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| In pulsed [[radar]] and [[sonar]] signal processing, an '''ambiguity function''' is a two-dimensional function of time delay and Doppler frequency
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| <math>\chi(\tau,f)</math> showing 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) due to the [[Doppler shift]] of the return from a moving target. The ambiguity
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| 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; Some 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>).
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| For a given [[Complex number|complex]] [[baseband]] pulse <math>s(t)</math>, the narrowband ambiguity function is given by
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| :<math>\chi(\tau,f)=\int_{-\infty}^\infty s(t)s^*(t-\tau) e^{i 2 \pi f t} \, dt</math>
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| 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
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| ambiguity function consists of examining the one-dimensional
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| zero-delay and zero-Doppler "cuts"; that is, <math>\chi(0,f)</math> and
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| <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 delay cut, with constant frequency given by the target's Doppler shift: <math>\chi(\tau,f_D)</math>.
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| ==Relationship to time–frequency distributions==
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| The ambiguity function plays a key role in the field of [[time–frequency signal processing]], as it is related to the [[Wigner–Ville distribution]] by a 2 Dimensional [[Fourier transform]]. This relationship is fundamental to the formulation of other [[time–frequency distribution]]s which are obtained by a 2-dimensional filtering in the ambiguity domain (that is, the ambiguity function of the signal), leading to the definition of a class of TFDs that are better adapted to the signals considered.<ref>B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0-08-044335-4</ref>
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| ==Wideband ambiguity function==
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| The wideband ambiguity function of <math>s \in L^2(R)</math> is:<ref name="Weiss"/><ref>L. Sibul, L. Ziomek, "Generalised wideband crossambiguity function", IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP '81.01/05/198105/1981; 6:1239–1242.</ref>
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| :<math>WB_{ss}(\tau,\alpha)=\sqrt{|{\alpha}|}\int_{-\infty}^\infty s(t)s^*(\alpha (t-\tau)) \, dt</math>
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| where ''<math>{\alpha}</math>'' is a time scale factor of the received signal relative to the transmitted signal given by:
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| :<math>\alpha = \frac{c-v}{c+v}</math>
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| for a target moving with constant radial velocity ''v''. The reflection of the signal is represented with compression (or expansion) in time by the factor ''<math> \alpha </math>'', which is equivalent to a compression by the factor ''<math>\alpha^{-1}</math>'' in the frequency domain (with an amplitude scaling). When the wave speed in the medium is sufficiently faster than the target speed, as is common with radar, this '''compression''' in frequency is closely approximated by a '''shift''' in frequency Δf = f<sub>c</sub>*v/c (known as the [[doppler shift]]). For a narrow band signal, this approximation results in the narrowband ambiguity function given above, which can be computed efficiently by making use of the [[Fast Fourier transform|FFT]] algorithm.
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| ==Ideal ambiguity function==
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| An ambiguity function of interest is a 2-dimensional [[Dirac delta function]] or
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| "thumbtack" function; that is, a function which is infinite at (0,0) and
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| zero elsewhere.
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| :<math>\chi(\tau,f) = \delta(\tau) \delta(f) \, </math>
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| An ambiguity function of this kind would be somewhat of a misnomer; it
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| would have no ambiguities at all, and both the zero-delay and zero-Doppler cuts would be an [[Dirac delta function|impulse]]. However, any Doppler shift would make the target disappear. This is not desirable if a target has unknown velocity it will disappear from the radar picture, but if Doppler
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| processing is independently performed, knowledge of the precise
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| Doppler frequency allows ranging without interference
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| from any other targets which are not also moving at exactly the same
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| velocity.
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| This type of ambiguity function is not physically realizable; that is, 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 binary phase-shift keyed waveforms using maximal-length sequences are the best known performers in this regard
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| .<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>
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| == Properties of the ambiguity function ==
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| (1) Maximum value
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| :<math>|\chi(\tau,f)|^2 \le |\chi(0,0)|^2</math>
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| (2) Symmetry about the origin
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| :<math>\chi(\tau,f) = \exp[j2\pi \tau f]\chi^{*}(-\tau,-f) \, </math>
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| (3) Volume invariance
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| :<math>\int_{-\infty}^\infty \int_{-\infty}^\infty |\chi(\tau,f)|^2 \, d\tau \,df=|\chi(0,0)|^2 = E^2</math>
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| (4) Modulation
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| : <math>\text{If } s(t) \rightarrow |\chi(\tau,f)| \text{ then }s(t) \exp[j\pi kt^2] {\rightarrow} |\chi(\tau,f+kt)| \, </math>
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| (5) Frequency energy spectrum
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| :<math>S(f)S^*(f) = \int_{-\infty}^\infty \chi(\tau,0) e^{-j2\pi\tau f} \, d\tau </math>
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| == Square pulse ==
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| [[Image:Square pulse ambiguity function.png|280px|thumb|right|Ambiguity function for a square pulse]]
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| Consider a simple square pulse of duration <math>\tau</math> and
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| amplitude <math>A</math>:
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| :<math>A (u(t)-u(t-\tau)) \, </math>
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| where <math>u(t)</math> is the [[Heaviside step function]]. The
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| matched filter output is given by the [[autocorrelation]] of the pulse, which is a triangular pulse of height <math>\tau A^2</math> and
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| duration <math>2 \tau</math> (the zero-Doppler cut). However, if the
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| measured pulse has a frequency offset due to Doppler shift, the
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| matched filter output is distorted into a [[sinc function]]. The
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| greater the Doppler shift, the smaller the peak of the resulting sinc,
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| and the more difficult it is to detect the target.
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| In general, the square pulse is not a desirable waveform from a pulse compression standpoint, because the autocorrelation function is too short in amplitude, making it difficult to detect targets in noise, and too wide in time, making it difficult to discern multiple overlapping targets.
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| == LFM pulse ==
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| [[Image:Lfm ambiguity function.png|280px|thumb|right|Ambiguity function for an LFM pulse]]
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| A commonly used [[radar]] or [[sonar]] pulse is the linear frequency modulated (LFM) pulse (or "chirp"). It has the advantage of greater bandwidth while keeping the pulse duration short and envelope constant. A [[constant envelope]] LFM pulse has an ambiguity function similar to that of the square pulse, except that it is skewed in the delay-Doppler plane. Slight Doppler mismatches for the LFM pulse do not change the general shape of the pulse and reduce the amplitude very little, but they do appear to shift the pulse
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| in time. Thus, an uncompensated Doppler shift changes the target's apparent range; this phenomenon is called range-Doppler coupling.
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| ==Multistatic ambiguity functions==
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| The ambiguity function can be extended to multistatic radars, which comprise multiple non-colocated transmitters and/or receivers (and can include [[bistatic radar]] as a special case).
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| For these types of radar, the simple linear relationship between time and range that exists in the monostatic case no longer applies, and is instead dependent on the specific geometry – i.e. the relative location of transmitter(s), receiver(s) and target. Therefore the multistatic ambiguity function is mostly usefully defined as a function of two- or three-dimensional position and velocity vectors for a given multistatic geometry and transmitted waveform.
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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>
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| == See also ==
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| * [[Matched filter]]
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| * [[Pulse compression]]
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| * [[Pulse-Doppler radar]]
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| * [[Digital signal processing]]
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| == References ==
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| {{reflist|1}}
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| == Further reading ==
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| * Richards, Mark A. ''Fundamentals of Radar Signal Processing''. McGraw–Hill Inc., 2005. ISBN 0-07-144474-2.
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| * Ipatov, Valery P. ''Spread Spectrum and CDMA''. Wiley & Sons, 2005. ISBN 0-470-09178-9
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| * Chernyak V.S. ''Fundamentals of Multisite Radar Systems'', CRC Press, 1998.
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| {{DEFAULTSORT:Ambiguity Function}}
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| [[Category:Time–frequency analysis]]
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| [[Category:Signal processing]]
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