  Neutrino oscillation experiments show that neutrinos have mass; however, the
absolute mass scale is exceedingly difficult to measure and is currently
unknown. A promising approach is to measure the energies of the electrons
released during the radioactive decay of tritium. The energies of interest are
within a few eV of the 18.6 keV end point, and so are mildly relativistic. By
capturing the electrons in a static magnetic field and measuring the frequency
of the cyclotron radiation emitted the initial energy can be determined, but
end-point events are infrequent, the observing times short, and the signal to
noise ratios low. To achieve a resolution of $<$ 10 meV, single-electron
emission spectra need to be recorded over large fields of view with highly
sensitive receivers. The principles of Cylotron Radiation Emission Spectroscopy
(CRES) have already been demonstrated by Project 8, and now there is
considerable interest in increasing the FoV to $>$ 0.1 m$^3$. We consider a
range of issues relating to the design and optimisation of inward-looking
quantum-noise-limited microwave receivers for single-electron CRES, and present
a single framework for understanding signal, noise and system-level behaviour.
Whilst there is a great deal of literature relating to the design of
outward-looking phased arrays for applications such as radar and
telecommunications, there is very little coverage of the new issues that come
into play when designing ultra-sensitive inward-looking phased arrays for
volumetric spectroscopy and imaging.
