On Bayesian Fisher Information Maximization for Distributed Vector Estimation

Abstract

In this paper, we consider the problem of bandwidth-constrained distributed estimation of a Gaussian vector with linear observation model. Each sensor makes a scalar noisy observation of the unknown vector, employs a multi-bit scalar quantizer to quantize its observation, and maps it to a digitally modulated symbol. Sensors transmit their symbols 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; and (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 refer to as FIM-max schemes). 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, in the sense that each sensor calculates its own transmit power using its local parameters. On the other hand, the solution to the log-determinant of Bayesian FIM maximization problem cannot be implemented in a distributed fashion and the FC needs to find the powers (using parameters of all sensors) and inform the active sensors of their transmit powers. 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 mean square error (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. Furthermore, comparing the performance of different receivers, our numerical results reveal that coherent receiver and noncoherent receiver with known channel statistics have the best and the worst performance, respectively.