Bayesian model selection for a broadband coprime array with unknown number of broadband sources

Abstract

Coprime microphone arrays use sparse sensing to achieve O(MN) degrees of freedom using only O(M + N) elements, where M and N are coprime integers. The benefit is a narrow beam at frequencies higher than the spatial Nyquist limit allows, with residual side lobes arising from aliasing. These side lobes can be mitigated when observing broadband sources [D. Bush and N. Xiang, J. Acoust. Soc. Am., 138, 447–456 (2015)]. Peak positions indicate directions of arrival in this case; however, uncertainties on number of concurrent sound sources in practical applications challenge classical approaches to direction-of-arrival estimations. One has to first resolve the uncertainty on how many sources are present. In this work, Bayesian inference is used to first select which model the data prefer from competing models before estimating model parameters, including the particular directions of arrival. The model is a linear combination of modified Laplacian distributions (one per sound source). The posterior probability function is explored over the entire parameter space by nested sampling in order to evaluate Bayesian evidence for each model. Bayesian evidence is crucial for resolving the uncertainties regarding number of sources, preferring simpler models while penalizing unnecessarily complicated models in an inherent implementation of Occam’s razor.