Achieving the Capacity of the $N$ -Relay Gaussian Diamond Network Within log $N$ Bits

We consider the
$N$
-relay Gaussian diamond network where a source node communicates to a destination node via
$N$
parallel relays through a cascade of a Gaussian broadcast (BC) and a multiple access (MAC) channel. Introduced in 2000 by Schein and Gallager, the capacity of this relay network is unknown in general. The best currently available capacity approximation, independent of the coefficients and the SNRs of the constituent channels, is within an additive gap of
$1.3 N$
bits, which follows from the recent capacity approximations for general Gaussian relay networks with arbitrary topology. In this paper, we approximate the capacity of this network within 2 log
$N$
bits. We show that two strategies can be used to achieve the information-theoretic cut-set upper bound on the capacity of the network up to an additive gap of
$O(\log N)$
bits, independent of the channel configurations and the SNRs. The first of these strategies is simple partial decode-and-forward. Here, the source node uses a superposition codebook to broadcast independent messages to the relays at appropriately chosen rates; each relay decodes its intended message and then forwards it to the destination over the MAC channel. A similar performance can be also achieved with compress-and-forward type strategies (such as quantize-map-and-forward and noisy network coding) that provide the
$1.3 N$
-bit approximation for general Gaussian networks, but only if the relays quantize their observed signals at a resolution inversely proportional to the number of relay nodes
$N$
. This suggests that the rule-of-thumb to quantize the received signals at the noise level in the current literature can be highly suboptimal.