Choi–Williams distribution function is one of the members of Cohen's class distribution function. 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 (interference terms) that plague the Wigner distribution function, 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:

• . This ensures that the marginal properties are satisfied (see Properties section).

• increases, the kernel value decays exponentially towards zero.

axes (which occurs when two components have the same center time or same center frequency), 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 (ZAM) 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

, which implies the Fourier transform (the distribution itself) will be real.

Time and Frequency Shift Invariance

, the resulting distribution is simply shifted by the same amounts in the time-frequency plane.

Parameter Selection

) controls the trade-off between auto-term resolution and cross-term suppression.

): 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.

): The kernel approaches unity (1).

• Pros: High resolution of auto-terms (approaching the Wigner Distribution).

• Cons: Poor suppression of cross-terms; interference becomes visible.

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.

See also

• Cone-shape distribution function

• Wigner distribution function

• Ambiguity function

• Short-time Fourier transform

References

• Time frequency analysis and wavelet transform class notes, Jian-Jiun Ding, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.

• S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.

• H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE. Trans. Acoustics, Speech, Signal Processing, vol. 37, no. 6, pp.862871, June 1989.

• Y. Zhao, L. E. Atlas, and R. J. Marks, “The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 38, no. 7, pp.10841091, July 1990.