Improving the performance of asymmetric M-PAM signal constellations in Euclidean space by embedding them in hyperbolic space

We address the problem of improving the performance of asymmetric M-PAM signal constellations in Euclidean space by embedding them in hyperbolic space when the communications system designer is bounded to using this modulation. We derive the probability density function of the additive noise in one dimensional hyperbolic space by showing that it is a log-normal random variable. From this, we determine the optimum receiver for the M-PAM signal constellations in one dimensional hyperbolic space, and show that it is equivalent to the optimum receiver in the Euclidean space. Finally, we compare the performance of the communication system using M-PAM signal constellations in one dimensional hyperbolic space with the corresponding system in the Euclidean space. We show that the symmetric M-PAM signal constellations in one dimensional hyperbolic space derived from the corresponding asymmetric M-PAM in the Euclidean space have asymptotic coding gains.