Uniform tight frames as optimal signals

Abstract

Non-orthogonal communication is a promising technique for future wireless networks (e.g., 6G and Wi-Fi 7). In the vector channel model, designing efficient non-orthogonal communication schemes amounts to the following extremum problem: \[ \max \min_k \frac{|v_k|^2}{σ^2 + \sum_{l \neq k} \langle v_k, v_l \rangle^2} \] where the maximum is taken among vector systems $(v_k{)}_1^N\subset {\mathbb{R}}^d$ satisfying $c_1\le |v_k{|}^2\le c_2$ for every $k$, and the parameter $\ddot{I}ƒ>0$ corresponds to the noise of the channel. We show that in the case $\ddot{I}ƒ=0$, uniform tight frames are the only optimal configurations. We also give quantitative bounds on the optimal capacity of vector channels with relatively small noise.

14 pages. Accepted for publication in Advances in Applied Mathematics