Bounds on the Distortion for Distributed Sensing of Slowly-Varying Random Fields

We consider a wireless sensor network deployed in an area to measure the realization of a finite multi-dimensional, slowly time-varying physical random field. Each sensor observes one noisy realization of the field, maps it linearly into a signal with a signature and sends it across a white Gaussian multiple access channel, under a constraint on the total energy given to all the sensors per field realization. The receiver or the ´collector node´ receives all the signals and tries to construct an estimate of the field within a certain mean distortion based on the MSE fidelity criterion. We derive, under the total energy constraint, a lower-bound on the distortion, an achievable one, and another lower-bound under a TDMA transmission scheme. In the case of the non-existence of the observation noise, we find the asymptotic decreasing behavior of the achievable distortion as a function of the number of sensors. Moreover, we derive a lower-bound on the distortion over all possible encoding techniques, assuming a free collaboration and information exchange between the sensors. We compare these bounds for a particular example with another bound on the achievable distortion [1].