Choi–Williams distribution function - Alchetron, the free social encyclopedia

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 η , τ 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 ) = ∫ − ∞ ∞ ∫ − ∞ ∞ A x ( η , τ ) Φ ( η , τ ) exp ⁡ ( j 2 π ( η t − τ f ) ) d η d τ ,

where

A x ( η , τ ) = ∫ − ∞ ∞ x ( t + τ / 2 ) x ∗ ( t − τ / 2 ) e − j 2 π t η d t ,

and the kernel function is:

Φ ( η , τ ) = exp ⁡ [ − α ( η τ ) 2 ] .

Following are the magnitude distribution of the kernel function in η , τ domain with different α 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 η and τ axes, this kernel function can do nothing about it.

References

Choi–Williams distribution function Wikipedia

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