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 $$\eta, \tau$$ 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:


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