Ambiguity function
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2008-01-12T11:25:53Z
Thunderbird2
3817137
/* LFM Pulse */ or sonar
In pulsed [[radar]] and [[sonar]] signal processing, an '''ambiguity function''' is
a two-dimensional function of time delay and Doppler frequency
<math>\chi(\tau,f)</math> showing the [[distortion]] of an uncompensated
[[matched filter]] (sometimes called [[pulse compression]]) due to the
[[Doppler shift]] of the return from a moving target. The ambiguity
function is determined by the properties of the [[Pulse (signal processing)|pulse]] used, 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).
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 delay cut, with constant frequency given by the target's Doppler shift: <math>\chi(\tau,f_{D})</math>.
==Wideband ambiguity function==
The wideband ambiguity function of <math>s \in L^2(R)</math> is (Sibul and Ziomek, 1981 in Weiss, 1994)
:<math>WB_{ss}(\tau,\alpha)=\sqrt{|{\alpha}|}\int_{-\infty}^{\infty}s(t)s^*({\alpha}(t-\tau)) dt</math>
where ''<math>{\alpha}</math>'' is a time scale factor of the received signal relative to the transmitted signal given by:
:<math>{\alpha} = \frac{c-v}{c+v}</math>
for an object moving with constant radial velocity ''v''.
==Ideal ambiguity function==
An ambiguity function of interest is a 2-dimensional [[Dirac delta function]] or
"thumbtack" function; that is, a function which is infinite at (0,0) and
zero elsewhere.
:<math>\chi(\tau,f) = \delta(\tau) \delta(f)</math>
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]]. 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
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 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
<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 of the ambiguity function ==
(1) Maximum value
:<math>|{\chi}(\tau,f)|^2\le|{\chi}(0,0)|^2</math>
(2) Symmetry about the origin
:<math>{\chi}(\tau,f) = {\chi}^{*}(-\tau,-f)</math>
(3) Volume invariance
:<math>\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|{\chi}(\tau,f)|^2d{\tau}df=|{\chi}(0,0)|^2=E^2</math>
(4) Modulation
:If s(t) <math> {\rightarrow} |{\chi}({\tau},f)| </math> then <math>s(t){\exp}^{j{\pi}kt^2} {\rightarrow} |{\chi}(\tau,v+kt)|</math>
(5) Frequency energy spectrum
:<math>S(f)S^*(f) = \int_{-\infty}^{\infty}{\chi}(\tau,0) e^{-j2\pi\tau f} d{\tau}</math>
== Square Pulse ==
[[Image:Square_pulse_ambiguity_function.png|280px|thumb|right|Ambiguity
function for a square pulse]]
Consider a simple square pulse of duration <math>\tau</math> and
amplitude <math>A</math>:
:<math>A (u(t)-u(t-\tau))</math>
where <math>u(t)</math> is the [[Heaviside step function]]. The
matched filter output is given by the [[autocorrelation]] of the pulse, which is a triangular pulse of height <math>\tau A^2</math> and
duration <math>2 \tau</math> (the zero-Doppler cut). However, if the
measured pulse has a frequency offset due to Doppler shift, the
matched filter output is distorted into a [[sinc function]]. The
greater the Doppler shift, the smaller the peak of the resulting sinc,
and the more difficult it is to detect the target.
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.
== LFM Pulse ==
[[Image:Lfm_ambiguity_function.png|280px|thumb|right|Ambiguity
function for an LFM pulse]]
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
in time. Thus, an uncompensated Doppler shift changes the target's apparent range;
this phenomenon is called range-Doppler coupling.
== References ==
{{reflist|1}}
*Richards, Mark A. ''Fundamentals of Radar Signal Processing''. McGraw-Hill Inc., 2005. ISBN 0-07-144474-2.
*Ipatov, Valery P. ''Spread Spectrum and CDMA''. Wiley & Sons, 2005. ISBN 0-470-09178-9
*Weiss, Lora G. "Wavelets and Wideband Correlation Processing". ''IEEE Signal Processing Magazine'', pp. 13-32, Jan 1994
*Woodward P.M. ''Probability and Information Theory with Applications to Radar'', Norwood, MA: Artech House, 1980.
== See also ==
*[[Matched filter]]
*[[Pulse compression]]
*[[Pulse-Doppler radar]]
*[[Digital signal processing]]
[[Category:Signal processing]]