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

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Abstract
The capacity C(ρa, ρp) of the discrete-time quadrature additive Gaussian channel (QAGC) with inputs subjected to (normalized) average and peak power constraints, ρa and ρp 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 ρp⩽7.8 (dB 4.8 (dB) per dimension). The capacity under a peak-power constraint is evaluated for a wide range of ρp, by incorporating the theoretical observations into a nonlinear dynamic programming procedure. Closed-form expressions for the asymptotic (low and large ρa and ρp) capacity and the corresponding capacity achieving distribution and for lower and upper bounds on the capacity C(ρa, ρp ) are developed. The capacity C(ρa, ρp ) 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 ρa/ρp and at moderate power values

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