Wigner-Ville distribution and cross Wigner-Ville distribution of noisy signals

The Wigner-Ville distribution (WVD) and the cross Wigner-Ville distribution (XWVD) have been shown to be efficient in the estimation of instantaneous frequency (IF).But the statistical result of the IF estimation from XWVD peak is much better than using WVD peak.The reason is given from a statistical point of view.Theoretical studies show that XWVD of the analyzed signal can be estimated from XWVD of the noise-contaminated signal.The estimation is unbiased,and the variance is equal to that of noise.In this case,WVD cannot be estimated from WVD of the noise-contaminated signal.Therefore,higher SNR is required when WVD is used to analyze signals.