed for the adaptive window length selection can be generalized for application on multicomponent signals with any distribution from the Cohen class. Simulations show that the developed algorithms are efficient, even for a very low value of the signal-to-noise ratio.