Optimal power allocation for parameter tracking in a distributed amplify-and-forward sensor network

Abstract—We consider the problem of optimal power allocation in a sensor network where the sensors observe a dynamic param-eter in noise and coherently amplify and forward their observa-tions to a fusion center (FC). The FC uses the observations in a Kalman filter to track the parameter, and we show how to find the optimal gain and phase of the sensor transmissions under both global and individual power constraints in order to minimize the mean squared error (MSE) of the parameter estimate. For the case of a global power constraint, a closed-form solution can be ob-tained. A numerical optimization is required for individual power constraints, but the problem can be relaxed to a semidefinite pro-gramming problem (SDP), and we show that the optimal result can be constructed from the SDP solution. We also study the dual problem of minimizing global and individual power consumption under a constraint on the MSE. As before, a closed-form solution can be found when minimizing total power, while the optimal so-lution is constructed from the output of an SDP when minimizing the maximum individual sensor power. For purposes of compar-ison, we derive an exact expression for the outage probability on the MSE for equal-power transmission, which can serve as an upper bound for the case of optimal power control. Finally, we present the results of several simulations to show that the use of optimal power control provides a significant reduction in either MSE or transmit power compared with a non-optimized approach (i.e., equal power transmission). Index Terms—Amplify-and-forward networks, distributed esti-mation, distributed tracking, wireless sensor networks