Sequential Bayesian Sparse Signal Reconstruction Using Array Data

In this paper, the sequential reconstruction of source waveforms under a sparsity constraint is considered from a Bayesian perspective. Let the wave field, which is observed by a sensor array, be caused by a spatially-sparse set of sources. A spatially weighted Laplace-like prior is assumed for the source field and the corresponding weighted Least Absolute Shrinkage and Selection Operator (LASSO) cost function is derived. After the weighted LASSO solution has been calculated as the maximum a posteriori estimate at time step $k$ , the posterior distribution of the source amplitudes is analytically approximated. The weighting of the Laplace-like prior for time step $k+1$ is then fitted to the approximated posterior distribution. This results in a sequential update for the LASSO weights. Thus, a sequence of weighted LASSO problems is solved for estimating the temporal evolution of a sparse source field. The method is evaluated numerically using a uniform linear array in simulations and applied to data which were acquired from a towed horizontal array during the long range acoustic communications experiment.