Parameter Tracking via Optimal Distributed Beamforming in an Analog Sensor Network

We consider the problem of optimal distributed beamforming in a sensor network where the sensors observe a dynamic parameter in noise and coherently amplify and forward their observations to a fusion center (FC). The FC uses a Kalman filter to track the parameter using the observations from the sensors, 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 obtained. A numerical optimization is required for individual power constraints, but the problem can be relaxed to a semidefinite programming problem (SDP), and we show how the optimal solution can be constructed from the solution to the SDP. Simulation results show that compared with equal power transmission, the use of optimized power control can significantly reduce the MSE.