Multihop analog network coding via amplify-and-forward: The high SNR regime

In the simplest relaying strategy, a network node amplifies and forwards a received signal over a wireless channel. Multihop amplify-and-forward allows for a (noisy) linear combination of signals simultaneously sent from multiple sources to be propagated through the network over multiple layers of relays. The performance of multihop amplify-and-forward is limited by noise propagated to the destination over multiple hops, and we expect this strategy to perform well only in high SNR. In this paper, this intuition is formalized and high-SNR conditions under which multihop amplify-and-forward approaches capacity in a layered relay network are determined. By relating the received signal power and the received power of the propagated noise at the nodes, the rate achievable with multihop amplify-and-forward is determined. In particular, when all received powers are lower bounded by 1/δ, the noise power propagated to the destination over L layers is of the order Lδ. The result demonstrates that multihop amplify-and-forward approaches the cut-set bound as received powers at relays increase. As all powers in the network increase at the same rate, the multihop amplify-and-forward rate and the upper bound are within a gap that is independent of channel gains. This gap grows linearly with the number of nodes.