# 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
${\displaystyle G_{x}(t,f)=\int _{-\infty }^{\infty }e^{-\pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )\,d\tau }$
• Wigner distribution function
${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}\,d\tau }$
• Gabor–Wigner transform
There are many different combinations to define the Gabor–Wigner transform. Here four different definitions are given.
1. ${\displaystyle D_{x}(t,f)=G_{x}(t,f)\times W_{x}(t,f)}$
2. ${\displaystyle D_{x}(t,f)=\min \left\{|G_{x}(t,f)|^{2},|W_{x}(t,f)|\right\}}$
3. ${\displaystyle D_{x}(t,f)=W_{x}(t,f)\times \{|G_{x}(t,f)|>0.25\}}$
4. ${\displaystyle D_{x}(t,f)=G_{x}^{2.6}(t,f)W_{x}^{0.7}(t,f)}$