مرکز منطقه ای اطلاع رساني علوم و فناوري - Some results on real-part/imaginary-part and magnitude-phase relations in ambiguity functions

Abstract :

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.