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 scenario. The ambiguity▼
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 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
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
