Joint pre-coder design and greedy power allocation for compressed spatial field estimation

In this paper, we propose a distributed beamforming scheme for the estimation of spatial fields (e.g. temperature, moisture) with wireless sensor networks. The pre-coding scheme allows for an over-the-air compressed representation of the correlated set of spatial observations, which are encoded in a number of consecutive sensor-to-gateway (GW) transmissions. The ultimate goal is to minimize the distortion in the reconstructed spatial field and, simultaneously, keep the number of transmissions low (i.e. the compression ratio high). However, the design of the set of normalized pre-coders, for which we derive a closed-form expression, and the corresponding power allocation problems turn out to be coupled. By resorting to a greedy power allocation strategy, both problems can be iteratively and jointly solved. The performance of the proposed pre-coding scheme is assessed by means of computer simulations. Other compressed beamforming schemes requiring channel inversion are used as a benchmark.