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 in the number of antennas to . 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.