Microwave, submillimetre-wave, and far-infrared phased arrays are ofconsiderable importance for astronomy. We consider the behaviour imaging phasedarrays and interferometric phased arrays from a functional perspective. It isshown that the average powers, field correlations, power fluctuations, andcorrelations between power fluctuations at the output ports of an imaging orinterferometric phased array can be found once the synthesised receptionpatterns are known. The reception patterns do not have to be orthogonal or evenlinearly independent. It is shown that the operation of phased arrays isintimately related to the mathematical theory of frames, and that the theory offrames can be used to determine the degree to which any class of intensity orfield distribution can be reconstructed unambiguously from the complexamplitudes of the travelling waves at the output ports. The theory can be usedto set up a likelihood function that can, through Fisher information, be usedto determine the degree to which a phased array can be used to recover theparameters of a parameterised source. For example, it would be possible toexplore the way in which a system, perhaps interferometric, might observe twowidely separated regions of the sky simultaneously.
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