Editing Ambiguity function (section)

{{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>). 
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 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>.