The Approximate Capacity of the Gaussian N-Relay Diamond Network

TL;DR

This paper provides a near-accurate approximation of the capacity of the Gaussian N-relay diamond network, showing that simple bursty amplify-and-forward schemes are nearly optimal across various regimes.

Contribution

It introduces a capacity approximation with bounds independent of the number of relays, improving previous results and demonstrating the effectiveness of bursty amplify-and-forward.

Findings

Capacity approximated within 1.8 bits and a factor of 14.

Bursty amplify-and-forward is uniformly approximately optimal.

Achieves near-capacity in both symmetric and asymmetric networks.

Abstract

We consider the Gaussian "diamond" or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N^2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log^4(N)) of capacity with pre-constant in the order notation independent of the channel gains.