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

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 , \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 derivatives of the phase function.