Multiple and single snapshot compressive beamforming

Abstract

For a sound field observed on a sensor array, compressive sensing (CS) reconstructs the directionof-arrival (DOA) of multiple sources using a sparsity constraint. The DOA estimation is posed as an underdetermined problem by expressing the acoustic pressure at each sensor as a phaselagged superposition of source amplitudes at all hypothetical DOAs. Regularizing with an 1-norm constraint renders the problem solvable with convex optimization, and promoting sparsity gives highresolution DOA maps. Here, the sparse source distribution is derived using maximum a posteriori (MAP) estimates for both single and multiple snapshots. CS does not require inversion of the data covariance matrix and thus works well even for a single snapshot where it gives higher resolution than conventional beamforming. For multiple snapshots, CS outperforms conventional high-resolution methods, even with coherent arrivals and at low signal-to-noise ratio. The superior resolution of CS is demonstrated with vertical array data from the SWellEx96 experiment for coherent multi-paths.

Block-sparse beamforming for spatially extended sources in a Bayesian formulation

“…Thus, the parsimonious LASSO estimate improves significantly the resolution in DOA estimation in the presence of only a few sources as shown in Ref. 22.…”

“…22 In Ref. 22, the LASSO problem [Eq. (16)] is solved in the dual domain, which yields k2A H ðy À AxÞk 1 l. This primal-dual correspondence dictates that for every active elementx i 6 ¼ 0 of the primal solutionx, the corresponding element 2a H ðy À AxÞ has amplitude equal to l. Thus, gradually reducing l from an initial value of l ¼ 2kA…”

“…In acoustic source localization, LASSO has superior performance compared to traditional methods, providing high-resolution DOA maps even in the presence of coherent sources or single-snapshot data. 21,22 The LASSO in its standard form does not promote block sparsity (i.e., grouping of correlated locations), hence, it is not suitable for recovering multipole or spatially extended sources (e.g., edge-noise in aeroacoustic applications 23 or submerged objects in sonar imaging 24 ). For such problems, which involve certain structural constraints rather than pure sparsity, a weighed LASSO 25,26 (WL) formulation is introduced.…”

Sub-antenna sparse processing for coherence loss in underwater source localization

“…In underwater acoustics, we can also mention the use of convex-relaxation based algorithms, such as the Least Absolute Shrinkage and Selection Operator (LASSO) [4] or the Simultaneous LASSO (SLASSO) when considering multiple measurements (snapshots) [7]. All these latter techniques offer a high resolution, but are intrinsically not designed to be robust to random environmental fluctuations such as internal waves or sound speed spatiotemporal dynamics.…”

“…Formally, the paper investigates a combination of a sub-antenna approach [9], [10] with sparse techniques [4], [7]. On the one hand, sub-antenna techniques are well suited to deal with coherence loss, but at the expense of a resolution loss.…”