Gabor–Wigner transform

The Gabor transform, named after Dennis Gabor, and the Wigner distribution function, named after Eugene Wigner, are both tools for time-frequency analysis. Since the Gabor transform does not have high clarity, and the Wigner distribution function has a "cross term problem" (i.e. is non-linear), a 2007 study by S. C. Pei and J. J. Ding proposed a new combination of the two transforms that has high clarity and no cross term problem.^[1] Since the cross term does not appear in the Gabor transform, the time frequency distribution of the Gabor transform can be used as a filter to filter out the cross term in the output of the Wigner distribution function.

Mathematical definition

• Gabor transform

$G_x(t,f) = \int_{-\infty}^\infty e^{-\pi(\tau-t)^2}e^{-j2\pi f\tau}x(\tau) \, d\tau$

• Wigner distribution function

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

• Gabor–Wigner transform

There are many different combinations to define the Gabor–Wigner transform. Here four different definitions are given.

1. $D_x(t,f)=G_x(t,f)\times W_x(t,f)$
2. $D_x(t,f)=\min\left\{|G_x(t,f)|^2,|W_x(t,f)|\right\}$
3. $D_x(t,f)=W_x(t,f)\times \{|G_x(t,f)|>0.25\}$
4. $D_x(t,f)=G_x^{2.6}(t,f)W_x^{0.7}(t,f)$

See also

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

References

1. S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans. Signal Processing, vol. 55, no. 10, pp. 4839–4850, Oct. 2007.