Yttria-stabilized zirconia

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 $\eta ,\tau$ 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:

$C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,$

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

• 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.