Wave physics of the frequency difference autoproduct: Bilinear time-frequency analysis

Abstract

Frequency difference beamforming (Abadi et al., 2012, JASA, 132, 3018-3029) and matched field processing (Worthmann et al., 2015, JASA, 138, 3549-3562) are array signal processing techniques that determine acoustic ray-path directions and source locations, respectively, from sparse array recordings by shifting the signal processing to a lower, out-of-band, difference-frequency bandwidth. This frequency down-shift is accomplished through the use of the frequency difference autoproduct (Δf-AP), a quadratic product of complex frequency-domain acoustic field amplitudes at two different frequencies. In this presentation, the physical and mathematical underpinnings of this quadratic product are discussed. Specifically, in the branch of applied mathematics termed bilinear time-frequency analysis, this Δf-AP is actually one of four quadratic time-frequency functions, each inter-related through forward and inverse Fourier transforms. The other three functions are the time-domain autoproduct, the Wigner distribution, and the ambiguity function. These four functions and several of their interesting properties are reviewed, including their relationship to the Wiener-Khintchine Theorem, and the spatial and spectral coherence and correlation functions. Additionally, the behavior of these functions when evaluated from simulated acoustic recordings from ideal single-path and multi-path acoustic environments is shown and compared. [Sponsored by ONR and NSF.]