A Fast Broadband Beamspace Transformation

We present a new computationally efficient method for multi-beamforming in the broadband setting. Our "fast beamspace transformation" forms $B$ beams from $M$ sensor outputs using a number of operations per sample that scales linearly (to within logarithmic factors) with $M$ when $B\sim M$. While the narrowband version of this transformation can be performed efficiently with a spatial fast Fourier transform, the broadband setting requires coherent processing of multiple array snapshots simultaneously. Our algorithm works by taking $N$ samples off of each of $M$ sensors and encoding the sensor outputs into a set of coefficients using a special non-uniform spaced Fourier transform. From these coefficients, each beam is formed by solving a small system of equations that has Toeplitz structure. The total runtime complexity is $\mathcal{O}(M\log N+B\log N)$ operations per sample, exhibiting essentially the same scaling as in the narrowband case and vastly outperforming broadband beamformers based on delay and sum whose computations scale as $\mathcal{O}(MB)$. Alongside a careful mathematical formulation and analysis of our fast broadband beamspace transform, we provide a host of numerical experiments demonstrating the algorithm's favorable computational scaling and high accuracy. Finally, we demonstrate how tasks such as interpolating to ``off-grid" angles and nulling an interferer are more computationally efficient when performed directly in beamspace.