imported>VulcanSphere: Adding short description: "Variation of Cohen's class distribution function"

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Adding short description: "Variation of Cohen's class distribution function"

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{{Short description|Variation of Cohen's class distribution function}}
The '''cone-shape distribution function,''' also known as the '''Zhao–Atlas–Marks time-frequency distribution''',<ref name=ZAM>Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994)</ref> (acronymized as the ZAM <ref>{{cite journal|author1=L.M. Khadra |author2=J. A. Draidi |author3=M. A. Khasawneh |author4=M. M. Ibrahim. |title=Time-frequency distributions based on generalized cone-shaped kernels for the representation of nonstationary signals|journal=Journal of the Franklin Institute|volume=335|issue=5|pages=915–928|doi=10.1016/s0016-0032(97)00023-9|year=1998 }}</ref><ref>{{cite journal|author1=Deze Zeng |author2=Xuan Zeng |author3=G. Lu |author4=B. Tang |title=Automatic modulation classification of radar signals using the generalised time-frequency representation of Zhao, Atlas and Marks|journal=IET Radar, Sonar & Navigation|date=2011|volume=5|issue=4|pages=507–516|doi=10.1049/iet-rsn.2010.0174}}</ref><ref>{{cite journal|author1=James R. Bulgrin |author2=Bernard J. Rubal |author3=Theodore E. Posch |author4=Joe M. Moody |title=Comparison of binomial, ZAM and minimum cross-entropy time-frequency distributions of intracardiac heart sounds|journal=Signals, Systems and Computers, 1994. 1994 Conference Record of the Twenty-Eighth Asilomar Conference on|volume=1|pages=383–387}}</ref> distribution<ref>{{cite journal|last1=Christos, Skeberis, Zaharias D. Zaharis, Thomas D. Xenos, and Dimitrios Stratakis.|title=ZAM distribution analysis of radiowave ionospheric propagation interference measurements|journal=Telecommunications and Multimedia (TEMU), 2014 International Conference on|date=2014|pages=155–161}}</ref> or ZAMD<ref name=ZAM />), is one of the members of [[Cohen's class distribution function]].<ref name=ZAM /><ref>{{cite journal|last1=Leon Cohen|title=Time-frequency distributions-a review|journal=Proceedings of the IEEE|date=1989|volume=77|issue=7|pages=941–981|doi=10.1109/5.30749|citeseerx=10.1.1.1026.2853}}</ref> It was first proposed by Yunxin Zhao, Les E. Atlas, and [[Robert J. Marks II]] in 1990.<ref>{{cite journal|author1=Y. Zhao |author2=L. E. Atlas |author3=R. J. Marks II |title=The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals|journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|date=July 1990|volume=38|issue=7|pages=1084–1091|doi=10.1109/29.57537|citeseerx=10.1.1.682.8170 }}</ref> The distribution's name stems from the twin cone shape of the distribution's kernel function on the <math>t, \tau</math> plane.<ref>{{cite book|last1=R.J. Marks II|title=Handbook of Fourier analysis & its applications|date=2009|publisher=Oxford University Press}}</ref> 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.<ref>{{cite journal|author1=Patrick J. Loughlin |author2=James W. Pitton |author3=Les E. Atlas |title=Bilinear time-frequency representations: New insights and properties|journal=IEEE Transactions on Signal Processing|date=1993|volume=41|issue=2|pages=750–767|doi=10.1109/78.193215|bibcode=1993ITSP...41..750L }}</ref><ref>{{cite journal|author1=Seho Oh |author2=R. J. Marks II|title=Some properties of the generalized time frequency representation with cone-shaped kernel|journal=IEEE Transactions on Signal Processing|date=1992|volume=40|issue=7|pages=1735–1745|doi=10.1109/78.143445|bibcode=1992ITSP...40.1735O}}</ref>

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

:<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 is seen in 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.

The cone-shape distribution function is in the [[MATLAB]] Time-Frequency Toolbox<ref>[http://tftb.nongnu.org/refguide.pdf] Time-Frequency Toolbox For Use with MATLAB</ref> and [[National Instruments]]' LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis <ref>[http://www.ni.com/pdf/products/us/4msw69-70.pdf] National Instruments. LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis. [http://zone.ni.com/reference/en-XX/help/372656A-01/lvtimefreqtk/tfa_cone_shaped_distribution/] TFA Cone-Shaped Distribution VI</ref>

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

==References==
{{Reflist}}

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

Cone-shape distribution function - Revision history

imported>VulcanSphere: Adding short description: "Variation of Cohen's class distribution function"