  The recently developed technique of Cyclotron Radiation Emission Spectroscopy
(CRES) uses frequency information from the cyclotron motion of an electron in a
magnetic bottle to infer its kinetic energy. Here we derive the expected radio
frequency signal from an electron in a waveguide CRES apparatus from first
principles. We demonstrate that the frequency-domain signal is rich in
information about the electron's kinematic parameters, and extract a set of
measurables that in a suitably designed system are sufficient for disentangling
the electron's kinetic energy from the rest of its kinematic features. This
lays the groundwork for high-resolution energy measurements in future CRES
experiments, such as the Project 8 neutrino mass measurement.
