# On the Equivalence of Semidifinite Relaxations for MIMO Detection with General Constellations

## TL;DR

This paper investigates the theoretical relationship between different semidefinite relaxation methods for MIMO detection, establishing their equivalence and providing insights into their tightness under certain conditions.

## Contribution

It proves the equivalence of two distinct SDRs for MIMO detection, enhancing understanding of their theoretical properties and performance.

## Key findings

- Two SDRs for MIMO detection are equivalent.
- The tightness of one SDR can be guaranteed under a specific condition.
- Theoretical insight into the relationship between different relaxation techniques.

## Abstract

The multiple-input multiple-output (MIMO) detection problem is a fundamental problem in modern digital communications. Semidefinite relaxation (SDR) based algorithms are a popular class of approaches to solving the problem because the algorithms have a polynomial-time worst-case complexity and generally can achieve a good detection error rate performance. In spite of the existence of various different SDRs for the MIMO detection problem in the literature, very little is known about the relationship between these SDRs. This paper aims to fill this theoretical gap. In particular, this paper shows that two existing SDRs for the MIMO detection problem, which take quite different forms and are proposed by using different techniques, are equivalent. As a byproduct of the equivalence result, the tightness of one of the above two SDRs under a sufficient condition can be obtained.