Tweedie distributions

The Wigner distribution (WD) was first proposed for corrections to classical statistical mechanics in 1932 by Eugene Wigner. The Wigner distribution, 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 :

$C_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{W}_x(\theta ,\nu )\mathrm{\Pi }(t-\theta ,f-\nu )d\theta d\nu =[W_x\ast \mathrm{\Pi }](t,f)$

where $\mathrm{\Pi }(t,f)$ 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

$W_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau f}d\tau$

Cohen's kernel function : $\mathrm{\Pi }(t,f)={\delta }_{(0,0)}(t,f)$

• Spectrogram

$S{P}_x(t,f)=|S{T}_x(t,f){|}^2=S{T}_x(t,f)S{T}_x^{*}(t,f)$

where $S{T}_x$ is the short-time Fourier transform of $x$.

$S{T}_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(\tau )w^{*}(t-\tau )e^{-j2\pi f\tau }d\tau$

Cohen's kernel function : $\mathrm{\Pi }(t,f)=W_h(t,f)$ 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.

• Pseudo Wigner distribution

$P{W}_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}w(\tau /2)w^{*}(-\tau /2)x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau f}d\tau$

Cohen's kernel function : $\mathrm{\Pi }(t,f)={\delta }_0(t)W_h(t,f)$ 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

$P{W}_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S{T}_x(t,f+\nu /2)S{T}_x^{*}(t,f-\nu /2)e^{j2\pi \nu t}d\nu$

• Smoothed pseudo Wigner distribution :

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 $q$ :

$SP{W}_x(t,f)=[q\ast P{W}_x(.,f)](t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}q(t-u)\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}w(\tau /2)w^{*}(-\tau /2)x(u+\tau /2)x^{*}(u-\tau /2)e^{-j2\pi \tau f}d\tau du$

Cohen's kernel function : $\mathrm{\Pi }(t,f)=q(t)W(f)$ where $W$ is the Fourier transform of the window $w$.

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.

• S-method

$SM(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S{T}_x(t,f+\nu /2)S{T}_x^{*}(t,f-\nu /2)G(\nu )e^{j2\pi \nu t}d\nu$

Cohen's kernel function : $\mathrm{\Pi }(t,f)=g(t)W_h(t,f)$

The S-method limits the range of the integral of the PWD with a low-pass windowing function $g(t)$ of Fourier transform $G(f)$. 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 $P{W}_x$ [$g(t)=1$] and the power spectrogram $S{P}_x$ [$g(t)={\delta }_0(t)$].

Note that in the original 1994 paper, Stankovic defines the S-methode with a modulated version of the short-time Fourier transform :

$SM(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\widetilde{ST}}_x(t,f+\nu ){\widetilde{ST}}_x^{*}(t,f-\nu )P(\nu )d\nu$

where

${\widetilde{ST}}_x(t,f)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t+\tau )w^{*}(\tau )e^{-j2\pi f\tau }d\tau =S{T}_x(t,f)e^{j2\pi ft}$

Even in this case we still have

$\mathrm{\Pi }(t,f)=p(2t)W_h(t,f)$

See also

• Time-frequency representation
• Wigner distribution function
• Bilinear time–frequency distribution
• Short-time Fourier transform
• Gabor transform

References

• P. Gonçalves and R. Baraniuk, “Pseudo Affine Wigner Distributions : Definition and Kernel Formulation”, IEEE Trans. on Signal Processing, vol. 46, no. 6, Jun. 1998
• L. Stankovic, “A Method for Time-Frequency Signal Analysis”, IEEE Trans. on Signal Processing, vol. 42, no. 1, Jan. 1994