On the capacity of large Gaussian relay networks

We determine the capacity of a particular large Gaussian relay network in the limit as the number of relays tends to infinity. The upper bounds follow from a cut-set argument, and the lower bound follows from an argument involving uncoded transmission. We prove that in many cases of interest, upper and lower bounds coincide in the limit as the number of relays tends to infinity. Hence, this paper gives one more example where the cut-set bound is achievable, and one more example where uncoded transmission achieves optimal performance. In the latter sense, the result is an extension to \cite{GastparRV:01b}. To illustrate our findings, we first apply them to a sensor network situation. The comparison of our results with the CEO problem leads to a new instance of the fact that the source-channel separation paradigm does {\em not} extend to networks in general. Then, we show how to extend our approach to include certain ad-hoc wireless networks, which leads to a capacity result: When all nodes act purely as relays for a single source-destination pair, capacity grows with the logarithm of the number of nodes.