Modified Wigner distribution function

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 formula_2 is Cohen's kernel function, which is often a low-pass function, and normally serves to mask out the interference in the original Wigner representation.
Mathematical definition.
Cohen's kernel function : formula_4
where formula_6 is the short-time Fourier transform of formula_7. 
Cohen's kernel function : formula_9 which is the WD of the window function itself. This can be verified by applying the convolution property of the Wigner distribution function.
The spectrogram cannot produce interference since it is a positive-valued quadratic distribution.
formula_10
Can't solve the cross term problem, however it can solve the problem of 2 components time difference larger than window size B.
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formula_12
Where L is any integer greater than 0
Increase L can reduce the influence of cross term (however it can't eliminate completely )
For example, for L=2, the dominant third term is divided by 4 ( which is equivalent to 12dB ).
This gives a significant improvement over the Wigner Distribution.
Properties of L-Wigner Distribution:
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formula_38 
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When formula_40 and formula_41, it becomes the original Wigner distribution function.
It can avoid the cross term when the order of phase of the exponential function is no larger than formula_42
However the cross term between two components cannot be removed.
formula_43should be chosen properly such that
formula_44
formula_45
formula_46
If formula_47
when formula_48 , formula_49
formula_50
formula_51
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Cohen's kernel function : formula_54 which is concentred on the frequency axis.
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 formula_56 :
Cohen's kernel function : formula_58 where formula_59 is the Fourier transform of the window formula_60.
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 : formula_62
The S-method limits the range of the integral of the PWD with a low-pass windowing function formula_63 of Fourier transform formula_64. This results in the cross-term removal, without blurring the auto-terms that are well-concentred along the frequency axis.
The S-method strikes a balance in smoothing between the pseudo-Wigner distribution formula_65 [formula_66] and the power spectrogram formula_67 [formula_68].
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