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.