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
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:
Cx(t,f)=
Ax(η,\tau)\Phi(η,\tau)\exp(j2\pi(ηt-\tauf))dηd\tau,
where
Ax(η,\tau)=
x(t+\tau/2)x*(t-\tau/2)e-j2\pidt,
and the kernel function is:
\Phi\left(η,\tau\right)=\exp\left[-\alpha\left(η\tau\right)2\right].
See also
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
- 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.