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

2015/04/20 by Varun Jog, Jog, Varun, Venkat Anantharam +1
Computer Science · Engineering · Mathematics · #Antenna Design and Analysis #Antenna Design and Optimization #FOS: Computer and information sciences #Information Theory (cs.IT) #Structural Analysis and Optimization #cs.IT #math.IT

paper · pdf · doi:10.48550/arxiv.1504.05182

55 pages, 5 figures. Contains results submitted to ISIT 2014 and ISIT 2015. Submitted to IEEE Transactions on Information Theory

arxiv created 2015/04/20 · openalex publication_date 2015/04/20 · arxiv updated 2015/04/22 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

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≥ 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) ⊆ ℝⁿ, 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) ⊕ B_n(√(nν)), which is the Minkowski sum of \cal 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 \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.

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