# Steepest Gradient-Based Orthogonal Precoder For Integer-Forcing MIMO

## TL;DR

This paper introduces an orthogonal precoding scheme for integer-forcing MIMO systems using a steepest gradient algorithm, which outperforms existing methods in rate, error, and complexity.

## Contribution

It proposes a novel orthogonal precoding method with an efficient steepest gradient algorithm, improving performance and reducing complexity over unitary precoding in integer-forcing MIMO.

## Key findings

- Orthogonal precoding outperforms unitary precoding in rate and error metrics.
- The proposed algorithm has low complexity and is adaptable to various MIMO configurations.
- Numerical results show superiority over UPIF type II and X-precoder in certain scenarios.

## Abstract

In this paper, we develop an orthogonal precoding scheme for integer-forcing (IF) linear receivers using the steepest gradient algorithm. Although this scheme can be viewed as a special case of the unitary precoded integer-forcing (UPIF), it has two major advantages. First, the orthogonal precoding outperforms its unitary counterpart in terms of achievable rate, outage probability, and error rate. We verify this advantage via theoretical and numerical analyses. Second, it exhibits lower complexity as the dimension of orthogonal matrices is half that of unitary matrices in the real-valued domain. For finding ``good'' orthogonal precoder matrices, we propose an efficient algorithm based on the steepest gradient algorithm that exploits the geometrical properties of orthogonal matrices as a Lie group. The proposed algorithm has low complexity and can be easily applied to an arbitrary MIMO configuration. We also confirm numerically that the proposed orthogonal precoding outperforms UPIF type II in some scenarios and the X-precoder in high-order QAM schemes, e.g., $64$- and $256$-QAM.