The impact of constellation cardinality on Gaussian channel capacity

Abstract—Denote by Cm(snr) the Gaussian channel capacity with signal-to-noise ratio snr and input cardinality m. We show that as m grows, Cm(snr) approaches C(snr) = 12 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. I