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Ambiguity function
In pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of time delay and Doppler frequency
$\chi\left(\tau,f\right)$ showing the distortion of a returned pulse due to the receiver matched filterWoodward P.M. ''Probability and Information Theory with Applications to Radar'', Norwood, MA: Artech House, 1980.〕 (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 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〔Weiss, Lora G. "Wavelets and Wideband Correlation Processing". ''IEEE Signal Processing Magazine'', pp. 13–32, Jan 1994〕).
For a given complex baseband pulse $s\left(t\right)$, the narrowband ambiguity function is given by
:$\chi\left(\tau,f\right)=\int_^\infty s\left(t\right)s^$
*(t-\tau) e^ \, dt
where $^$
* denotes the complex conjugate and $i$ is the imaginary unit. Note that for zero Doppler shift ($f=0$) this reduces to the autocorrelation of $s\left(t\right)$. A more concise way of representing the
ambiguity function consists of examining the one-dimensional
zero-delay and zero-Doppler "cuts"; that is, $\chi\left(0,f\right)$ and
$\chi\left(\tau,0\right)$, 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: $\chi\left(\tau,f_D\right)$.
==Relationship to time–frequency distributions==

The ambiguity function plays a key role in the field of time–frequency signal processing,〔E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” ''Digital Signal Processing'', vol. 19, no. 1, pp. 153-183, January 2009.〕 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 distributions: the bilinear time–frequency distributions are obtained by a 2-dimensional filtering in the ambiguity domain (that is, the ambiguity function of the signal). This class of distribution may be better adapted to the signals considered.〔B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0-08-044335-4〕
Moreover, the ambiguity distribution can be seen as the short-time Fourier transform of a signal using the signal itself as the window function. This remark has been used to define an ambiguity distribution over the time-scale domain instead of the time-frequency domain.〔Shenoy, R.G.; Parks, T.W., "Affine Wigner distributions," IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP-92., pp.185-188 vol.5, 23-26 Mar 1992, (doi: 10.1109/ICASSP.1992.226539 )〕

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