  We address the problem of recovering a sparse signal observed by a resource
constrained wireless sensor network under channel fading. 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 nonuniform
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 in order 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 sub-exponential
with a given sub-exponential norm. Numerical results are provided to
corroborate the theoretical findings.
