Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Capacity of the Binary Energy Harvesting Channel

Eli Shemuel, Oron Sabag, Haim H. Permuter

TL;DR

This work resolves the open problem of computing the capacity of the Binary Energy Harvesting Channel (BEHC) with a unit-sized battery by formulating computable single-letter lower and upper bounds through Q-graphs and convex optimization. By modeling the BEHC as a finite-state channel with causal state information at the encoder and leveraging a sequence of $N$-node Q-graphs, the authors show that the bounds converge to the true capacity $C_{BEHC}$ with a gap that scales as $O(N)$. They prove convexity of both bound formulations and demonstrate that, for harvesting probabilities ${η\in\{0.1,0.2,\dots,0.9\}}$, the capacity can be computed to ${1e-6}$ precision, outperforming prior bounds. The framework is extended to noisy BEHCs with feedback, where a Markov decision process yields numerical achievable rates for a binary symmetric channel, with the notable result that the rate matches $1- H(p)$ when ${η=1}$. Overall, the paper provides a practical, principled method to obtain highly accurate BEHC capacity values and paves the way for efficient analysis of broader EH models.

Abstract

The capacity of a channel with an energy-harvesting (EH) encoder and a finite battery remains an open problem, even in the noiseless case. A key instance of this scenario is the binary EH channel (BEHC), where the encoder has a unit-sized battery and binary inputs. Existing capacity expressions for the BEHC are not computable, motivating this work, which determines the capacity to any desired precision via convex optimization. By modeling the system as a finite-state channel with state information known causally at the encoder, we derive single-letter lower and upper bounds using auxiliary directed graphs, termed $Q$-graphs. These $Q$-graphs exhibit a special structure with a finite number of nodes, $N$, enabling the formulation of the bounds as convex optimization problems. As $N$ increases, the bounds tighten and converge to the capacity with a vanishing gap of $O(N)$. For any EH probability parameter $η\in \{0.1,0.2, \dots, 0.9\}$, we compute the capacity with a precision of ${1e-6}$, outperforming the best-known bounds in the literature. Finally, we extend this framework to noisy EH channels with feedback, and present numerical achievable rates for the binary symmetric channel using a Markov decision process.

Capacity of the Binary Energy Harvesting Channel

TL;DR

This work resolves the open problem of computing the capacity of the Binary Energy Harvesting Channel (BEHC) with a unit-sized battery by formulating computable single-letter lower and upper bounds through Q-graphs and convex optimization. By modeling the BEHC as a finite-state channel with causal state information at the encoder and leveraging a sequence of -node Q-graphs, the authors show that the bounds converge to the true capacity with a gap that scales as . They prove convexity of both bound formulations and demonstrate that, for harvesting probabilities , the capacity can be computed to precision, outperforming prior bounds. The framework is extended to noisy BEHCs with feedback, where a Markov decision process yields numerical achievable rates for a binary symmetric channel, with the notable result that the rate matches when . Overall, the paper provides a practical, principled method to obtain highly accurate BEHC capacity values and paves the way for efficient analysis of broader EH models.

Abstract

The capacity of a channel with an energy-harvesting (EH) encoder and a finite battery remains an open problem, even in the noiseless case. A key instance of this scenario is the binary EH channel (BEHC), where the encoder has a unit-sized battery and binary inputs. Existing capacity expressions for the BEHC are not computable, motivating this work, which determines the capacity to any desired precision via convex optimization. By modeling the system as a finite-state channel with state information known causally at the encoder, we derive single-letter lower and upper bounds using auxiliary directed graphs, termed -graphs. These -graphs exhibit a special structure with a finite number of nodes, , enabling the formulation of the bounds as convex optimization problems. As increases, the bounds tighten and converge to the capacity with a vanishing gap of . For any EH probability parameter , we compute the capacity with a precision of , outperforming the best-known bounds in the literature. Finally, we extend this framework to noisy EH channels with feedback, and present numerical achievable rates for the binary symmetric channel using a Markov decision process.

Paper Structure

This paper contains 27 sections, 13 theorems, 75 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and the Channel Model
  3. Notation and Definitions
  4. Preliminary on Quantized-graphs
  5. The Energy-Harvesting Communication Model
  6. A Lower Bound on the Feedback Capacity of FSCs With SI Available Causally at the Encoder
  7. Main Results
  8. Convex Optimization Formulations
  9. Convex Lower Bound on $C_{\text{BEHC}}$
  10. Stationary Distribution
  11. FSC Law
  12. Policy
  13. PMF
  14. Convex Upper Bound on $C_{\text{BEHC}}$
  15. Modified FSC Law Constraint Functions
  16. ...and 12 more sections

Key Result

Theorem 1

( shemuel2024finite): For any $Q$-graph, given a fixed finite cardinality $\left|\mathcal{U}\right|$ ($U^+,U \in {\mathcal{U}}$) and a function $f: \mathcal{U} \times \mathcal{S} \to \mathcal{X}$, the feedback capacity is lower bounded by for all $P(u^+|u,q)\in\mathcal{P}_\pi$ that are BCJR-invariant eq:BCJR_Def, where the joint distribution is given by

Figures (5)

  • Figure 1: The energy-harvesting model with a finite battery.
  • Figure 2: FSC with feedback and SI available causally to the encoder. At time $i$, the current state $S_{i-1}$ influences output $Y_i$.
  • Figure 3: An instance of a $2$-node $Q$-graph with ternary output.
  • Figure 4: A $Q$-graph used for lower bounding $C_{\text{BEHC}}$ in Theorem \ref{['thr:EH_cvx_lower']}.
  • Figure 5: A $Q$-graph for upper bounding $C_{\text{BEHC}}$ in Theorem \ref{['thr:EH_cvx_upper']}.

Theorems & Definitions (23)

  • Definition 1: Connectivity
  • Theorem 1
  • Theorem 2: Convex Lower Bound
  • Theorem 3: Convex Upper Bound
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 13 more