Ambiguity function: Difference between revisions

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In{{Short pulsed [[radar]] and [[sonar]] signal processing, an '''ambiguity function''' is a two-dimensional functiondescription|Function of timepropagation 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 showingrepresents 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) dueof tothe return from a moving target. The ambiguity function is defined by the properties of the [[DopplerPulse shift(signal processing)|pulse]] and of the returnfilter, fromand anot movingany particular target. The ambiguityscenario.

function is determined by the properties of the [[Pulse (signal processing)|pulse]] and the [[matched filter]], and not any particular target scenario. Many definitions of the ambiguity function exist; Somesome are restricted to narrowband signals and others are suitable to describe the [[propagation 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>).

For a given [[Complex number|complex]] [[baseband]] pulse <math>s(t)</math>, the narrowband ambiguity function is given by

:<math>\chi(\tau,f)=\int_{-\infty}^\infty s(t)s^*(t-\tau) e^{i 2 \pi f t} \, dt</math>

where <math>^*</math> denotes the [[complex conjugate]] and <math>i</math> is the [[imaginary unit]]. Note that for zero Doppler shift (<math>f=0</math>), this reduces to the [[autocorrelation]] of <math>s(t)</math>. A more concise way of representing the

ambiguity function consists of examining the one-dimensional

zero-delay and zero-Doppler "cuts"; that is, <math>\chi(0,f)</math> and

<math>\chi(\tau,0)</math>, respectively. The matched filter output as a function of a time (the signal one would observe in a radar system) is a Doppler cut, with the constant frequency given by the target's Doppler shift: <math>\chi(\tau,f_D)</math>.

==Background and motivation==

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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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Just as the monostatic ambiguity function is naturally derived from the matched filter, the multistatic ambiguity function is derived from the corresponding optimal ''multistatic'' detector – i.e. that which maximizes the probability of detection given a fixed probability of false alarm through joint processing of the signals at all receivers. The nature of this detection algorithm depends on whether or not the target fluctuations observed by each bistatic pair within the multistatic system are mutually correlated. If so, the optimal detector performs phase coherent summation of received signals which can result in very high target ___location accuracy.<ref>T. Derham, S. Doughty, C. Baker, K. Woodbridge, [http://sites.google.com/site/thomasderham/Home/AmbiguityFunctionsforSpatiallyCoherentandIncoherentMultistaticRadar.pdf?attredirects=0 "Ambiguity Functions for Spatially Coherent and Incoherent Multistatic Radar,"] IEEE Trans. Aerospace and Electronic Systems (in press).</ref> If not, the optimal detector performs incoherent summation of received signals which gives diversity gain. Such systems are sometimes described as ''MIMO radars'' due to the information theoretic similarities to [[MIMO]] communication systems.<ref>G. San Antonio, D. Fuhrmann, F. Robey, "MIMO radar ambiguity functions," IEEE Journal of Selected Topics in Signal Processing, Vol. 1, No. 1 (2007).</ref>

[[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

stationary random process.

==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>

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

:<math>= \int^\infty_{-\infty}e^{-\alpha\pi (t+\tau/2-t_{0})^{2}+j2\pi f_{0}(t+\tau/2)}+e^{-\alpha\pi (t-\tau/2-t_{0})^{2}-j2\pi f_{0}(t-\tau/2)}e^{-j2\pi tn}dt</math>

:<math>= \int^\infty_{-\infty}e^{-\alpha\pi [2(t-t_{0})^{2}+\tau^{2}/2]+j2\pi f_{0}\tau}e^{-j2\pi tn}dt</math>

:<math>= \int^\infty_{-\infty}e^{-\alpha\pi [2t^{2}-\tau^{2}/2]+j2\pi f_{0}\tau}e^{-j2\pi tn}e^{-j2\pi t_{0}n}dt</math>

:<math>A_{x}(\tau,n) = \sqrt\frac{1}{2\alpha}exp[-\pi (\frac{\alpha\tau^{2}}{2}+\frac{n^{2}}{2\alpha})]exp[j2\pi (f_{0}\tau-t_{0}n)]</math>

<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>

<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>,

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

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

*<math>t_{d} = t_{1}+t_{2}</math>,

*<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>

[[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

== See also ==

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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. "[httphttps://ieeexplore.ieee.org/xpldocument/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]

*[http://djj.ee.ntu.edu.tw/TFW_Writing4.pdf 4 National Taiwan University, Time-Frequency Analysis and Wavelet Transform 2021, Professor of Jian-Jiun Ding, Department of Electrical Engineering]

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