On $2 \times 2$ MIMO Gaussian Channels with a Small Discrete-Time Peak-Power Constraint
Alex Dytso, Luca Barletta, Gerhard Kramer
TL;DR
This paper analyzes the capacity of a $2\times2$ MIMO Gaussian channel under a peak-power constraint by reformulating the problem as maximizing mutual information over input vectors restricted to an ellipse $\mathcal{E}(r_p,r_m)$. In the small-peak-power regime ($r_m \le r_p \le \sqrt{2}$), the authors show the capacity-achieving input concentrates on the ellipse boundary, derive a KKT-based condition, and establish symmetry properties of the optimal distribution. A concrete sufficient condition is given under which a two-point distribution at $(\pm r_p,0)$ is optimal, and they prove discreteness of the optimal input when $r_p\neq r_m$, with the special case $r_p=r_m$ yielding a uniform distribution on the circle $\partial\mathcal{E}(r_p,r_p)$. The results illuminate the structure of capacity-achieving distributions for low-peak-power MIMO channels and connect to classical SISO insights, while outlining open questions for higher dimensions and the $r_m\to r_p$ transition.
Abstract
A multi-input multi-output (MIMO) Gaussian channel with two transmit antennas and two receive antennas is studied that is subject to an input peak-power constraint. The capacity and the capacity-achieving input distribution are unknown in general. The problem is shown to be equivalent to a channel with an identity matrix but where the input lies inside and on an ellipse with principal axis length $r_p$ and minor axis length $r_m$. If $r_p \le \sqrt{2}$, then the capacity-achieving input has support on the ellipse. A sufficient condition is derived under which a two-point distribution is optimal. Finally, if $r_m < r_p \le \sqrt{2}$, then the capacity-achieving distribution is discrete.