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 ${\displaystyle \eta ,\tau }$ 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:

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

where

${\displaystyle A_{x}(\eta ,\tau )=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi t\eta }\,dt,}$

and the kernel function is:

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

See also

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