Choi–Williams distribution function explained

Choi–Williams distribution function is one of the members of Cohen's class distribution function.^[1] It was first proposed by Hyung-Ill Choi and William J. Williams in 1989. This distribution function adopts exponential kernel to suppress the cross-term. However, the kernel gain does not decrease along the

η , \tau axes in the ambiguity domain. Consequently, the kernel function of Choi–Williams distribution function can only filter out the cross-terms that result from the components that differ in both time and frequency center.

Mathematical definition

The definition of the cone-shape distribution function is shown as follows:

C _{x(t,} f) = \int ^{inf ty} _{-inf ty} \int ^{inf ty} _{-inf ty} A _{x(η,\tau)} \Phi( η ,\tau) \exp (j 2 \pi( η t - \tau f)) d η d\tau,

where

A _{x(η,\tau)} = \int ^{inf ty} _{-inf ty} x(t + \tau / 2 )x ^{*(t-\tau} / 2 ) e ^{-j2\pi} dt,

and the kernel function is:

\Phi \left( η ,\tau \right) = \exp \left[ - \alpha \left( η \tau \right) ^{2} \right].

See also

• • Cone-shape distribution function Wigner distribution function
• Ambiguity function
• Short-time Fourier transform

References

• Time frequency analysis and wavelet transform class notes, Jian-Jiun Ding, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
• S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
• H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE. Trans. Acoustics, Speech, Signal Processing, vol. 37, no. 6, pp. 862 - 871, June 1989.
• Y. Zhao, L. E. Atlas, and R. J. Marks, “The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 38, no. 7, pp. 1084 - 1091, July 1990.

Notes and References

1. E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009.