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A Geometric Analysis of the AWGN channel with a $(σ, ρ)$-Power Constraint

In this paper, we consider the AWGN channel with a power constraint called the $(σ, ρ)$-power constraint, which is motivated by energy harvesting communication systems. Given a codeword, the constraint imposes a limit of $σ+ k ρ$ on the total power of any $k\geq 1$ consecutive transmitted symbols. Such a channel has infinite memory and evaluating its exact capacity is a difficult task. Consequently, we establish an $n$-letter capacity expression and seek bounds for the same. We obtain a lower bound on capacity by considering the volume of ${\cal S}_n(σ, ρ) \subseteq \mathbb{R}^n$, which is the set of all length $n$ 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 ${\cal S}_n(σ, ρ) \oplus B_n(\sqrt{nν})$, which is the Minkowski sum of ${\cal S}_n(σ, ρ)$ and the $n$-dimensional Euclidean ball of radius $\sqrt{nν}$. 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 ${\cal S}_n(σ, ρ)$. We show that as the dimension $n$ increases, the logarithm of the sequence of intrinsic volumes of $\{{\cal S}_n(σ, ρ)\}$ converges to a limit function under an appropriate scaling. The upper bound on capacity is then expressed in terms of this limit function. We derive the asymptotic capacity in the low and high noise regime for the $(σ, ρ)$-power constrained AWGN channel, with strengthened results for the special case of $σ= 0$, which is the amplitude constrained AWGN channel.