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We consider the N-relay Gaussian diamond network where a source nodecommunicates to a destination node via N parallel relays through a cascade of aGaussian broadcast (BC) and a multiple access (MAC) channel. Introduced in 2000by Schein and Gallager, the capacity of this relay network is unknown ingeneral. The best currently available capacity approximation, independent ofthe coefficients and the SNR's of the constituent channels, is within anadditive gap of 1.3 N bits, which follows from the recent capacityapproximations for general Gaussian relay networks with arbitrary topology.  In this paper, we approximate the capacity of this network within 2 log Nbits. We show that two strategies can be used to achieve theinformation-theoretic cutset upper bound on the capacity of the network up toan additive gap of O(log N) bits, independent of the channel configurations andthe SNR's. The first of these strategies is simple partial decode-and-forward.Here, the source node uses a superposition codebook to broadcast independentmessages to the relays at appropriately chosen rates; each relay decodes itsintended message and then forwards it to the destination over the MAC channel.A similar performance can be also achieved with compress-and-forward typestrategies (such as quantize-map-and-forward and noisy network coding) thatprovide the 1.3 N-bit approximation for general Gaussian networks, but only ifthe relays quantize their observed signals at a resolution inverselyproportional to the number of relay nodes N. This suggest that therule-of-thumb to quantize the received signals at the noise level in thecurrent literature can be highly suboptimal.

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