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Nobody Expects a Differential Equation: Minimum Energy-Per-Bit for the Gaussian Relay Channel with Rank-1 Linear Relaying

Oliver Kosut, Michelle Effros, Michael Langberg

TL;DR

This work provides an upper bound on the minimum energy-per-bit for the Gaussian relay channel under a rank-1 linear-relaying constraint by transforming a non-convex energy optimization into a tractable differential equation problem. The key idea is to connect optimality conditions to a running-sum differential system, solve a reduced two-parameter (A,B) dynamic with an invariant $AB+\frac{1}{A}-\frac{1}{B}=c_2$, and then map the solution back to a rank-1 linear-relay code. The resulting bound, denoted $\mathcal{E}^*_{\text{1LR}}$, improves upon prior 2D and block-Markov bounds in many parameter regimes and provides insight into how low-complexity relaying can approach fundamental energy efficiency limits. The approach combines rank-1 covariance structure, strictly causal linear relaying, and a rigorous differential-equation framework to yield practical, low-complexity schemes for energy-constrained wireless networks.

Abstract

Motivated by the design of low-complexity low-power coding solutions for the Gaussian relay channel, this work presents an upper bound on the minimum energy-per-bit achievable on the Gaussian relay channel using rank-1 linear relaying. Our study addresses high-dimensional relay codes and presents bounds that outperform prior known bounds using 2-dimensional schemes. A novelty of our analysis ties the optimization problem at hand to the solution of a certain differential equation which, in turn, leads to a low energy-per-bit achievable scheme.

Nobody Expects a Differential Equation: Minimum Energy-Per-Bit for the Gaussian Relay Channel with Rank-1 Linear Relaying

TL;DR

This work provides an upper bound on the minimum energy-per-bit for the Gaussian relay channel under a rank-1 linear-relaying constraint by transforming a non-convex energy optimization into a tractable differential equation problem. The key idea is to connect optimality conditions to a running-sum differential system, solve a reduced two-parameter (A,B) dynamic with an invariant , and then map the solution back to a rank-1 linear-relay code. The resulting bound, denoted , improves upon prior 2D and block-Markov bounds in many parameter regimes and provides insight into how low-complexity relaying can approach fundamental energy efficiency limits. The approach combines rank-1 covariance structure, strictly causal linear relaying, and a rigorous differential-equation framework to yield practical, low-complexity schemes for energy-constrained wireless networks.

Abstract

Motivated by the design of low-complexity low-power coding solutions for the Gaussian relay channel, this work presents an upper bound on the minimum energy-per-bit achievable on the Gaussian relay channel using rank-1 linear relaying. Our study addresses high-dimensional relay codes and presents bounds that outperform prior known bounds using 2-dimensional schemes. A novelty of our analysis ties the optimization problem at hand to the solution of a certain differential equation which, in turn, leads to a low energy-per-bit achievable scheme.
Paper Structure (11 sections, 6 theorems, 153 equations, 1 figure)

This paper contains 11 sections, 6 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Prior Bounds
  4. Main Result
  5. Transforming the Optimization Problem
  6. From Optimality Conditions to a Differential Equation
  7. Proof of Lemma \ref{['lemma:A0c1']}
  8. Proof of Lemma \ref{['lemma:optimality_conditions']}
  9. Proof of Lemma \ref{['lemma:Euler']}
  10. Proof of Lemma \ref{['lemma:DE']}
  11. Proof of Lemma \ref{['lemma:final_achievability']}

Key Result

Lemma 1

Fix any $A_f,B_f>0$ where $A_f/B_f\le a^2$. LetNote that $f$ implicitly depends on $A_f,B_f$ through $\phi$. There exists a unique pair $(A_0,\psi)$ such that $A_0\ge A_f$, $\psi>0$ and

Figures (1)

  • Figure 1: Comparison between bounds on the minimum energy-per-bit for the Gaussian relay channel. Bounds are computed for channel parameters $a=1.1$ and $b\in[0,10]$. The normalized energy-per-bit $\mathcal{E}/(2\ln 2)$ is plotted for each bound. The block-Markov achievable bound is \ref{['block_Markov']}, the $2\times 2$ bound is from \ref{['2x2_scheme']}, the rank 1 bound is from Thm. \ref{['thm:main']}, and the cut-set bound is \ref{['cutset_bound']}.

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5