The uniqueness theorem for ambiguity functions states that ff waveforms <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v(t)</tex> have the same ambiguity function, i.e., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\chi_{u}(\tau, \Delta) = X_{\upsilon , \Delta)</tex> , then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v(t)</tex> are identical except for a rotation, i.e., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v(t) = e^{i\lambda}u(t)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> is a real constant. Through the artifice of treating the even and odd parts of the waveforms, denoted <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(t)</tex> , respectively, correlative results have been obtained for the real and imaginary parts of ambiguity functions. Thus, if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re \{\chi_{u}(\tau, \Delta)\} = Re \{\chi_{\upsilon}(\tau, \Delta)\}</tex> , then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e_{\upsilon}(t) = e^{i \lambda e}e_{u}(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o_{\upsilon}(t) = e^{i \lambda o}o_{u}(t)</tex> . From <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re \{\chi_{u}(\tau, \Delta)\}</tex> , the waveform class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t) = e^{i\lambda} [e_{u}(t) + e^{ik}o_{u}(t)]</tex> may be constructed, but because of the arbitrary rotation, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e^{ik}</tex> , a unique <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\chi_{u}</tex> -function is not determinable, in general. An important exception to this statement is the case when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\chi_{u}(\tau, \Delta)</tex> is real, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re \{\chi_{u}\} = \chi_{u}</tex> determines a unique waveform (within a rotation) and this waveform can only be even or odd. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Im \{\chi_{u}(\tau, \Delta)\} = Im \{\chi_{\upsilon(\tau, \Delta)\}</tex> then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e_{\upsilon}(t) =ae^{i \gamma}e_{u}(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o_{\upsilon}(t) = 1/ae^{ir}o_{u}(t)</tex> . If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Im \{\chi_{u}(\tau, \Delta)\}</tex> is given, {\em and} <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t)</tex> is known to have unit energy, then within rotations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e^{i \lambda}</tex> , only two possible waveform choices are possible for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t)</tex> . If it also is known which of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e_{u}(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o_{u}(t)</tex> has the greater energy, the function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Im \{\chi_{u}(\tau, \Delta)\}</tex> uniquely determines <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u(t)</tex> (within a rotation) and the complete <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\chi_{u}</tex> -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 derivatives of the phase function.