Editing Ambiguity function (section)

==Wideband ambiguity function==

The wideband ambiguity function of <math>s \in L^2(R)</math> is:<ref name="Weiss"/><ref>L. Sibul, L. Ziomek, "Generalised wideband crossambiguity function", IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP '81.01/05/198105/1981; 6:1239–1242.</ref>

:<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 a target moving with constant radial velocity ''v''. The reflection of the signal is represented with compression (or expansion) in time by the factor ''<math> \alpha </math>'', which is equivalent to a compression by the factor ''<math>\alpha^{-1}</math>'' in the frequency domain (with an amplitude scaling). When the wave speed in the medium is sufficiently faster than the target speed, as is common with radar, this '''compression''' in frequency is closely approximated by a '''shift''' in frequency Δf = f<sub>c</sub>*v/c (known as the [[doppler shift]]). For a narrow band signal, this approximation results in the narrowband ambiguity function given above, which can be computed efficiently by making use of the [[Fast Fourier transform|FFT]] algorithm.