A Modified Wigner distribution function is a variation of the Wigner distribution function (WD) with reduced or removed cross-terms.
The Wigner distribution (WD) was first proposed for corrections to classical statistical mechanics in 1932 by Eugene Wigner. The Wigner distribution function, or Wigner–Ville distribution (WVD) for analytic signals, also has applications in time frequency analysis. The Wigner distribution gives better auto term localisation compared to the smeared out spectrogram (SP). However, when applied to a signal with multi frequency components, cross terms appear due to its quadratic nature. Several methods have been proposed to reduce the cross terms. For example, in 1994 L. Stankovic proposed a novel technique, now mostly referred to as S-method, resulting in the reduction or removal of cross terms. The concept of the S-method is a combination between the spectrogram and the Pseudo Wigner Distribution (PWD), the windowed version of the WD.
The original WD, the spectrogram, and the modified WDs all belong to the Cohen's class of bilinear time-frequency representations :
where
Mathematical definition
Cohen's kernel function :
where
Cohen's kernel function :
The spectrogram cannot produce interference since it is a positive-valued quadratic distribution.
Cohen's kernel function :
Note that the pseudo Wigner can also be written as the Fourier transform of the “spectral-correlation” of the STFT
In the pseudo Wigner the time windowing acts as a frequency direction smoothing. Therefore, it suppresses the Wigner distribution interference components that oscillate in the frequency direction. Time direction smoothing can be implemented by a time-convolution of the PWD with a lowpass function
Cohen's kernel function :
Thus the kernel corresponding to the smoothed pseudo Wigner distribution has a separable form. Note that even if the SPWD and the S-Method both smoothes the WD in the time domain, they are not equivalent in general.
Cohen's kernel function :
The S-method limits the range of the integral of the PWD with a low-pass windowing function
Note that in the original 1994 paper, Stankovic defines the S-methode with a modulated version of the short-time Fourier transform :
where
Even in this case we still have