# A simple bound on the BER of the MAP decoder for massive MIMO systems

## TL;DR

This paper derives a simple, approximately tight upper bound on the BER of the MAP detector in massive MIMO systems, linking it to the replica method and providing insights at high-SNR.

## Contribution

It introduces a novel, simple upper bound on BER for massive MIMO, connecting it with the replica method and high-SNR analysis, supported by numerical validation.

## Key findings

- The BER bound is approximately tight at high-SNR.
- Under certain conditions, the replica prediction matches the matched filter lower bound.
- Numerical evidence supports the theoretical results.

## Abstract

The deployment of massive MIMO systems has revived much of the interest in the study of the large-system performance of multiuser detection systems. In this paper, we prove a non-trivial upper bound on the bit-error rate (BER) of the MAP detector for BPSK signal transmission and equal-power condition. In particular, our bound is approximately tight at high-SNR. The proof is simple and relies on Gordon's comparison inequality. Interestingly, we show that under the assumption that Gordon's inequality is tight, the resulting BER prediction matches that of the replica method under the replica symmetry (RS) ansatz. Also, we prove that, when the ratio of receive to transmit antennas exceeds $0.9251$, the replica prediction matches the matched filter lower bound (MFB) at high-SNR. We corroborate our results by numerical evidence.