Optimal uncoded regeneration for binary antipodal signaling

We derive, for a binary antipodal input signal, the optimal uncoded regenerator function when the channels at the ingress and at the egress of the regenerator are degraded by AWGN. We show that the optimal function is a Lambert W function parametrized on the energies of the noises and the input. For comparison, we derive the performance of systems in which the regenerator uses a hard limiter or an amplifier.