Wikipedia / Modified Wigner distribution function
Wikipedia / Modified Wigner distribution function
Note: the Wigner distribution function is abbreviated here as WD rather than WDF as used at 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 WignerVille 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 Ljubiša Stanković 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 :
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
Wigner distribution
Spectrogram
is the short-time Fourier transform.
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.
Modified form I
Can't solve the cross term problem, however it can solve the problem of 2 components time difference larger than window size B.
Modified form II
Modified form III (Pseudo L-Wigner Distribution)
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:
The L-Wigner Distribution is always real.
,
is limited in the frequency domainas well.
:
The integral over time is:
, where the distribution reaches its essential supremum:
Modified form IV (Polynomial Wigner Distribution Function)
, it becomes the original Wigner distribution function.
However the cross term between two components cannot be removed.
should be chosen properly such that
Pseudo Wigner distribution
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
Smoothed pseudo Wigner distribution :
:
.
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 smooths the WD in the time domain, they are not equivalent in general.
S-method
].
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
See also
References
P. Gonçalves and R. Baraniuk, “Pseudo Affine Wigner Distributions : Definition and Kernel Formulation”, IEEE Transactions on Signal Processing, vol. 46, no. 6, Jun. 1998
L. Stankovic, “A Method for Time-Frequency Analysis”, IEEE Transactions on Signal Processing, vol. 42, no. 1, Jan. 1994
L. J. Stankovic, S. Stankovic, and E. Fakultet, “An analysis of instantaneous frequency representation using time frequency distributions-generalized Wigner distribution,” IEEE Trans. on Signal Processing, pp. 549-552, vol. 43, no. 2, Feb. 1995