Algorithms for FFT Beamforming Radio Interferometers

Radio interferometers consisting of identical antennas arranged on a regular lattice permit fast Fourier transform beamforming, which reduces the correlation cost from O(n(2)) in the number of antennas to O(n( )log n). We develop a formalism for describing this process and apply this formalism to derive a number of algorithms with a range of observational applications. These include algorithms for forming arbitrarily pointed tied-array beams from the regularly spaced Fourier transform-formed beams, sculpting the beams to suppress sidelobes while only losing percent-level sensitivity, and optimally estimating the position of a detected source from its observed brightness in the set of beams. We also discuss the effect that correlations in the visibility-space noise, due to cross talk and sky contributions, have on the optimality of Fourier transform beamforming, showing that it does not strictly preserve the sky information of the n(2) correlation, even for an idealized array. Our results have applications to a number of upcoming interferometers, in particular the Canadian Hydrogen Intensity Mapping Experiment-Fast Radio Burst (CHIME/FRB) project.