Wireless Sensor Networks (WSNs) enable a wealth of new applications where remote estimation is essential. Individual sensors simultaneously sense a dynamic process and transmit measured information over a shared channel to a central fusion center. The fusion center computes an estimate of the process state by means of a Kalman filter. In this technical note we assume that the WSN admits a tree topology with one fusion center at the root. At each time step only a subset of sensors can be selected to transmit observations to the fusion center due to a limited energy budget. We propose a stochastic sensor selection algorithm that randomly selects a subset of sensors according to a certain probability distribution, which is opportunely designed to minimize the asymptotic expected covariance matrix of the estimation error. We show that the optimal stochastic sensor selection problem can be relaxed into a convex optimization problem and thus efficiently solved. We also provide a possible implementation of our algorithm which does not introduce any communication overhead. The technical note ends with some numerical examples that show the effectiveness of the proposed approach.

The Problem 4 can be solved directly via semidefinite programming. As a result, Problem 5 is no longer needed. For details, please see my notes.