Signal Reconstruction in Multidimensional Sensor Fields

Sensor networks are often used for environmental monitoring. In this context, sampling and reconstruction of a physical field is a fundamental issue to solve. We consider a bandlimited, multidimensional field and study the quality level of its reconstruction when the sensor measurements are noisy and the number of available sensors varies. We find that, for an exact analysis of the problem, we would need the closed-form expression of the eigenvalue distribution of the reconstruction matrix, which, to the best of our knowledge, is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. We then apply our findings to the study of the MSE of the reconstructed field, when linear reconstruction techniques are used, and we derive an approximation that is very tight already for a 3-dimensional field.