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

Abstract

The uniqueness theorem for ambiguity functions states that ff waveformsu(t)andv(t)have the same ambiguity function, i.e.,chi_{u}(tau, Delta) = X_{upsilon , Delta), thenu(t)andv(t)are identical except for a rotation, i.e.,v(t) = e^{ilambda}u(t), wherelambdais a real constant. Through the artifice of treating the even and odd parts of the waveforms, denotede(t)ando(t), respectively, correlative results have been obtained for the real and imaginary parts of ambiguity functions. Thus, ifRe {chi_{u}(tau, Delta)} = Re {chi_{upsilon}(tau, Delta)}, thene_{upsilon}(t) = e^{i lambda e}e_{u}(t)ando_{upsilon}(t) = e^{i lambda o}o_{u}(t). FromRe {chi_{u}(tau, Delta)}, the waveform classu(t) = e^{ilambda} [e_{u}(t) + e^{ik}o_{u}(t)]may be constructed, but because of the arbitrary rotation,e^{ik}, a uniquechi_{u}-function is not determinable, in general. An important exception to this statement is the case whenchi_{u}(tau, Delta)is real, andRe {chi_{u}} = chi_{u}determines a unique waveform (within a rotation) and this waveform can only be even or odd. IfIm {chi_{u}(tau, Delta)} = Im {chi_{upsilon(tau, Delta)}thene_{upsilon}(t) =ae^{i gamma}e_{u}(t)ando_{upsilon}(t) = 1/ae^{ir}o_{u}(t). IfIm {chi_{u}(tau, Delta)}is given, {em and}u(t)is known to have unit energy, then within rotations of the forme^{i lambda}, only two possible waveform choices are possible foru(t). If it also is known which ofe_{u}(t)ando_{u}(t)has the greater energy, the functionIm {chi_{u}(tau, Delta)}uniquely determinesu(t)(within a rotation) and the completechi_{u}-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. Self-reciprocal two-dimensional Fourier transforms are demonstrated for the product of the squared-magnitude function and either of the first partial derivat- ives of the phase function.