Computation alignment: Capacity approximation without noise accumulation

Consider several source nodes communicating across a wireless network to a destination node with the help of several layers of relay nodes. Recent work by Avestimehr et al. has approximated the capacity of this network up to an additive gap. The communication scheme achieving this capacity approximation is based on compress-and-forward, resulting in noise accumulation as the messages traverse the network. As a consequence, the approximation gap increases linearly with the network depth. This paper derives a capacity approximation for multi-layer wireless relay networks with an approximation gap that is independent of the network depth and dependent only on the number of source nodes and the fading statistics. This is achieved by a new communication strategy termed computation alignment. This strategy is based on the compute-and-forward framework, which enables relays to decode deterministic functions of the transmitted messages. Alone, compute-and-forward is insufficient to approach the capacity as it incurs a penalty for approximating the wireless channel with complex-valued coefficients by a channel with integer coefficients. Here, this penalty is avoided by breaking the wireless channel into several subchannels combined with a signal-alignment strategy to ensure that these subchannels have integer channel coefficients.