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{{Short description|Variation of the Wigner distribution function}}
{{No footnotes|date=February 2025}}
:''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 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 [[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 :
:<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.

<br />

* '''Modified form I'''

<math> W_x(t,f) = \int_{-B}^B w(\tau)x(t+\tau/2) x^*(t-\tau/2) e^{-j2\pi\tau f} \, d\tau
</math>

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'''

<math> W_x(t,f) = \int_{-B}^B w(\eta)X(f+\eta/2) X^*(f-\eta/2) e^{j2\pi t \eta} \, d\eta

</math>

* '''Modified form III (Pseudo L-Wigner Distribution)'''

<math> W_x(t,f) = \int_{-\infty}^\infty w(\tau)x^L(t+\tau/(2L)) \overline{x^{*L}(t-\tau/(2L))} e^{-j2\pi \tau f} \, d\tau

</math>

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.
# If the signal is time shifted <math> x(t-t0)

</math>'','' then its LWD is time shifted as well, <math> LWD: W_x(t-t0,f)

</math>
# The LWD of a modulated signal <math> x(t)\exp(j\omega_0 t)

</math>is shifted in frequency <math> LWD: W_x(t,f-f0)

</math>
# Is the signal <math> x(t)

</math>is time limited, i.e.,<math> x(t)=0

</math> <math> for \left\vert t \right\vert >T,

</math> then the L-Wigner distribution is time limited, <math> LWD: W_x(t,f)=0

</math> <math> for\left\vert t \right\vert >T

</math>
# If the signal <math> x(t)

</math>is band limited with <math> f_m

</math>(<math> F(f)=0

</math><math> for \left\vert f \right\vert > f_m

</math>), then <math> LWD: W_x(t,f)

</math>is limited in the frequency domain by <math> f_m

</math>as well.
# Integral of L-Wigner distribution over frequency is equal to the generalized signal power: <math> \int_{-\infty}^\infty W_x(t, f)df = \left\vert x(t) \right\vert ^{2L}

</math>
# Integral of <math> LWD: W_x(t,f)

</math>over time and frequency is equal to the <math> 2L^{th}

</math>power of the <math> 2L^{th}

</math>norm of signal <math> x(t)

</math>:

<math> \int_{-\infty}^\infty \int_{-\infty}^\infty W_x(t,f)dtdf = \int_{-\infty}^\infty \left\vert x(t) \right\vert ^{2L} dt = \lVert x(t) \rVert _{2L} ^{2L}

</math>
# The integral over time is:

<math> \int_{-\infty}^\infty W_x(t,f)dt = \left\vert F_L(f) \right\vert ^2=\left\vert \underbrace{ F(L_f)*F(L_f)*\cdots*F(L_f) }_{L times} \right\vert ^2

</math>
# For a large value of <math> L(L\rightarrow \infty)

</math>We may neglect all values of <math> LWD: W_x(t,f)

</math>, Comparing them to the one at the points <math> (t_m, f_m)

</math>, where the distribution reaches its essential supremum:

<math> \lim_{L \to \infty} (W_x(t,f)/W_x(t_m,f_m)) = \begin{cases} 0, & \text{if }f \neq f_m\text{ or }t\neq t_m\text{ } \\ 1, & \text{if }f=f_m\text{ and } t=t_m\end{cases}

</math> <br />

* '''Modified form IV (Polynomial Wigner Distribution Function)'''

<math> W_x(t,f) = \int_{-B}^B [\textstyle \prod_{l=1}^{q/2} \displaystyle x(t+d_l \tau) x^*(t- d_{-l} \tau)] e^{-j2\pi\tau f} \, d\tau

</math>

When <math> q=2

</math> and <math> d_l=d_{-l}=0.5

</math>, 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 <math> q/2+1

</math>

However the cross term between two components cannot be removed.

<math> d_l

</math>should be chosen properly such that

<math> \textstyle \prod_{l=1}^{q/2} \displaystyle x(t+d_l \tau) x^*(t-d_{-l}\tau)=\exp\big(j2\pi\textstyle \sum_{n=1}^{q/2+1} n a_n t^{n-1}\tau \displaystyle\big)

</math>

<math> W_x(t,f) = \int_{-\infty}^\infty \exp\Bigl(-j2\pi (f-\sum_{n=1}^{q/2+1}na_nt^{n-1})\tau\Bigr)d\tau

</math>

<math> \cong \delta\bigl(f-\sum_{n=1}^{q/2+1}na_nt^{n-1}\bigr)

</math>

If <math> x(t)=\exp\bigl(j2\pi\sum_{n=1}^{q/2+1}a_nt^n\bigr)

</math>

when <math> q=2

</math> , <math> x(t+d_l \tau) x^*(t-d_{-l}\tau)=\exp\bigl(j2\pi\sum_{n=1}^{q/2+1}na_nt^{n-1}\tau\bigr)

</math>

<math> a_2(t+d_l\tau)^2+a_1(t+d_l\tau)-a_2(t-d_{-l}\tau)^2-a_1(t-d_{-l}\tau)=2a_2t\tau+a_1\tau

</math>

<math> \Longrightarrow d_l+d_{-l}=1, d_l-d_{-l}=0

</math>

<math> \Longrightarrow d_l=d_{-l}=1/2

</math>

*'''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 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

[[Category:Signal processing]]
[[Category:Transforms]]

Modified Wigner distribution function - Revision history

imported>Yodo9000: added no-footnotes template