Modified Wigner distribution function

Wigner distribution function (WDF) was first proposed for corrections to classical statistical mechanics in 1932 by Eugene Wigner. But Wigner distribution function also can be used for time frequency analysis. Comparing to short-time Fourier transform such as Gabor transform, Wigner distribution function can give higher clarity in some case. However, the cross term problem limit its application. In 1995, the study of L. J. Stankovic and S. Stankovic proposes a modified form of the Wigner distribution function. The modified Wigner distribution function can have similar performance of Wigner distribution function in time frequency analysis and much less corss term problem.

Mathematical definition

The concept of modified Wigner distribution function is adding a mask function to reduce the effects of the cross term.

• Original Wigner distribution function

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

• Modified Wigner distribution function

$W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau /2)w^{*}(-\tau /2)x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau$

w(t) is the mask function.The most simple choice is a rectangular function with different width. The width is definded by parameter B.

$w(t)={\begin{cases}1\ \ \ if\ |t|<B\\0\ \ \ otherwise\end{cases}}$

• Other expression

$W_{x}(t,f)=\int _{-\infty }^{\infty }P(\theta )Y(t,f+\theta /2)Y^{*}(t,f-\theta /2)d\theta \ \ \ where\ \ Y(t,f)=\int _{-\infty }^{\infty }w(\tau )x(t+\tau )e^{-j2\pi f\tau }d\tau$

$P(\theta )\approx 1\ \ when\ |\theta |\ is\ small$

$P(\theta )\approx 0\ \ when\ |\theta |\ is\ large$

Performance of modified Wigner distribution function

Here gives some examples to show that the modified WDF really can reduce the cross term problem.

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

Gabor vs WDF

Gabor vs WDF

• $x(t)=e^{jt^{3}}$

Gabor vs WDF

Gabor vs WDF

From the above examples we can see that the modified version of WDF can give more clear time frequency distriution due to the less cross term effect.

References

• Jian-Jiun Ding, Time frequency analysis and wavelet transform class note,the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
• 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