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. 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.   formula_11   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:   formula_33  formula_34  formula_38    formula_39  When formula_40 and formula_41, it becomes the original Wigner distribution function.