Coprime microphone arrays use sparse sensing to achieve greater degrees of freedom, while the coprimality of the microphone subarrays help resolve grating lobe ambiguities. The result 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, as shown by Bush and Xiang [J. Acoust. Soc. Am. 138, 447-456 (2015)]. Peak positions may indicate directions of arrival in this case; however, one must first ask how many sources are present. In answering this question, this work employs a model describing scenes with potentially multiple concurrent sound sources. Bayesian inference is used to first select which model the data prefer from competing models before estimating model parameters, including the particular source locations. The model is a linear combination of Laplace distribution functions (one per sound source). The likelihood function is explored by a Markov Chain Monte Carlo method called nested sampling in order to evaluate Bayesian evidence for each model. These values increase monotonically with model complexity; however, diminished returns are penalized via an implementation of Occam's razor.

Coprime arrays represent a form of sparse sensing which can achieve narrow beams using relatively few elements, exceeding the spatial Nyquist sampling limit. The purpose of this paper is to expand on and experimentally validate coprime array theory in an acoustic implementation. Two nested sparse uniform linear subarrays with coprime number of elements ( M and N) each produce grating lobes that overlap with one another completely in just one direction. When the subarray outputs are combined it is possible to retain the shared beam while mostly canceling the other superfluous grating lobes. In this way a small number of microphones ( N+M-1) creates a narrow beam at higher frequencies, comparable to a densely populated uniform linear array of MN microphones. In this work beampatterns are simulated for a range of single frequencies, as well as bands of frequencies. Narrowband experimental beampatterns are shown to correspond with simulated results even at frequencies other than the arrays design frequency. Narrowband side lobe locations are shown to correspond to the theoretical values. Side lobes in the directional pattern are mitigated by increasing bandwidth of analyzed signals. Direction of arrival estimation is also implemented for two simultaneous noise sources in a free field condition.

Coprime linear microphone arrays allow for narrower beams with fewer sensors. A coprime microphone array consists of two staggered uniform linear subarrays with M and N microphones, where M and N are coprime with each other. By applying spatial filtering to both subarrays and combining their outputs, M+N−1 microphones yield M⋅N directional bands. In this work, the coprime sampling theory is implemented in the form of a linear microphone array of 16 elements with coprime numbers of 9 and 8. This coprime microphone array is experimentally tested to validate the coprime array theory. Both predicted and measured results are discussed. Experimental results confirm that narrow beampatterns as predicted by the coprime sampling theory can be obtained by the coprime microphone array.

Until now, coprime sensor arrays have used two sparsely spaced subarrays to emulate the performance of a single uniform array with many more sensors (generally on the order of the product of each subarrays' number of sensors). This allows for similar results with fewer sensors, or the observation of higher frequencies (above the Nyquist limit) with a similar number of sensors. The theory rests on the cross-referencing (using directional filter banks) or cancellation (using product processing) of the M grating lobes in one subarray's beampattern and N grating lobes in the other, where M and N are coprime integers. Sets of coprime integers can consist of more than two integers, however, and introducing another coprime factor theoretically multiplies observable frequency (or further decreases the number of array elements needed for the same frequency). Any amount, n, of coprime integers and corresponding subarrays may be used. In this work, “n-tuple coprime sensor array” theory is expounded and implemented. Experimentally measured beampattern results of a triple coprime sensor array (with three subarrays) are shown, using an extension of the authors' previously established product processing. Results also confirm that the usable range of an n-tuple coprime array extends below its design frequency.

Coprime linear microphone arrays consist of subarrays, each with inter-element spacing of half the wavelength observed times an integer factor. These two factors, M and N, corresponding to each subarray, are coprime, which ensures that their sensitivity completely overlaps only in the direction of the main beam. This implies a single observable wavelength, thus frequency; however, the grating lobe-mitigating effect can also be achieved for broadband sources [D. Bush, and N. Xiang, J. Acoust. Soc. Am., 138, 447-456 (2015)]. A modified Laplacian function provides a phenomenological model for broadband noise array responses, but real-world signals vary in spectral content making it prudent to develop a model which incorporates finer-resolution frequency dependence. This work also explores spatial filtering/source separation techniques for coprime linear microphone arrays. Multichannel experimental impulse response measurements with differing angles of incidence are convolved with independent speech signals. Subsequently, coprime beamforming is applied to the results in order to directionally filter, thus separating the source signals.