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

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.