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 {\displaystyle \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 ) = ∫ − ∞ ∞ ∫ − ∞ ∞ A x ( η , τ ) Φ ( η , τ ) exp ( j 2 π ( η t − τ f ) ) d η d τ , {\displaystyle 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 ( η , τ ) = ∫ − ∞ ∞ x ( t + τ / 2 ) x ∗ ( t − τ / 2 ) e − j 2 π t η d t , {\displaystyle 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:
Φ ( η , τ ) = exp [ − α ( η τ ) 2 ] . {\displaystyle \Phi \left(\eta ,\tau \right)=\exp \left[-\alpha \left(\eta \tau \right)^{2}\right].}See also
- Cone-shape distribution function
- Wigner distribution function
- Ambiguity function
- Short-time Fourier transform
- 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. 862–871, 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. 1084–1091, July 1990.
References
E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009. ↩