Sufficient and Necessary Conditions of Distributed Compressed Sensing with Prior Information

This paper considers the recovery problem of distributed compressed sensing (DCS), where J (J≥2) signals all have sparse common component and sparse innovation components. The decoder attempts to jointly recover each component based on {Mj} random noisy measurements (j=1,…,J) with the prior information on the support probabilities, i.e., the probabilities that the entries in each component are nonzero. We give both the sufficient and necessary conditions on the total number of measurements $\sum\nolimits_{j = 1}^J M_j$ that is needed to recover the support set of each component perfectly. The results show that when the number of signal J increases, the required average number of measurements $\sum\nolimits_{j = 1}^J M_j/J$ decreases. Furthermore, we propose an extension of one existing algorithm for DCS to exploit the prior information, and simulations verify its improved performance.