A geometric analysis of the AWGN channel with a (σ, ρ)-power constraint

We consider the additive white Gaussian noise (AWGN) channel with a (σ, ρ)-power constraint, which is motivated by energy harvesting communication systems. This constraint imposes a limit of σ + kρ on the total power of any k ≥ 1 consecutive transmitted symbols in a codeword. We analyze the capacity of this channel geometrically, by considering the set S_n(σ, ρ) ⊆ ℝⁿ which is the set of all n-length sequences satisfying the (σ, ρ)-power constraints. For a noise power of ν, we obtain an upper bound on capacity by considering the volume of the Minkowski sum of S_n(σ, ρ) and the n-dimensional Euclidean ball of radius √(nν). We analyze this bound using a result from convex geometry known as Steiner's formula, which gives the volume of this Minkowski sum in terms of the intrinsic volumes of S_n(σ, ρ). We show that as n increases, the logarithms of the intrinsic volumes of {S_n(σ, ρ)} converge to a limit function under an appropriate scaling. An upper bound on capacity is obtained in terms of the limit function, thus pinning down the asymptotic capacity of the (σ, ρ)-power constrained AWGN channel in the low-noise regime. We derive stronger results when σ = 0, corresponding to the amplitude-constrained AWGN channel.