A Geometric Analysis of the AWGN Channel With a $(\sigma , \rho )$ -Power Constraint.

In this paper, we consider the additive white Gaussian noise (AWGN) channel with a power constraint called the (\sigma , \rho ) -power constraint, which is motivated by energy harvesting communication systems. Given a codeword, the constraint imposes a limit of \sigma + k \rho 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 , which is the set of all length n sequences satisfying the (\sigma , \rho ) -power constraints. For a noise power of \nu , which is the Minkowski sum of {\mathcal{ S}}_{n}(\sigma , \rho ) and the n -dimensional Euclidean ball of radius \sqrt n\nu . 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 \mathcal Sn(\sigma , \rho ) . We show that as the dimension n increases, the logarithm of the sequence of intrinsic volumes of \{ {\mathcal{ S}}_{n}(\sigma , \rho )\} 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 (\sigma , \rho )$ -power constrained AWGN channel, with strengthened results for the special case of $\sigma = 0$ , which is the amplitude constrained AWGN channel. [ABSTRACT FROM PUBLISHER]