The impact of constellation cardinality on Gaussian channel capacity

Denote by C
_m
(snr) the Gaussian channel capacity with signal-to-noise ratio SNR and input cardinality m. We show that as m grows, C
_m
(snr) approaches C(snr) = 1/2 log(1 + snr) exponentially fast. Lower and upper bounds on the exponent are given as functions of SNR. We propose a family of input constellations based on the roots of the Hermite polynomials which achieves exponential convergence.