Gabor-Wigner transform

The Gabor transform and the Wigner distribution function 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 [#refDJJ2007| [2] , 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 [#refDJJ2007| [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:$G\_x(t,f)\; =\; int\_\{-infty\}^\{infty\}e^\{-pi(\; au-t)^2\}e^\{-j2pi\; f\; au\}x(\; au)d\; au$
*Wigner distribution function:$W\_x(t,f)=int\_\{-infty\}^\{infty\}x(t+\; au/2)x^*(t-\; au/2)e^\{-j2pi\; au,f\}d\; au$
*Gabor-Wigner transform:There are many different combinations to define the Gabor-Wigner transform. Here four different definitions are given.
#$D\_x(t,f)=G\_x(t,f)\; imes\; W\_x(t,f)$
#$D\_x(t,f)=minleft\{|G\_x(t,f)|^2,|W\_x(t,f)|\; ight\}$
#$D\_x(t,f)=W\_x(t,f)\; imes\; \{|G\_x(t,f)|>0.25\}$
#$D\_x(t,f)=G\_x^\{2.6\}(t,f)W\_x^\{0.7\}(t,f)$

Performance of Gabor-Wigner transform

Here some examples are given to show the performance of four Gabor-Wigner transform comparing to Gabor transform and Wigner distribution function.
*$x(t)=cos(8pi\; t)+cos(16pi\; t)$

*$x(t)=e^\{jt^3\}$

:The above examples illustrate that the Gabor-Wigner transform has less cross term and higher clarity than Gabor transform.

ee also

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

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. 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.