Choi–Williams distribution function: Difference between revisions

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{{Short description|Variation of Cohen's class distribution function}}

'''Choi–Williams distribution function''' is one of the members of [[Cohen's class distribution function]].<ref>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.</ref> 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 <math>\eta, \tau</math> 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 ==

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:<math>\Phi \left(\eta,\tau \right) = \exp \left[-\alpha \left(\eta \tau \right)^2 \right].</math>

Following are the magnitude distribution of the kernel function in <math>\eta, \tau</math> ___domain with different <math>\alpha</math> 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 <math>\eta</math> and <math>\tau</math> axes, this kernel function can do nothing about it.

== See also ==

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==References==

{{Reflist}}

*[http://djj.ee.ntu.edu.tw/TFW.htm 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.