The capacity of average and peak-power-limited quadrature Gaussian channels

The capacity C(ρ_a, ρ_p) of the discrete-time quadrature additive Gaussian channel (QAGC) with inputs subjected to (normalized) average and peak power constraints, ρ_a and ρ_p respectively, is considered. By generalizing Smith´s results for the scalar average and peak-power-constrained Gaussian channel, it is shown that the capacity achieving distribution is discrete in amplitude (envelope), having a finite number of mass-points, with a uniformly distributed independent phase and it is geometrically described by concentric circles. It is shown that with peak power being solely the effective constraint, a constant envelope with uniformly distributed phase input is capacity achieving for ρ_p⩽7.8 (dB 4.8 (dB) per dimension). The capacity under a peak-power constraint is evaluated for a wide range of ρ_p, by incorporating the theoretical observations into a nonlinear dynamic programming procedure. Closed-form expressions for the asymptotic (low and large ρ_a and ρ_p) capacity and the corresponding capacity achieving distribution and for lower and upper bounds on the capacity C(ρ_a, ρ_p ) are developed. The capacity C(ρ_a, ρ_p ) provides an improved ultimate upper bound on the reliable information rates transmitted over the QAGC with any communication systems subjected to both average and peak-power limitations, when compared to the classical Shannon formula for the capacity of the QAGC which does not account for the peak-power constraint. This is in particular important for systems that operate with restrictive (close to 1) average-to-peak power ratio ρ_a/ρ_p and at moderate power values