Editing Ambiguity function (section)

==Background and motivation==

[[Pulse-Doppler radar]] equipment sends out a series of [[radio frequency]] pulses. Each pulse has a certain shape (waveform)—how long the pulse is, what its frequency is, whether the frequency changes during the pulse, and so on. If the waves reflect off a single object, the detector will see a signal which, in the simplest case, is a copy of the original pulse but delayed by a certain time <math>\tau</math>—related to the object's distance—and shifted by a certain frequency <math>f</math>—related to the object's velocity ([[Doppler shift]]). If the original emitted pulse waveform is <math>s(t)</math>, then the detected signal (neglecting noise, attenuation, and distortion, and wideband corrections) will be:

:<math>s_{\tau,f}(t) \equiv s(t-\tau)e^{i 2\pi f t}.</math>

The detected signal will never be ''exactly'' equal to any <math>s_{\tau,f}</math> because of noise. Nevertheless, if the detected signal has a high correlation with <math>s_{\tau,f}</math>, for a certain delay and Doppler shift <math>(\tau,f)</math>, then that suggests that there is an object with <math>(\tau,f)</math>. Unfortunately, this procedure may yield [[false positive]]s, i.e. wrong values <math>(\tau',f')</math> which are nevertheless highly correlated with the detected signal. In this sense, the detected signal may be ''ambiguous''.

The ambiguity occurs specifically when there is a high correlation between <math>s_{\tau,f}</math> and <math>s_{\tau',f'}</math> for <math>(\tau,f) \neq (\tau',f')</math>. This motivates the ''ambiguity function'' <math>\chi</math>. The defining property of <math>\chi</math> is that the correlation between <math>s_{\tau,f}</math> and <math>s_{\tau',f'}</math> is equal to <math>\chi(\tau-\tau', f-f')</math>.

Different pulse shapes (waveforms) <math>s(t)</math> have different ambiguity functions, and the ambiguity function is relevant when choosing what pulse to use.

The function <math>\chi</math> is complex-valued; the degree of "ambiguity" is related to its magnitude <math>|\chi(\tau,f)|^2</math>.