Thecone-shape distribution function, also known as theZhao–Atlas–Marks time-frequency distribution,^[1] (acronymized as the ZAM^[2][3][4] distribution^[5] or ZAMD^[1]), is one of the members ofCohen's class distribution function.^[1][6] It was first proposed by Yunxin Zhao, Les E. Atlas, andRobert 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:

${\displaystyle C_x(t,f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{A}_x(\eta ,\tau )\mathrm{\Phi }(\eta ,\tau )\exp (j2\pi (\eta t-\tau f))d\eta d\tau ,}$

where

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

and the kernel function is

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

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

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 theChoi-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 theMATLAB Time-Frequency Toolbox^[11] andNational Instruments'LabVIEW Tools for Time-Frequency, Time-Series, andWavelet Analysis^[12]

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

• Time–frequency analysis
• Transforms

• CS1 maint: multiple names: authors list
• Articles with short description
• Short description matches Wikidata