Choi–Williams distribution function
- 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 does not decrease along the η,τ 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:
where
and the kernel function is:
Following are the magnitude distribution of the kernel function in η,τ domain with different α 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 η and τ axes, this kernel function can do nothing about it.
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.
Categories:
- Time–frequency analysis
- Transforms
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