Structured random measurements in signal processing

Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements.Comment: 22 pages, 2 figure

Simultaneous Sparse Approximation Using an Iterative Method with Adaptive Thresholding

This paper studies the problem of Simultaneous Sparse Approximation (SSA). This problem arises in many applications which work with multiple signals maintaining some degree of dependency such as radar and sensor networks. In this paper, we introduce a new method towards joint recovery of several independent sparse signals with the same support. We provide an analytical discussion on the convergence of our method called Simultaneous Iterative Method with Adaptive Thresholding (SIMAT). Additionally, we compare our method with other group-sparse reconstruction techniques, i.e., Simultaneous Orthogonal Matching Pursuit (SOMP), and Block Iterative Method with Adaptive Thresholding (BIMAT) through numerical experiments. The simulation results demonstrate that SIMAT outperforms these algorithms in terms of the metrics Signal to Noise Ratio (SNR) and Success Rate (SR). Moreover, SIMAT is considerably less complicated than BIMAT, which makes it feasible for practical applications such as implementation in MIMO radar systems

Matrix Completion in Colocated MIMO Radar: Recoverability, Bounds & Theoretical Guarantees

It was recently shown that low rank matrix completion theory can be employed for designing new sampling schemes in the context of MIMO radars, which can lead to the reduction of the high volume of data typically required for accurate target detection and estimation. Employing random samplers at each reception antenna, a partially observed version of the received data matrix is formulated at the fusion center, which, under certain conditions, can be recovered using convex optimization. This paper presents the theoretical analysis regarding the performance of matrix completion in colocated MIMO radar systems, exploiting the particular structure of the data matrix. Both Uniform Linear Arrays (ULAs) and arbitrary 2-dimensional arrays are considered for transmission and reception. Especially for the ULA case, under some mild assumptions on the directions of arrival of the targets, it is explicitly shown that the coherence of the data matrix is both asymptotically and approximately optimal with respect to the number of antennas of the arrays involved and further, the data matrix is recoverable using a subset of its entries with minimal cardinality. Sufficient conditions guaranteeing low matrix coherence and consequently satisfactory matrix completion performance are also presented, including the arbitrary 2-dimensional array case.Comment: 19 pages, 7 figures, under review in Transactions on Signal Processing (2013