Wigner distribution of noisy signals

The effect of noise on the Wigner distribution is analyzed. The mean and variance of the Wigner distribution of signals contaminated by additive noise are derived. It is shown that in the case of white noise the Wigner distribution calculated by definition calculated by definition cannot be used for estimation. For estimation, windowing in the time domain is not sufficient; smoothing in the time domain is necessary. Very simple expressions for variance and bias estimation are obtained. If the signal and noise are analytic, only one window is sufficient to make the variance finite. The results are demonstrated by numerical examples with linear and sinusoidal chirp pulses.