ChoiâÂÂWilliams 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 axes in the ambiguity domain. Consequently, the kernel function of ChoiâÂÂWilliams distribution function can only filter out the cross-terms that result from the components that differ in both time and frequency center.
The definition of the ChoiâÂÂWilliams distribution function is shown as follows:
where
and the kernel function is:
The primary motivation behind the ChoiâÂÂWilliams distribution is to suppress the cross-terms (interference terms) that plague the Wigner distribution function. In the ambiguity function domain (defined by variables ), the "auto-terms" of a signal (the actual signal components) are typically concentrated near the origin , while the cross-terms are located away from the origin.
The ChoiâÂÂWilliams kernel function is designed as a low-pass filter in the ambiguity domain:
Analysis of this kernel reveals its filtering characteristics:
Geometrically, this creates a "cross-shaped" passband in the ambiguity domain. The kernel preserves the signal components that lie on the axes (auto-terms) while attenuating components that are far from the axes (cross-terms). However, a known limitation is that if the cross-terms lie exactly on the or axes (which occurs when two components have the same center time or same center frequency), the ChoiâÂÂWilliams distribution cannot filter them out.
The ChoiâÂÂWilliams distribution possesses several desirable mathematical properties that make it attractive for analyzing non-stationary signals.
Unlike the Cone-Shape Distribution (ZAM) which only satisfies the time marginal, the ChoiâÂÂWilliams distribution satisfies both the time and frequency marginals. This suggests that the projection of the time-frequency distribution onto the time or frequency axis yields the correct instantaneous power or energy spectrum, respectively.
The distribution function is always real-valued. This is guaranteed because the kernel function is real and even: , which implies the Fourier transform (the distribution itself) will be real.
The CWD is invariant to time and frequency shifts. If the signal is shifted in time by and in frequency by , the resulting distribution is simply shifted by the same amounts in the time-frequency plane.
The parameter (often denoted as in some literature where the kernel is ) controls the trade-off between auto-term resolution and cross-term suppression.
In practical applications, the value of is usually chosen empirically based on the signal characteristics, typically ranging between 0.1 and 10. For signals with complex multicomponent structures, an intermediate value is selected to balance the clarity of the components against the "ghost" interference patterns.