Optimal Distributed Decision Over Wireless Sensor Networks Affected by Multipath Fading

The design of sensor networks capable of reaching a consensus on a globally optimal decision test, without the need for a fusion center, is a problem that has received considerable attention in the last years. Many consensus algorithms have been proposed, with convergence conditions depending on the graph describing the interaction among the nodes. In most works, the graph is undirected and there are no propagation delays. Only recently, the analysis has been extended to consensus algorithms incorporating a homogeneous delay. The more realistic case of inhomogeneous delays has been considered only for agreement algorithms, whose goal is to make all nodes able to reach a common value, but without the need to force such a final value to coincide with a prescribed globally optimal decision function. In this work, we propose a consensus algorithm where each node converges to a globally optimal decision statistic, valid for a wideband wireless network, where the link between each pair of nodes is a multipath, frequency-selective, channel. The main contribution of the paper is to derive necessary and sufficient conditions on the network topology and sufficient conditions on the channel transfer functions guaranteeing the exponential convergence of the proposed algorithm to a globally optimal decision value, for any bounded delay condition.