Choi–Williams distribution function

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 $\eta, \tau$ 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:

$C_x(t, f) = \int_{-\infty}^\infty \int_{-\infty}^\infty A_x(\eta,\tau) \Phi(\eta,\tau) \exp (j2\pi(\eta t-\tau f))\, d\eta\, d\tau,$

where

$A_x(\eta,\tau) = \int_{-\infty}^\infty x(t+\tau /2)x^*(t-\tau /2) e^{-j2\pi t\eta}\, dt,$

and the kernel function is:

$\Phi \left(\eta,\tau \right) = \exp \left[-\alpha \left(\eta \tau \right)^2 \right].$

Following are the magnitude distribution of the kernel function in $\eta, \tau$ 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 $\tau$ axes, this kernel function can do nothing about it.

See also

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

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.

• 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.