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

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The uniqueness theorem for ambiguity functions states that ff waveforms u(t) and v(t) have the same ambiguity function, i.e., \chi_{u}(\tau , \Delta ) = X_{\upsilon}(\tau , \Delta ) , then u(t) and v(t) are identical except for a rotation, i.e., v(t) = e^{i\lambda }u(t) , where \lambda is a real constant. Through the artifice of treating the even and odd parts of the waveforms, denoted e(t) and o(t) , respectively, correlative results have been obtained for the real and imaginary parts of ambiguity functions. Thus, if \Re {\chi _{u}(\tau , \Delta )} = \Re {\chi _{\upsilon }(\tau , \Delta )} , then e_{\upsilon }(t) = e^{i \lambda e}e_{u}(t) and o_{\upsilon }(t) = e^{i \lambda o}o_{u}(t) . From \Re {\chi _{u}(\tau , \Delta )} , the waveform class u(t) = e^{i\lambda } [e_{u}(t) + e^{ik}o_{u}(t)] may be constructed, but because of the arbitrary rotation, e^{ik} , a unique \chi _{u} -function is not determinable, in general. An important exception to this statement is the case when \chi _{u}(\tau , \Delta ) is real, and \Re {\chi _{u}} = \chi _{u} determines a unique waveform (within a rotation) and this waveform can only be even or odd. If Im \{ \chi_{u}(\tau , \Delta ) \} = Im \{ \chi_{\upsilon} (\tau , \Delta ) \} then e_{\upsilon }(t) =ae^{i \gamma }e_{u}(t) and o_{\upsilon }(t) = 1/ae^{ir}o_{u}(t) . If \Im {\chi _{u}(\tau , \Delta )} is given, {em and} u(t) is known to have unit energy, then within rotations of the form e^{i \lambda } , only two possible waveform choices are possible for u(t) . If it also is known which of e_{u}(t) and o_{u}(t) has the greater energy, the function \Im {\chi _{u}(\tau , \Delta )} uniquely determines u(t) (within a rotation) and the complete \chi _{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.

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Information Theory, IEEE Transactions on  (Volume:10 ,  Issue: 4 )