Some results on real-part/imaginary-part and magnitude-phase relations in ambiguity functions

The uniqueness theorem for ambiguity functions states that if waveforms u(f) and u(f) have the same ambiguity function, i.e., X%(‘S, A) = X.(‘S, A), then u(f) and u(f) are identical except for a rotation, i.e., u(f) = e%(f), where ;i is a real constant. Through the artifice of treating the even and odd parts of the waveforms, denoted e(f) and o(f), respectively, correlative results have been obtained for the real and imaginary parts of ambiguity functions. Thus, if Re (~JT, A) ] = Re (x&z, A) ), then e,(f) = eixeeU(f) and o,(f) = eiAootL(f). From Re (~~(7, A) ), the waveform class u(f) = eax [e%(f) + e”koJf)] may be constructed, but because of the aritrary rotation, eik, a unique Xu-function is not determinable, in general. An important exception to this statement is the case when ~~(7, A) is real, and Re ( xu } = xti determines a unique waveform (within a rotation) and this waveform can only be even or odd. If Im (x~(T, A)] = Zm (xv(s, A)) then e,(f) = aeiYe,(f) and o,(f) = l/aeiVoU(f). If Zm (~~(7, A) ) is given, and u(f) is known to have unit energy, then within rotations of the form eih, only two possible waveform choices are possible for u(f). If it also is known which of e%(f) and o,(f) has the greater energy, the function Zm (x%(T, A) ] uniquely determines u(f) (within a rotation) and the complete ,y,,-function. The results on magnitude/phase relationships include a formula which enables one to compute the squared magnitude of an ambiguity function as an ordinary two-dimensional correlation function. Selfreciprocal two-dimensional Fourier transforms are demonstrated for the product of the squared-magnitude function and either of the first partial derivatives of the phase function.