Wigner distribution function

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The Wigner distribution function (WDF), named after Eugene Wigner, was first proposed for corrections to classical statistical mechanics in 1932 by Eugene Wigner. But the Wigner distribution function is also a good transform for time frequency analysis. Comparing to a short-time Fourier transform such as the Gabor transform, the Wigner distribution function can give higher clarity in some case. Furthermore, the Wigner distribution function has many properties that other time frequency analysis transforms have.

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[edit] Mathematical definition

Also there are many different kinds of definition for the Wigner distribution function, the definition gives here is special for time frequency analysis. Wigner distribution function Wx(t,f) is defined as:

Definition 1: W_x(t,f)=\int_{-\infty}^{\infty}x(t+\tau/2)x^*(t-\tau/2)e^{-j2\pi\tau\,f}\,d\tau.

Definition 2: W_x(t,\omega)=\int_{-\infty}^{\infty}x(t+\tau/2)x^*(t-\tau/2)e^{-j2\pi\tau\omega}\,d\tau.

From the definition we can know that the WDF is essentially the Fourier transform of the input signal’s auto-correlation function.

[edit] Time frequency analysis example

Here gives some example to illustrate that the WDF really can be used for time frequency analysis.

[edit] Constant input signal

When the input signal is constant, its time-frequency distribution should be a vertical line lay on the time-axis. For example, if x(t)=1, then

W_x(t,f)=\int_{-\infty}^{\infty}e^{-j2\pi\tau\,f}\,d\tau=\delta(f),

which is exactly what we want.

[edit] Sinusoidal input signal

When the input signal is a sinusoidal function, its time frequency distribution should be a horizontal line cross the frequency-axis at its sinusoidal frequency. For example, if x(t) = ej2πht, then

\begin{align} W_x(t,f)& {} = \int_{-\infty}^{\infty}e^{j2\pi h(t+\tau/2)}e^{-j2\pi h(t-\tau/2)}e^{-j2\pi\tau\,f}\,d\tau \\ & {} = \int_{-\infty}^{\infty}e^{-j2\pi\tau(f-h)}\,d\tau\\ & {} = \delta(f-h) \end{align}

which is exactly what we want.

[edit] Chirp input signal

When the input signal is a chirp function, the instantaneous frequency is a linear function. That is the time frequency distribution should be a straight line. For example, if x(t)=e^{j2\pi kt^2}, then its instantaneous frequency is \frac{1}{2\pi}\frac{d(2\pi kt^2)}{dt}=2kt, and by WDF

\begin{align} W_x(t,f) & {} = \int_{-\infty}^{\infty}e^{j2\pi k(t+\tau/2)^2}e^{-j2\pi k(t-\tau/2)^2}e^{-j2\pi\tau\,f}d\tau \\ & {} = \int_{-\infty}^{\infty}e^{j4\pi kt\tau}e^{-j2\pi\tau f}\,d\tau \\ & {} = \int_{-\infty}^{\infty}e^{-j2\pi\tau(f-2kt)}\,d\tau\\ & {} = \delta(f-2kt), \end{align}

which is exactly what we want.

[edit] Delta input signal

When the input signal is a delta function, since it is only non-zero at t=0 and contain infinite frequency component, its time frequency distribution should be a vertical line cross the origin. That is the time frequency distribution of delta function should also be a delta function. And by WDF

\begin{align} W_x(t,f) & {} = \int_{-\infty}^{\infty}\delta(t+\tau/2)\delta(t-\tau/2) e^{-j2\pi\tau\,f}\,d\tau \\ & {}= 4\int_{-\infty}^{\infty}\delta(2t+\tau)\delta(2t-\tau)e^{-j2\pi\tau f}\,d\tau \\ & {} = 4\delta(4t)e^{j4\pi tf}\\ & {} = \delta(t)e^{j4\pi tf} \\ & {} = \delta(t) \end{align}

which is also exactly what we want.

Using the Wigner distribution function for time frequency is best suitable for input signal's phase is 2nd order or lower. For those signals, WDF can exactly generate the time frequency distribution of the input signal.

[edit] Performance of Wigner distribution function

Here are some examples to show that the performance of the Wigner distribution function is better than Gabor transform.

  • x(t) = cos(2πt)


  • x(t)=e^{j\pi t^2}
  • x(t)=\begin{cases} 1 \ \ \ |t|<2 \\ 0\ \ \ \text{otherwise} \end{cases} rectangular function


[edit] Cross term problem

The Wigner distribution function is not a linear transform. The cross term problem will occur when there is more than one component in the input signal while short-time Fourier transform doesn't have this problem. The followings are some examples that show the cross term problem of Wigner distribution function.

  • x(t)=\begin{cases} \cos(2\pi t) \ \ \ t\le-2 \\ \cos(4\pi t)\ \ \ -2<t\le2 \ \ \ \\ \cos(3\pi t) \ \ \ t>2 \end{cases}


  • x(t)=e^{jt^3}

In order to reduce the cross term problem, many other transforms have been proposed, including the modified Wigner distribution function, the Gabor-Wigner transform, and Cohen’s class distribution.

[edit] Properties of the Wigner distribution function

Wigner distribution function has many properties. These properties are listed in the following table.

Remarks
1 Projection property |x(t)|^2=\int_{-\infty}^{\infty}W_x(t,f) df \ \ \ |X(f)|^2=\int_{-\infty}^{\infty}W_x(t,f) dt
2 Energy property \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_x(t,f)dfdt = \int_{-\infty}^{\infty} |x(t)|^2dt=\int_{-\infty}^{\infty} |X(f)|^2df
3 Recovery property \int_{-\infty}^{\infty} W_x(t/2,f) e^{j2\pi ft}df =x(t)x^*(0) \ \ \ \int_{-\infty}^{\infty} W_x(t,f/2) e^{j2\pi ft}dt =X(f)X^*(0)
4 Mean condition frequency and mean condition time \begin{matrix}X(f)=|X(f)|e^{j2\pi\psi(f)}\ \ \ x(t)=|x(t)|e^{j2\pi\phi(t)} \ \ \ \\ If\ \phi'(t)=|x(t)|^{-2}\int_{-\infty}^{\infty}fW_x(t,f)df \ \\ and\ -\psi'(f)=|X(f)|^{-2}\int_{-\infty}^{\infty}tW_x(t,f)dt \end{matrix}
5 Moment properties \begin{matrix} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}t^nW_x(t,f)dtdf=\int_{-\infty}^{\infty}t^n|x(t)|^2dt \\ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^nW_x(t,f)dtdf=\int_{-\infty}^{\infty}f^n|X(f)|^2df \end{matrix}
6 Real properties W^*_x(t,f)=W_x(t,f)
7 Region properties \begin{matrix}If\ x(t)=0\ for\ t>t_0\ then\ W_x(t,f)=0\ for\ t>t_0 \\ If\ x(t)=0\ for\ t<t_0\ then\ W_x(t,f)=0\ for\ t<t_0 \end{matrix}
8 Multiplication theory \begin{matrix}If\ y(t)=x(t)h(t)\ then\\ W_y(t,f)=\int_{-\infty}^{\infty}W_x(t,\rho)W_h(t,f-\rho)d\rho \end{matrix}
9 Convolution theory \begin{matrix}If\ y(t)=\int_{-\infty}^{\infty}x(t-\tau)h(\tau)d\tau\ then \\ W_y(t,f)=\int_{-\infty}^{\infty}W_x(\rho,f)W_h(t-\rho,f)d\rho \end{matrix}
10 Correlation theory \begin{matrix}If\ y(t)=\int_{-\infty}^{\infty}x(t+\tau)h^*(\tau)d\tau\ then \\ W_y(t,\omega)=\int_{-\infty}^{\infty}W_x(\rho,\omega)W_h(-t+\rho,\omega)d\rho \end{matrix}
11 Time-shifting property \begin{matrix}If\ y(t)=x(t-t_0)\ then \\ W_y(t,f)=W_x(t-t_0,f) \end{matrix}
12 Modulation property \begin{matrix}If\ y(t)=e^{j2\pi f_0t}x(t)\ then \\ W_y(t,f)=W_x(t,f-f_0) \end{matrix}

[edit] See also

[edit] References

  • Jian-Jiun Ding, Time frequency analysis and wavelet transform class note,the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
  • S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
  • E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev., vol. 40, pp. 749-759, 1932.
  • T. A. C. M. Classen and W. F. G. Mecklenbrauker, “The Wigner distributiona tool for time-frequency signal analysis; Part I,” Philips J. Res., vol. 35, pp. 217-250, 1980.
  • F. Hlawatsch, G. F. BoudreauxBartels: “Linear and quadratic time-frequency signal representation,” IEEE Signal Processing Magazine, pp. 21-67, Apr. 1992.
  • R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience, NJ, 2004.