We consider the problem of optimal power allocation 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 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 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 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 solution is constructed from the output of an SDP when minimizing
the maximum individual sensor power. For purposes of comparison, 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).Comment: 28 pages, 6 figures, accepted by IEEE Transactions on Signal
Processing, Jan. 201