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.

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

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.

Gabor–Wigner transform

Mathematical definition

See also

References