Choi-Williams distribution function

Choi-Williams distribution function is one of the members of Cohen's class distribution function. 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 dose not decrease along the $eta,\; au$ axes in the ambiguity domain. Consequently, the kernel function of Choi-Williams distribution function can only filter out the cross-terms result form the components differ in both time and frequency center.

Mathematical definition

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

:$C\_x(t,\; f)\; =\; int\_\{-infty\}^infty\; int\_\{-infty\}^infty\; A\_x(eta,\; au)\; Phi(eta,\; au)\; exp\; (j2pi(eta\; t-\; au\; f)),\; deta,\; d\; au,$

where

:$A\_x(eta,\; au)\; =\; int\_\{-infty\}^infty\; x(t+\; au\; /2)x^*(t-\; au\; /2)\; e^\{-j2pi\; teta\},\; dt,$

and the kernel function is:

:$Phi\; left(eta,\; au\; ight)\; =\; exp\; left\; [-alpha\; left(eta\; au\; ight)^2\; ight]\; .$

Following are the magnitude distribution of the kernel function in $eta,\; au$ domain with different $alpha$ values.

As we can see from the figure above, the kernel function indeed suppress the interference which is away from the origin, but for the cross-term locates on the $eta$ and $au$ axes, this kernel function can do nothing about it.

See also

*Cohen's class distribution function
*Cone-shape distribution function
*Wigner distribution function
*Ambiguity function
*Short-time Fourier transform

Reference

*Jian-Jiun Ding, Time frequency analysis and wavelet transform class note,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.

External links

* [http://djj.ee.ntu.edu.tw/TFW.htm Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.]