  We consider the problem of distributed estimation of a Gaussian vector with
linear observation model. Each sensor makes a scalar noisy observation of the
unknown vector, quantizes its observation, maps it to a digitally modulated
symbol, and transmits the symbol over orthogonal power-constrained fading
channels to a fusion center (FC). The FC is tasked with fusing the received
signals from sensors and estimating the unknown vector. We derive the Bayesian
Fisher Information Matrix (FIM) for three types of receivers: (i) coherent
receiver (ii) noncoherent receiver with known channel envelopes (iii)
noncoherent receiver with known channel statistics only. We also derive the
Weiss-Weinstein bound (WWB). We formulate two constrained optimization
problems, namely maximizing trace and log-determinant of Bayesian FIM under
network transmit power constraint, with sensors transmit powers being the
optimization variables. We show that for coherent receiver, these problems are
concave. However, for noncoherent receivers, they are not necessarily concave.
The solution to the trace of Bayesian FIM maximization problem can be
implemented in a distributed fashion. We numerically investigate how the
FIM-max power allocation across sensors depends on the sensors observation
qualities and physical layer parameters as well as the network transmit power
constraint. Moreover, we evaluate the system performance in terms of MSE using
the solutions of FIM-max schemes, and compare it with the solution obtained
from minimizing the MSE of the LMMSE estimator (MSE-min scheme), and that of
uniform power allocation. These comparisons illustrate that, although the WWB
is tighter than the inverse of Bayesian FIM, it is still suitable to use
FIM-max schemes, since the performance loss in terms of the MSE of the LMMSE
estimator is not significant.
