The origins of frequency-difference and frequency-sum beamforming

Beamforming the signals recorded by an array enables the determination of sound source location(s) or the arrival directions of ray paths between a sound source and the receiving array. Frequency-difference and frequency-sum beamforming are beamforming techniques that provide out-of-band information from in-band signal frequencies. Interestingly, the out-of-band frequencies can be chosen by the user, within limits set by the signal recordings, to achieve desired properties of the beamformed output, such as: increased resolution, reduced sidelobes, or greater robustness to random scattering. Both techniques are general and are not limited to any particular acoustic environment, frequency range, or array geometry. Frequency-sum beamforming, generates higher-frequency information from lower frequency signal components, enhancing beamforming results in scenarios with random scattering between the source and the receivers. However, it is limited by artifacts arising from cross-terms when multiple source signals are present in the same bandwidth. Conversely, frequency-difference beamforming manufactures lower-frequency information from higher frequency signal components, effectively mitigating the impact of spatial aliasing in situations where the receiving array is sparse. This presentation delves into the origins of frequency-difference and frequency-sum beamforming, presents the fundamental mathematics underlying their algorithms, and showcases their performance via simulations and experimental results. Work supported by ONR.