Tweedie distributions: Difference between revisions

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The '''Wigner distribution (WD)''' was first proposed for corrections to classical statistical mechanics in 1932 by [[Eugene Wigner]]. The [[Wigner distribution function|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.
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The original WD, the spectrogram, and the modified WDs all belong to the [[Cohen's class]] of bilinear time-frequency representations :
:<math>C_x(t, f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_x(\theta,\nu) \Pi(t - \theta,f - \nu)\, d\theta\, d\nu \quad = [W_x\,\ast\,\Pi] (t,f)</math>
where <math>\Pi \left(t, f\right)</math> 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'''
:<math> W_x(t,f) = \int_{-\infty}^\infty x(t+\tau/2) x^*(t-\tau/2) e^{-j2\pi\tau\,f} \, d\tau</math>
Cohen's kernel function : <math>\Pi (t,f) = \delta_{(0,0)} (t,f) </math>
 
*'''Spectrogram'''
:<math>SP_x (t,f) = |ST_x (t,f)|^2 = ST_x (t,f)\,ST_x^* (t,f)</math>
where <math>ST_x</math> is the [[short-time Fourier transform]] of <math>x</math>.  
:<math> ST_x(t,f) = \int_{-\infty}^\infty x(\tau) w^*(t-\tau) e^{-j2\pi f\tau} \, d\tau</math>
 
Cohen's kernel function : <math>\Pi (t,f) = W_h(t,f) </math> 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'''
:<math> PW_x(t,f) = \int_{-\infty}^\infty w(\tau/2) w^*(-\tau/2) x(t+\tau/2) x^*(t-\tau/2) e^{-j2\pi\tau\,f} \, d\tau</math>
Cohen's kernel function : <math>\Pi (t,f) = \delta_0 (t)\,W_h(t,f) </math> 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
:<math> PW_x(t,f) = \int_{-\infty}^\infty ST_x(t, f+\nu/2) ST_x^*(t, f-\nu/2) e^{j2\pi\nu\,t} \, d\nu</math>
 
*'''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 <math>q</math> :
:<math> SPW_x(t,f) = [ q\,\ast\, PW_x (.,f)] (t) =  \int_{-\infty}^\infty q(t-u) \int_{-\infty}^\infty w(\tau/2) w^*(-\tau/2) x(u+\tau/2) x^*(u-\tau/2) e^{-j2\pi\tau\,f} \, d\tau\, du</math>
Cohen's kernel function : <math>\Pi (t,f) = q(t)\, W(f) </math> where <math>W</math> is the Fourier transform of the window <math>w</math>.
 
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'''
:<math> SM(t,f) = \int_{-\infty}^\infty ST_x(t, f+\nu/2) ST_x^*(t, f-\nu/2) G(\nu) e^{j2\pi\nu\,t} \, d\nu</math>
 
Cohen's kernel function : <math>\Pi (t,f) = g(t)\, W_h(t,f) </math>
 
The S-method limits the range of the integral of the PWD with a low-pass windowing function <math>g(t)</math> of Fourier transform <math>G(f)</math>. 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 <math>PW_x</math> [<math>g(t) = 1</math>] and the power spectrogram <math>SP_x</math> [<math>g(t) = \delta_0 (t)</math>].
 
Note that in the original 1994 paper, Stankovic defines the S-methode with a modulated version of the short-time Fourier transform :
:<math> SM(t,f) = \int_{-\infty}^\infty \tilde{ST}_x(t,f+\nu) \tilde{ST}_x^*(t,f-\nu) P(\nu)\, d\nu </math>
 
where
 
:<math> \tilde{ST}_x(t,f) = \int_{-\infty}^\infty x(t+\tau) w^*(\tau) e^{-j2\pi f\tau} \, d\tau \quad = ST_x(t,f)\,e^{j2\pi f t}</math>
 
Even in this case we still have
 
: <math>\Pi (t,f) = p(2t)\, W_h(t,f) </math>
 
== 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
 
[[Category:Signal processing]]
[[Category:Transforms]]

Latest revision as of 03:31, 7 February 2014

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