# 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.[2] 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. $D_x(t,f)=G_x^{2.6}(t,f)W_x^{0.7}(t,f)$