# Diversity Order in ISI Channels with Single-Carrier Frequency-Domain Equalizers

## TL;DR

This paper investigates the diversity gain of single-carrier frequency-domain equalizers in frequency-selective channels, revealing how diversity depends on channel memory, spectral efficiency, and block length, with specific results for MMSE and zero forcing receivers.

## Contribution

It provides a detailed analysis of diversity order in SC-FDE systems, showing the dependence on rate, channel memory, and block length, and contrasting MMSE with zero forcing methods.

## Key findings

- MMSE SC-FDE achieves full diversity at low rates.
- Diversity decreases with higher spectral efficiency.
- Zero forcing SC-FDE always has diversity order one.

## Abstract

This paper analyzes the diversity gain achieved by single-carrier frequency-domain equalizer (SC-FDE) in frequency selective channels, and uncovers the interplay between diversity gain $d$, channel memory length $\nu$, transmission block length $L$, and the spectral efficiency $R$. We specifically show that for the class of minimum means-square error (MMSE) SC-FDE receivers, for rates $R\leq\log\frac{L}{\nu}$ full diversity of $d=\nu+1$ is achievable, while for higher rates the diversity is given by $d=\lfloor2^{-R}L\rfloor+1$. In other words, the achievable diversity gain depends not only on the channel memory length, but also on the desired spectral efficiency and the transmission block length. A similar analysis reveals that for zero forcing SC-FDE, the diversity order is always one irrespective of channel memory length and spectral efficiency. These results are supported by simulations.