Cone-shape distribution function

{{Short description|Variation of Cohen's class distribution function}}

The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,Leon Cohen, Time Frequency Analysis: Theory and Applications, Prentice Hall, (1994) (acronymized as the ZAM {{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 }}{{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}}{{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}} distribution{{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}} or ZAMD), is one of the members of Cohen's class distribution function.{{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}} It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990.{{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 }} The distribution's name stems from the twin cone shape of the distribution's kernel function on the $t, \tau$ plane.{{cite book|last1=R.J. Marks II|title=Handbook of Fourier analysis & its applications|date=2009|publisher=Oxford University Press}} 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.{{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 }}{{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}}

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:

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

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

Image:cone shape 1.jpg

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

Image:cone shape 2.jpg

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[http://tftb.nongnu.org/refguide.pdf] Time-Frequency Toolbox For Use with MATLAB and National Instruments' LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis [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

References

Category:Time–frequency analysis

Category:Transforms

Cone-shape distribution function

Mathematical definition

See also

References