  We consider the scenario in which multiple sensors send spatially correlated
data to a fusion center (FC) via independent Rayleigh-fading channels with
additive noise. Assuming that the sensor data is sparse in some basis, we show
that the recovery of this sparse signal can be formulated as a compressive
sensing (CS) problem. To model the scenario in which the sensors operate with
intermittently available energy that is harvested from the environment, we
propose that each sensor transmits independently with some probability, and
adapts the transmit power to its harvested energy. Due to the probabilistic
transmissions, the elements of the equivalent sensing matrix are not Gaussian.
Besides, since the sensors have different energy harvesting rates and different
sensor-to-FC distances, the FC has different receive signal-to-noise ratios
(SNRs) for each sensor. This is referred to as the inhomogeneity of SNRs. Thus,
the elements of the sensing matrix are also not identically distributed. For
this unconventional setting, we provide theoretical guarantees on the number of
measurements for reliable and computationally efficient recovery, by showing
that the sensing matrix satisfies the restricted isometry property (RIP), under
reasonable conditions. We then compute an achievable system delay under an
allowable mean-squared-error (MSE). Furthermore, using techniques from large
deviations theory, we analyze the impact of inhomogeneity of SNRs on the
so-called k-restricted eigenvalues, which governs the number of measurements
required for the RIP to hold. We conclude that the number of measurements
required for the RIP is not sensitive to the inhomogeneity of SNRs, when the
number of sensors n is large and the sparsity of the sensor data (signal) k
grows slower than the square root of n. Our analysis is corroborated by
extensive numerical results.
