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In mathematics, '''cone-shape distribution function''' is one of the members of [[Cohen's class distribution function]]. It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks in 1990. The reason why this distribution is so named is because its kernel function in <math>t, \tau</math> domain looks like two cones. The advantage of this special kernel function is that it can completely remove the cross-term between two components that have same center frequency, but on the other hand, the cross-term results form components with the same time center can not be removed by the cone-shape kernel. 

== Mathematical definition ==
The definition of the cone-shape distribution function is shown as follows:

:<math>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,</math>

where

:<math>A_x(\eta,\tau)=\int_{-\infty}^{\infty}x(t+\tau /2)x^*(t-\tau /2)e^{-j2\pi t\eta}\, dt,</math>

and the kernel function is

:<math>\Phi \left(\eta,\tau \right) = \frac{\sin \left(\pi \eta \tau \right)}{ \pi \eta \tau }\exp \left(-2\pi \alpha \tau^2  \right). </math>

The kernel function in <math>t, \tau</math> domain is defined as:

:<math>\phi \left(t,\tau \right) = \begin{cases} \frac{1}{\tau} \exp \left(-2\pi \alpha \tau^2 \right), & |\tau | \ge 2|t|, \\ 0, & \mbox{otherwise}. \end{cases} </math>

Following are the magnitude distribution of the kernel function in <math>t, \tau</math> domain. 

[[Image:cone shape 1.jpg|300px|center|]]

Following are the magnitude distribution of the kernel function in <math>\eta, \tau</math> domain with different <math>\alpha</math> values.

[[Image:cone shape 2.jpg|600px|center|]]

As we can see from the figure above, a properly chosen kernel of cone-shape distribution function can filter out the interference on the <math>\tau</math> axis in the <math>\eta, \tau</math> domain, or the ambiguity domain. Therefore, unlike the [[Choi-Williams distribution function]], the cone-shape distribution function can effectively reduce the cross-term results form two component with same center frequency. However, the cross-terms on the <math>\eta</math> axis are still preserved.

== See also ==
*[[Cohen's class distribution function]]
*[[Choi-Williams distribution function]] 
*[[Wigner distribution function]]
*[[Ambiguity function]]
*[[Short-time Fourier transform]]

==References==
*[http://djj.ee.ntu.edu.tw/TFW.htm 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.

== External links ==
* Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.]

[[Category:Time–frequency analysis]]
[[Category:Transforms]]