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 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 Choi–Williams distribution function is shown as follows:
where
and the kernel function is:

Kernel Design and Cross-Term Suppression

The primary motivation behind the Choi–Williams distribution is to suppress the cross-terms that plague the Wigner distribution function. In the ambiguity function domain, the "auto-terms" of a signal are typically concentrated near the origin, while the cross-terms are located away from the origin.
The Choi–Williams kernel function is designed as a low-pass filter in the ambiguity domain:
Analysis of this kernel reveals its filtering characteristics:
  • On the axes, the kernel value is. This ensures that the marginal properties are satisfied.
  • Away from the axes, as the product increases, the kernel value decays exponentially towards zero.
Geometrically, this creates a "cross-shaped" passband in the ambiguity domain. The kernel preserves the signal components that lie on the axes while attenuating components that are far from the axes. However, a known limitation is that if the cross-terms lie exactly on the or axes, the Choi–Williams distribution cannot filter them out.

Properties

The Choi–Williams distribution possesses several desirable mathematical properties that make it attractive for analyzing non-stationary signals.

Marginals

Unlike the Cone-Shape Distribution which only satisfies the time marginal, the Choi–Williams 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.Time Marginal: Integration over frequency recovers the instantaneous power.Frequency Marginal: Integration over time recovers the power spectral density.

Real-Valuedness

The distribution function is always real-valued. This is guaranteed because the kernel function is real and even:, which implies the Fourier transform will be real.

Time and Frequency Shift Invariance

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.

Parameter Selection

The parameter controls the trade-off between auto-term resolution and cross-term suppression.Large : The kernel decays very quickly away from the axes.
  • * Pros: Strong suppression of cross-terms.
  • * Cons: The auto-terms may be smeared or distorted because the kernel might filter out some high-frequency contents of the auto-terms themselves.Small : The kernel approaches unity.
  • * Pros: High resolution of auto-terms.
  • * Cons: Poor suppression of cross-terms; interference becomes visible.
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.