Sensor selection is a useful method to help reduce data throughput, as well as computational, power, and hardware requirements, while still maintaining acceptable performance. Although minimizing the Cramér-Rao bound has been adopted previously for sparse sensing, it did not consider multiple targets and unknown source models. In this work, we propose to tackle the sensor selection problem for angle of arrival estimation using the worst-case Cramér-Rao bound of two uncorrelated sources. To do so, we cast the problem as a convex semi-definite program and retrieve the binary selection by randomized rounding. Through numerical examples related to a linear array, we illustrate the proposed method and show that it leads to the natural selection of elements at the edges plus the center of the linear array. This contrasts with the typical solutions obtained from minimizing the single-target Cramér-Rao bound.

Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complex-valued. We study the popular recovery method ofl1-regularized least squares or LASSO. While several studies have shown that LASSO provides desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to solve the complex-valued LASSO problem and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP. Our theoretical results are concerned with the case of i.i.d. Gaussian sensing matrices. Simulations confirm that our results hold for a larger class of random matrices.