Multi-relay regeneration for long haul communications

A concatenated additive white gaussian noise (AWGN) relay channel is considered. The different relays are only allowed to carry out simple symbol manipulation (e.g. hard limiter, linear amplifier or any symbol-wise analog scheme). The decay of the overall capacity as a function of the number of concatenated relays is explored. Contrary to point-to-point channels, where using a refined constellation can only increase the capacity, we show that the larger the constellation the faster the capacity decays. Since the initial point-to-point capacity of refined constellations is larger and their decay is faster, there is a tradeoff that leads to an optimal constellation size which is a function of the SNR and the number of relays.