Gabor–Wigner transform

Gobor transform and Wigner distribution function are both good tool for time frequency analysis. However, each of them have their own problem. The Gabor transform dose not have high clarity and the Wigner distribution function have cross term problem.The study by S. C. Pei and J. J. Ding in 2007 propose a new transform that combines the advantage of the two transform that is high clarity and no cross term.

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 dedinitions 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. $D_{x}(t,f)=G_{x}^{2.6}(t,f)W_{x}^{0.7}(t,f)$

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.