# Explicit Lower Bounds on the Outage Probability of Integer Forcing over Nrx2 Channels

## TL;DR

This paper derives explicit lower bounds on the outage probability for integer-forcing equalization in MIMO channels, providing benchmarks for system performance and assessing various precoding schemes.

## Contribution

It introduces a simple explicit lower bound on outage probability for 2xN systems using Jacobi ensemble properties, extending to space-time precoding and single-antenna MAC scenarios.

## Key findings

- Derived explicit lower bounds for outage probability.
- Extended bounds to random space-time precoding schemes.
- Showed integer-forcing with space-time coding approaches these bounds.

## Abstract

The performance of integer-forcing equalization for communication over the compound multiple-input multipleoutput channel is investigated. An upper bound on the resulting outage probability as a function of the gap to capacity has been derived previously, assuming a random precoding matrix drawn from the circular unitary ensemble is applied prior to transmission. In the present work a simple and explicit lower bound on the worst-case outage probability is derived for the case of a system with two transmit antennas and two or more receive antennas, leveraging the properties of the Jacobi ensemble. The derived lower bound is also extended to random space-time precoding, and may serve as a useful benchmark for assessing the relative merits of various algebraic space-time precoding schemes. We further show that the lower bound may be adapted to the case of a $1 \times N_t$ system. As an application of this, we derive closed-form bounds for the symmetric-rate capacity of the Rayleigh fading multiple-access channel where all terminals are equipped with a single antenna. Lastly, we demonstrate that the integer-forcing equalization coupled with distributed space-time coding is able to approach these bounds.