Cone-shape distribution function

The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution, (acronymized as the ZAM distribution or ZAMD), 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 II in 1990. The distribution's name stems from the twin cone shape of the distribution's kernel function on the $t,\tau$ plane. 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.

Mathematical definition

The definition of the cone-shape distribution function is:

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

where

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

and the kernel function is

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

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

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

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

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

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

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

Properties

The Cone-Shape Distribution Function (ZAM) possesses specific mathematical properties that distinguish it from other members of Cohen's class. These properties determine the distribution's accuracy in representing signal energy in time and frequency.

Marginals

For a time-frequency distribution $C_{x}(t,f)$ to interpret the signal energy correctly, it is often desired to satisfy the marginal properties:

$\int _{-\infty }^{\infty }C_{x}(t,f)df=|x(t)|^{2}$ (Time Marginal)

$\int _{-\infty }^{\infty }C_{x}(t,f)dt=|X(f)|^{2}$ (Frequency Marginal)

In the case of the Cone-Shape Distribution:

• Time Marginal: The kernel in the ambiguity domain $\Phi (\eta ,\tau )$ must satisfy $\Phi (\eta ,0)=1$. For the ZAM kernel, $\Phi (\eta ,0)=\lim _{\tau \to 0}{\frac {\sin(\pi \eta \tau )}{\pi \eta \tau }}e^{-2\pi \alpha \tau ^{2}}=1$. Thus, the ZAM distribution preserves the time marginal, meaning the summation over frequency at a given time $t$ yields the instantaneous power of the signal.
• Frequency Marginal: The condition is $\Phi (0,\tau )=1$. For the ZAM kernel, substituting $\eta =0$ gives $\Phi (0,\tau )=e^{-2\pi \alpha \tau ^{2}}$. Since this expression depends on $\tau$ and $\alpha$ and is not strictly 1 (unless $\alpha =0$, which trivializes the kernel), the ZAM distribution does not preserve the frequency marginal. This implies that integrating the distribution over time does not perfectly recover the energy spectrum $|X(f)|^{2}$.

Time and Frequency Shift Invariance

The ZAM distribution satisfies both time and frequency shift invariance.

• If $y(t)=x(t-t_{0})$, then $C_{y}(t,f)=C_{x}(t-t_{0},f)$.
• If $y(t)=x(t)e^{j2\pi f_{0}t}$, then $C_{y}(t,f)=C_{x}(t,f-f_{0})$.

These properties are crucial for analyzing signals where events may occur at arbitrary times or frequencies without distorting the representation shape.

Real-Valuedness

Since the kernel function in the time-lag domain $\phi (t,\tau )$ satisfies the conjugate symmetry $\phi (t,-\tau )=\phi ^{*}(t,\tau )$, the resulting distribution $C_{x}(t,f)$ is always real-valued. This allows for a straightforward physical interpretation of signal energy, although negative values may still appear (a common trait in Cohen's class excluding the Spectrogram).

Performance and Cross-Term Suppression

A primary motivation for the Cone-Shape Distribution is the suppression of cross-terms (interference terms) that arise in the Wigner Distribution Function (WDF) when analyzing multi-component signals.

Mechanism of Suppression

In the Ambiguity Domain ($\eta ,\tau$), the auto-terms of a signal components are concentrated near the origin $(0,0)$, while cross-terms between components are located away from the origin.

• The Wigner Distribution has an all-pass kernel $\Phi _{WDF}(\eta ,\tau )=1$, preserving all cross-terms.
• The Choi-Williams Distribution uses an exponential kernel that covers both axes but decays away from them.
• The Cone-Shape Distribution has a unique kernel support region. In the $t,\tau$ domain, the kernel is non-zero only within the cone defined by $|\tau |\geq 2|t|$. This geometric constraint in the $t-\tau$ domain translates to a low-pass filtering effect on the cross-terms in the Time-Frequency domain.

Specifically, the ZAM kernel strongly attenuates interference components that result from signals separated in frequency (which appear on the $\tau$ axis in the ambiguity domain) while preserving the resolution of components separated in time. This makes ZAM particularly effective for signals composed of short-duration events occurring at different frequencies.

Comparison with Other Distributions

Comparison of Time-Frequency Distributions

Distribution	Kernel $\Phi (\eta ,\tau )$	Cross-Term Suppression	Marginal Properties
Wigner-Ville (WVD)	$1$	None	Satisfies both
Choi-Williams (CWD)	$e^{-{\frac {(\pi \eta \tau )^{2}}{\sigma }}}$	Good (depends on $\sigma$)	Satisfies both
Cone-Shape (ZAM)	${\frac {\sin(\pi \eta \tau )}{\pi \eta \tau }}e^{-2\pi \alpha \tau ^{2}}$	Excellent for frequency-separated components	Time only

While the ZAM distribution excels at removing cross-terms formed by components with the same center time but different frequencies ("vertical" cross-terms in the TF plane), it is less effective at removing cross-terms from components with the same frequency but different times ("horizontal" cross-terms), as the kernel $\Phi (\eta ,\tau )$ does not decay along the $\eta$ axis (frequency shift axis) as strongly as it does along the $\tau$ axis.

See also

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