Partial Decode-Forward Relaying for the Gaussian Two-Hop Relay Network

Abstract

The multicast capacity of the Gaussian two-hop relay network with one source, $N$ relays, and $L$ destinations is studied. It is shown that a careful modification of the partial decode-forward coding scheme, whereby the relays recover and coherently transmit degraded sets of message parts, achieves the cutset upper bound within $(1/2)\log N$ bits regardless of the channel gains and power constraints. This scheme improves upon a previous scheme by Chern and Ozgur, which is also based on partial decode-forward yet has an unbounded gap from the cutset bound for $L \ge 2$ destinations. When restricted to noncoherent transmission among the relays, the proposed partial decode-forward scheme achieves a slightly larger gap of $\log N$ bits from the cutset bound. The computation of this relaxed achievable rate involves evaluating mutual information across $L(N+1)$ cuts out of the total $L 2^{N}$ possible cuts, providing a very simple linear-complexity algorithm to approximate the single-source multicast capacity of the Gaussian two-hop relay network.