# On the Ergodic Rate Lower Bounds with Applications to Massive MIMO

## TL;DR

This paper introduces two new ergodic rate lower bounds for massive MIMO systems that are effective in scenarios where traditional channel hardening assumptions do not hold, such as highly correlated channels or cell-free setups.

## Contribution

The authors propose two novel bounds on achievable ergodic rates that complement the classical bound, especially in non-hardening or interference-limited massive MIMO scenarios.

## Key findings

- The new bounds perform well when channel hardening is weak and the coherence block is large.
- They are effective in interference-limited systems with finite antennas.
- The bounds improve understanding of system performance under various realistic conditions.

## Abstract

A well-known lower bound widely used in the massive MIMO literature hinges on channel hardening, i.e., the phenomenon for which, thanks to the large number of antennas, the effective channel coefficients resulting from beamforming tend to deterministic quantities. If the channel hardening effect does not hold sufficiently well, this bound may be quite far from the actual achievable rate. In recent developments of massive MIMO, several scenarios where channel hardening is not sufficiently pronounced have emerged. These settings include, for example, the case of small scattering angular spread, yielding highly correlated channel vectors, and the case of cell-free massive MIMO. In this short contribution, we present two new bounds on the achievable ergodic rate that offer a complementary behavior with respect to the classical bound: while the former performs well in the case of channel hardening and/or when the system is interference-limited (notably, in the case of finite number of antennas and conjugate beamforming transmission), the new bounds perform well when the useful signal coefficient does not harden but the channel coherence block length is large with respect to the number of users, and in the case where interference is nearly entirely eliminated by zero-forcing beamforming. Overall, using the most appropriate bound depending on the system operating conditions yields a better understanding of the actual performance of systems where channel hardening may not occur, even in the presence of a very large number of antennas.