Gabor-Wigner transform

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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[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[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.

Contents

[edit] 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)

[edit] 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(8πt) + cos(16πt)
Gabor vs WDF
  • x(t)=e^{jt^3}
Gabor vs WDF
The above examples illustrate that the Gabor-Wigner transform has less cross term and higher clarity than Gabor transform.

[edit] See Also

[edit] 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.