Ambiguity function: Difference between revisions

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In pulsed [[radar]] and [[sonar]] signal processing, an '''ambiguity function''' is a two-dimensional function of [[propagation delay]] <math>\tau</math> and [[Doppler frequency]] <math>f</math>, <math>\chi(\tau,f)</math>. It represents the [[distortion]] of a returned pulse due to the receiver [[matched filter]]<ref>[[Philip Woodward|Woodward P.M.]] ''Probability and Information Theory with Applications to Radar'', Norwood, MA: Artech House, 1980.</ref> (commonly, but not exclusively, used in [[pulse compression]] radar) of the return from a moving target. The ambiguity function is defined by the properties of the [[Pulse (signal processing)|pulse]] and of the filter, and not any particular target scenario.

Many definitions of the ambiguity function exist; some are restricted to narrowband signals and others are suitable to describe the delay and Doppler relationship of wideband signals. Often the definition of the ambiguity function is given as the magnitude squared of other definitions (Weiss<ref name="Weiss">Weiss, Lora G. "Wavelets and Wideband Correlation Processing". ''IEEE Signal Processing Magazine'', pp. 13–32, Jan 1994</ref>).

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(6) Upper bounds for <math> p>2 </math> and lower bounds for <math> p<2 </math> exist <ref>E. H. Lieb,

"Integral Bounds for Radar Ambiguity Functions and Wigner Distributions", J. Math. Phys., vol. 31, pp.594-599 (1990)

</ref> for the <math> p^{th} </math> power integrals

:<math>\int_{-\infty}^\infty \int_{-\infty}^\infty |\chi(\tau,f)|^p \, d\tau \,df </math>.

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[[File:Ambiguity function plane.png|thumb|Ambiguity function plane]]

==Ambiguity function plane==

An ambiguity function plane can be viewed as a combination of an infinite

number of radial lines. <br>

Each radial line can be viewed as the fractional Fourier transform of a

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[[File:Ambiguity function figure.png|thumb|Ambiguity function]]

The Ambiguity function (AF) is the operators that are related to the WDF.<br>

:<math>A_{x}(\tau,n) = \int^\infty_{-\infty}x(t+\frac{\tau}{2}) x^{*}(t-\frac{\tau}{2}) e^{-j 2 \pi tn} dt</math><br>

(1)If <math>x(t) = exp[-\alpha\pi{(t-t_{0})^2} + j2\pi f_{0}t]</math><br>

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<br>

[[File:Wdf Ambiguity function plane.png|thumb|Wdf Ambiguity function plane]]

WDF and AF for the signal with only 1 term<br>

(2) If <math>x(t) = exp[-\alpha_{1}\pi (t-t_{1})^{2}+j2\pi f_{1}t] + exp[-\alpha_{2}\pi (t-t_{2})^{2}+j2\pi f_{2}t]</math>

:<math>A_{x}(\tau,n)</math>

:<math>= \int^\infty_{-\infty}x_{1}(t+\tau/2)x_{1}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> +<br>

:<math>\int^\infty_{-\infty}x_{2}(t+\tau/2)x_{2}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> + <br>

:<math>\int^\infty_{-\infty}x_{1}(t+\tau/2)x_{2}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> +<br>

:<math> \int^\infty_{-\infty}x_{2}(t+\tau/2)x_{1}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math><br>

:<math>A_{x}(\tau,n) = A_{x1}(\tau,n) + A_{x2}(\tau,n) + A_{x1x2}(\tau,n) + A_{x2x1}(\tau,n)</math><br>

<br>

:<math>A_{x}(\tau,n) = A_{x1}(\tau,n) + A_{x2}(\tau,n) + A_{x1x2}(\tau,n) + A_{x2x1}(\tau,n)</math><br>

<br>

:<math>A_{x}(\tau,n) = \sqrt\frac{1}{2\alpha_{1}}exp[-\pi (\frac{\alpha_{1}\tau^{2}}{2}+\frac{n^{2}}{2\alpha_{1}})]exp[j2\pi (f_{1}\tau-t_{1}n)]</math>

:<math>A_{x}(\tau,n) = \sqrt\frac{1}{2\alpha_{2}}exp[-\pi (\frac{\alpha_{2}\tau^{2}}{2}+\frac{n^{2}}{2\alpha_{1}})]exp[j2\pi (f_{2}\tau-t_{2}n)]</math><br>

<br>

When <math>\alpha_{1} = \alpha_{2}</math>

:<math>A_{x1x2}(\tau,n) = \sqrt\frac{1}{2\alpha_{u}}exp[-\pi (\alpha_{u}\frac{(\tau -t_{d})^{2}}{2}+\frac{(n-f_{d})^{2}}{2\alpha_{u}})]exp[j2\pi (f_{u}\tau-t_{u}n+f_{d}t_{u})]</math><br>

where<br>

*<math>t_{u} = (t_{1}+t_{2}/2)</math>,

*<math>f_{u} = (f_{1}+f_{2})/2</math>,

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*<math>f_{d} = f_{1}-f_{2}</math>,

*<math>\alpha_{d} = \alpha_{1}-\alpha_{2}</math>

*:<math>A_{x2x1}(\tau,n) = A_{x1x2}^{*}(-\tau,-n)</math>

When <math>\alpha_{1}</math> ≠ <math>\alpha_{2}</math>

:<math>A_{x1x2}(\tau,n) = \sqrt\frac{1}{2\alpha_{u}}exp[-\pi \frac{[(n-f_{d})+j(\alpha_{1}t_{1}+\alpha_{2}t_{2})-j\alpha_{d}\tau /2]^{2}}{2\alpha_{u}}exp[-\pi(\alpha_{1}(t_{1}-\frac{\tau}{2})^{2})+\alpha_{2}(t_{2}-\frac{\tau}{2})^{2})]exp[j2\pi

f_{u}\tau]</math><br>

[[File:WDF AF 2.png|thumb|WDF and AF for the signal with 2 terms]]

*:<math>A_{x2x1}(\tau,n) = A_{x1x2}^{*}(-\tau,-n)</math>

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WDF and AF for the signal with 2 terms<br>

<br>

For the ambiguity function:<br>

*The auto term is always near to the origin

*The auto term is always near to the origin

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* Augusto Aubry, Antonio De Maio, Bo Jiang, and Shuzhong Zhang. "[https://ieeexplore.ieee.org/document/6563125 Ambiguity function shaping for cognitive radar via complex quartic optimization]." IEEE Transactions on Signal Processing 61 (2013): 5603-5619.

* Mojtaba Soltanalian, and Petre Stoica. "[http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6142119 Computational design of sequences with good correlation properties]." IEEE Transactions on Signal Processing, 60.5 (2012): 2180-2193.

* G. Krötzsch, M. A. Gómez-Méndez, Transformada Discreta de Ambigüedad, Revista Mexicana de Física, Vol. 63, pp. 505--515&nbsp;505–515 (2017). "[https://rmf.smf.mx/pdf/rmf/63/6/63_6_505.pdf Transformada Discreta de Ambigüedad]".

*[http://djj.ee.ntu.edu.tw/TFW_Writing2.pdf 2 National Taiwan University, Time-Frequency Analysis and Wavelet Transform 2021, Professor of Jian-Jiun Ding, Department of Electrical Engineering]

*[http://djj.ee.ntu.edu.tw/TFW_Writing3.pdf 3 National Taiwan University, Time-Frequency Analysis and Wavelet Transform 2021, Professor of Jian-Jiun Ding, Department of Electrical Engineering]