Wireless Compressive Sensing over Fading Channels with Distributed Sparse Random Projections

We address the problem of recovering a sparse signal observed by a resource constrained wireless sensor network with fading channels. Sparse random matrices are exploited to reduce the communication cost in forwarding information to a fusion center. The presence of channel fading leads to inhomogeneity and non-Gaussian statistics in the effective measurement matrix that relates the measurements collected at the fusion center and the sparse signal being observed. We analyze the impact of channel fading on recovery of a given sparse signal by leveraging the properties of heavy-Tailed random matrices. We quantify the additional number of measurements required to ensure reliable signal recovery in the presence of nonidentical fading channels compared to that is required with identical Gaussian channels. Our analysis provides insights into how to control the probability of sensor transmissions at each node based on the channel fading statistics to minimize the number of measurements collected at the fusion center for reliable sparse signal recovery. We further discuss recovery guarantees of a given sparse signal with any random projection matrix where the elements are subexponential with a given subexponential norm. Numerical results are provided to corroborate the theoretical findings.