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Wikipedia / 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. 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:

where

and the kernel function is

domain is defined as:

domain.

values.

domain, or the ambiguity domain. Therefore, unlike the Choi-Williams distribution function 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

to interpret the signal energy correctly, it is often desired to satisfy the marginal properties:

  1. (Time Marginal)

  2. (Frequency Marginal)

In the case of the Cone-Shape Distribution:

  • Time Marginal. Thus, the ZAM distribution preserves the time marginal yields the instantaneous power of the signal.

  • Frequency Marginal, which trivializes the kernel), the ZAM distribution does not preserve the frequency marginal..

Time and Frequency Shift Invariance

The ZAM distribution satisfies both time and frequency shift invariance.

  • .

  • .

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

Real-Valuedness

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

, while cross-terms between components are located away from the origin.

  • The Wigner Distribution, 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 domain translates to a low-pass filtering effect on the cross-terms in the Time-Frequency domain.

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

axis.

See also

References