  Low-noise phased arrays are essential for the next generation of microwave
and submillimetre wave astronomy. We analyze their behaviour from a functional
perspective, and show that their operation is intimately related to the
mathematical theory of frames. No assumptions are made about the orthogonality
or linear independence of the synthesised beams. Frame theory allows an
unambiguous assessment of whether the outputs of an array can be used to
observe a field or brightness distribution within a given class. Image
reconstruction is carried out using dual beams. We identify the natural modes
of phased arrays, and carry out an analysis of noise. The scheme allows the
expectation values, the mean-square fluctuations, and the correlations between
fluctuations at the output ports of a phased array to be determined for a
source in any state of spatial coherence. Both classical and photon-counting
statistics are included. Our model is conceptually powerful, and suggests many
simulation and image recovery techniques.
