  In this paper we propose and analyze a distributed algorithm for achieving
globally optimal decisions, either estimation or detection, through a
self-synchronization mechanism among linearly coupled integrators initialized
with local measurements. We model the interaction among the nodes as a directed
graph with weights (possibly) dependent on the radio channels and we pose
special attention to the effect of the propagation delay occurring in the
exchange of data among sensors, as a function of the network geometry. We
derive necessary and sufficient conditions for the proposed system to reach a
consensus on globally optimal decision statistics. One of the major results
proved in this work is that a consensus is reached with exponential convergence
speed for any bounded delay condition if and only if the directed graph is
quasi-strongly connected. We provide a closed form expression for the global
consensus, showing that the effect of delays is, in general, the introduction
of a bias in the final decision. Finally, we exploit our closed form expression
to devise a double-step consensus mechanism able to provide an unbiased
estimate with minimum extra complexity, without the need to know or estimate
the channel parameters.
