On Spatial Aliasing in Microphone Arrays

Abstract

Abstract-Microphone arrays sample the sound field in both space and time with the major objective being the extraction of the signal propagating from a desired direction-of-arrival (DOA). In order to reconstruct a spatial sinusoid from a set of discrete samples, the spatial sampling must occur at a rate greater than a half of the wavelength of the sinusoid. This principle has long been adapted to the microphone array context: in order to form an unambiguous beampattern, the spacing between elements in a microphone array needs to conform to this spatial Nyquist criterion. The implicit assumption behind the narrowband beampattern is that one may use linearity and Fourier analysis to describe the response of the array to an arbitrary wideband plane wave. In this paper, this assumption is analyzed. A formula for the broadband beampattern is derived. It is shown that in order to quantify the spatial filtering abilities of a broadband array, the incoming signal's bifrequency spectrum must be taken into account, particularly for nonstationary signals such as speech. Multi-dimensional Fourier analysis is then employed to derive the broadband spatial transform, which is shown to be the limiting case of the broadband beampattern as the number of sensors tends to infinity. The conditions for aliasing in broadband arrays are then determined by analyzing the effect of computing the broadband spatial transform with a discrete spatial aperture. It is revealed that the spatial Nyquist criterion has little importance for microphone arrays. Finally, simulation results show that the well-known steered response power (SRP) method is formulated with respect to stationary signals, and that modifications are necessary to properly form steered beams in nonstationary signal environments.Index Terms-Beamforming, broadband beampattern, microphone arrays, spatial aliasing, spatial sampling, wavenumber-frequency spectrum.

Aliasing-Free Wideband Beamforming Using Sparse Signal Representation

“…Remark 4: It is interesting to note that many classical wideband beamforming works, e.g., [4] and [5], consider the results to be readily "aliasing-free" if (13) is satisfied. However, [4] does not give a rigorous formulation of the related choice of frequencies, but simply asserts that the case where (13) is not satisfied "is very unlikely".…”

“…However, [4] does not give a rigorous formulation of the related choice of frequencies, but simply asserts that the case where (13) is not satisfied "is very unlikely". In a more specific manner, [5] claims that all wideband signals will not suffer from spatial aliasing, but the wideband signals considered in [5] have a continuous frequency spectrum and thus it is trivial to find two discrete frequencies, which are very close to each other to comply with (12). Compared to [5], Theorem 1 does not only give a more rigorous formulation of the frequency choices, but it is also of practical significance.…”

“…The exhaling sound of the diver is so wideband that it can almost be viewed as white [1]. Wideband "aliasing-free" beamforming has been reported in, e.g., [2]- [5]. These works require statistical knowledge to reconstruct the signal/noise subspace, which is (for both signal and noise) not always available or reliable due to the volatile underwater acoustic environment.…”

“…For this reason, the delay-sum beamformer [6] is sometimes preferred in practical sonar applications, although it is more susceptible to spatial aliasing and renders a poor resolution. Most notably, [2]- [5] are not free of spatial aliasing in a strict sense. First, it is often assumed that the array manifolds of the target DOAs are independent for all the frequencies, a situation that lacks theoretical guarantees.…”

A reverberation-time-aware DNN approach leveraging spatial information for microphone array dereverberation

“…Clearly, spatial aliasing is somewhat of a misunderstood phenomenon [38], since the human binaural auditory system does not experience problems localizing broadband sounds with an average spacing of 20 cm (corresponding to aliasing above 850 Hz). And it has been revealed in [38] that the spatial Nyquist criterion has little importance for microphone arrays.…”