Wireless compressive sensing for energy harvesting sensor nodes over fading channels

We consider the scenario in which multiple sensors send spatially correlated data to the 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 the signal can be formulated as a compressive sensing (CS) problem. To model the scenario where 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, referred to as the inhomogeneity of SNRs. Thus, the elements of the sensing matrix are also not identically distributed. We provide theoretical guarantees on the number of measurements for reliable reconstruction, by showing that the sensing matrix satisfies the restricted isometry property (RIP), under some mild 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 the SNRs on the so-called k-restricted eigenvalues, which governs the number of measurements required for the RIP to hold. Our analysis is corroborated by numerical results.