On the low-rate Shannon limit for binary intersymbol interference channels

For a discrete-time, binary-input, Gaussian channel with finite intersymbol interference, we prove that reliable communication can be achieved if, and only if, E_b/N₀>log2/G_opt, for some constant G_opt that depends on the channel. To determine this constant, we consider the finite-state machine which represents the output sequences of the channel filter when driven by binary inputs. We then define G_opt as the maximum output power achieved by a simple cycle in this graph, and show that no other cycle or asymptotically long sequence can achieve an output power greater than this. We provide examples where the binary input constraint leads to a suboptimality, and other cases where binary signaling is just as effective as real signaling at very low signal-to-noise ratios.