Linear relaying for Gaussian diamond networks

Linear relaying for the Gaussian diamond network is studied as a natural extension of the amplify-forward relaying strategy by Schein and Gallager. A single-letter optimal rate is characterized, which is shown to be achieved by time sharing between four amplify-forward strategies at different power levels. This linear relaying capacity has a bounded gap from the cutset bound when the network is symmetric, but in general has an unbounded gap. The main idea of the proof is to transform a multiletter rate expression into an infinite-dimensional optimization problem, the relaxation of which matches the performance of time-shared amplify-forward. A similar proof technique can be applied to other relay networks with layered structure such as the N-relay Gaussian diamond network and the receiver frequency-division Gaussian relay channel.