  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.
