Wireless Compressive Sensing for Energy Harvesting Sensor Nodes

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.