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VulcanSphere (talk | contribs) m Adding local short description: "Function of propagation delay and Doppler frequency", overriding Wikidata description "function of propagation delay and Doppler frequency, representing the distortion of a returned pulse due to the receiver matched filter of the return from a moving target" |
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{{Short description|Function of propagation delay and Doppler frequency}}
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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An ambiguity function of this kind would be somewhat of a misnomer; it would have no ambiguities at all, and both the zero-delay and zero-Doppler cuts would be an [[Dirac delta function|impulse]]. This is not usually desirable (if a target has any Doppler shift from an unknown velocity it will disappear from the radar picture), but if Doppler processing is independently performed, knowledge of the precise Doppler frequency allows ranging without interference from any other targets which are not also moving at exactly the same velocity.
This type of ambiguity function is produced by ideal [[white noise]] (infinite in duration and infinite in bandwidth).<ref>Signal Processing in Noise Waveform Radar By Krzysztof Kulpa (Google Books)</ref> However, this would require infinite power and is not physically realizable. There is no pulse <math>s(t)</math> that will produce <math>\delta(\tau) \delta(f)</math> from the definition of the ambiguity function. Approximations exist, however, and noise-like signals such as binary phase-shift keyed waveforms using [[Maximum length sequence|maximal-length sequences]] are the best known performers in this regard.<ref>G. Jourdain and J. P. Henrioux, "Use of large bandwidth-duration binary phase shift keying signals in target delay Doppler measurements," J. Acoust. Soc. Am. 90, 299–309 (1991).</ref>
== Properties ==
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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.
Each radial line can be viewed as the fractional Fourier transform of a
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==Example==
[[File:Ambiguity function figure.png|thumb|Ambiguity function]]
The Ambiguity function (AF) is the operators that are related to the [[Wigner distribution function|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
(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
(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> +
:<math>\int^\infty_{-\infty}x_{2}(t+\tau/2)x_{2}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> +
:<math>\int^\infty_{-\infty}x_{1}(t+\tau/2)x_{2}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math> +
:<math> \int^\infty_{-\infty}x_{2}(t+\tau/2)x_{1}^{*}(t-\tau/2)e^{-j2\pi tn}dt</math
:<math>A_{x}(\tau,n) = A_{x1}(\tau,n) + A_{x2}(\tau,n) + A_{x1x2}(\tau,n) + A_{x2x1}(\tau,n)</math
▲:<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>
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
where
*<math>t_{u} = (t_{1}+t_{2}/2)
*<math>f_{u} = (f_{1}+f_{2})/2
*<math>\alpha_{u} = (\alpha_{1}+\alpha_{2})/2
*<math>t_{d} = t_{1}+t_{2}
*<math>f_{d} = f_{1}-f_{2}
*<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
[[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
WDF and AF for the signal with 2 terms<br>
<br>
For the ambiguity function:
*The auto term is always near to the origin
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* Ipatov, Valery P. ''Spread Spectrum and CDMA''. Wiley & Sons, 2005. {{ISBN|0-470-09178-9}}
* Chernyak V.S. ''Fundamentals of Multisite Radar Systems'', CRC Press, 1998.
* Solomon W. Golomb, and [[Guang Gong]]. [http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/signal-design-good-correlation-wireless-communication-cryptography-and-radar Signal design for good correlation: for wireless communication, cryptography, and radar]. Cambridge University Press, 2005.
* M. Soltanalian. [http://theses.eurasip.org/theses/573/signal-design-for-active-sensing-and/download/ Signal Design for Active Sensing and Communications]. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.
* Nadav Levanon, and Eli Mozeson. Radar signals. Wiley. com, 2004.
* 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. "[
* G. Krötzsch, M. A. Gómez-Méndez, Transformada Discreta de Ambigüedad, Revista Mexicana de Física, Vol. 63, pp.
*[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]
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