Acoustic and Bayesian analyses of n-tuple coprime arrays

Abstract

Owing to number theory which holds that pairwise coprime sets can have more than two numbers, this work also derives frequency-dependent coprime functions for multiple subarrays (more than two), denominated n-tuple coprime arrays. A triple coprime array and a quadruple coprime array are constructed and tested in a similar manner to the original double coprime array to validate the expanded theory.

This work has so far resulted in 4 peer-reviewed archival journal publications, 5 domestic and international conference proceedings, and 1 patent pending.

To tackle direction of arrival estimation in a rigorous way, a model-based Bayesian framework is employed using broadband data from the double coprime array experiment. This begins with deriving the Bayesian quantities from the principle of maximum entropy and ends with nested sampling, which is demonstrated to be capable of elucidating both which model the data prefer, as well as what the most likely parameters (directions of arrival) are.

Sensor arrays use differences among measured signals at different points in space to help locate the direction of the incoming waves as well as isolate and amplify their signal information. Many of the designs and algorithms developed can be implemented across most types of array elements, be they antennae, hydrophones, seismometers, or in this case microphones. Adding microphones tends to increase benefits, but there's always a trade-off (even if it's just the cost of manufacturing the additional microphones); therefore, an ever-present line of inquiry in microphone array research is how to maintain or increase localization, filtering, or other benefits while reducing the number of microphones required. In linear arrays the spatial Nyquist theorem limits inter-element spacing to be within half of the shortest wavelength being observed, which implies that for a given aperture, in order to further reduce the number of array elements, some aliasing will occur. In order to resolve this aliasing, coprime sensing proposed interleaving two such "subarrays" whose deterministic aliasing patterns only match exactly in one direction due to the coprimality of their inter-element spacings. In its original derivation a specific design frequency is implied; this work derives frequency-dependent functions and validates the theory via construction and testing of a 16-channel linear coprime microphone array system. Aliasing cancellation between the two coprime subarrays is shown to work at frequencies lower than the coprime design frequency, and also across broad frequency bands.