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 electrons kinematic parameters, and extract a set of measurables that in a suitably designed system are sufficient for disentangling the electrons 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.
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