correlation between and for . This motivates the ambiguity function . The defining property of is that the correlation between and is equal to . Different pulse shapes (waveforms) have different ambiguity functions, and the ambiguity function is relevant when choosing what pulse to use. The function is complex-valued; the degree of "ambiguity" is related to its magnitude . Relationship to time–frequency distributions The ambiguity function plays a key role in the field of time–frequency signal processing, 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. 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. Wideband ambiguity function The wideband ambiguity function of is: where is a time scale factor of the received signal relative to the transmitted signal given by: 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 , which is equivalent to a compression by the factor 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 = fc*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 FFT algorithm. Ideal ambiguity function An ambiguity function of interest is a 2-dimensional Dirac delta function or "thumbtack" function; that is, a function which is infinite at (0,0) and zero elsewhere. An ambiguity function of this kind would be somewhat of a misnomer; it would have no ambiguities at all, and both the zero-delay and zero-Doppler cuts would be an impulse. This is not usually desirable (if a target has any Doppler shift from an unknown velocity it will disappear from the radar picture), but if Doppler processing is independently