Cone-shape distribution function

The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,^[1] (acronymized as the ZAM ^[2][3][4] distribution^[5] or ZAMD^[1]), is one of the members of Cohen's class distribution function.^[1][6] It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990.^[7] The distribution's name stems from the twin cone shape of the distribution's kernel function on the ${\displaystyle t,\tau }$ plane.^[8] The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel.^[9][10]

The definition of the cone-shape distribution function is:

${\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,}$

where

${\displaystyle A_{x}(\eta ,\tau )=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi t\eta }\,dt,}$

and the kernel function is

${\displaystyle \Phi \left(\eta ,\tau \right)={\frac {\sin \left(\pi \eta \tau \right)}{\pi \eta \tau }}\exp \left(-2\pi \alpha \tau ^{2}\right).}$

The kernel function in ${\displaystyle t,\tau }$ domain is defined as:

${\displaystyle \phi \left(t,\tau \right)={\begin{cases}{\frac {1}{\tau }}\exp \left(-2\pi \alpha \tau ^{2}\right),&|\tau |\geq 2|t|,\\0,&{\mbox{otherwise}}.\end{cases}}}$

Following are the magnitude distribution of the kernel function in ${\displaystyle t,\tau }$ domain.

Following are the magnitude distribution of the kernel function in ${\displaystyle \eta ,\tau }$ domain with different ${\displaystyle \alpha }$ values.

As is seen in the figure above, a properly chosen kernel of cone-shape distribution function can filter out the interference on the ${\displaystyle \tau }$ axis in the ${\displaystyle \eta ,\tau }$ 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 ${\displaystyle \eta }$ axis are still preserved.

The cone-shape distribution function is in the MATLAB Time-Frequency Toolbox^[11] and National Instruments' LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis ^[12]

• Cohen's class distribution function
• Choi-Williams distribution function
• Wigner distribution function
• Ambiguity function
• Short-time Fourier transform