Performance of Linear Field Reconstruction Techniques with Noise and Correlated Field Spectrum

We consider a wireless sensor network that is deployed over an area of interest for environmental monitoring. As often in the practice, the sensor nodes are randomly distributed on the area and their measurements are noisy. Furthermore, the sensors measure a multidimensional physical field (signal), with correlated spectrum, which can be approximated as bandlimited. A central controller, the sink node, is in charge of reconstructing the field from the sensor measurements, which represent an irregular sampling of the signal. We assume that the sink uses a linear reconstructing technique, and we take as performance metric of the reconstruction quality the mean square error (MSE) of the estimate. We then carry out an asymptotic analysis as the number of sensors and the number of the field harmonics go to infinity, while their ratio is kept constant. In particular, we approximate the MSE on the reconstructed field as a function of the eigenvalues of the matrix representing the sampling system. We validate our approximation against numerical results, for some of the most common spectrum correlation models.