# Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems
## 7 Conclusions  
The growing demand for high data rate systems motivates researchers to explore the sophisticated signal processing tools for massive MIMO systems. However, system complexity is the main limitation. This paper discusses the effectiveness of the lattice reduction-aided receivers for massive MIMO systems. In this paper, we analyze the KZ- and LLR-based lattice reduction algorithms. The performance of the massive MIMO system is analyzed in terms of the improvement in achievable capacity using the lattice reduction-aided receivers. Through the mathematical calculation and simulated results, this study showed that the lattice reduction-aided receivers outperform the conventional linear receivers, to enhance the channel capacity under the correlated channel condition. Furthermore, the KZ reduction-aided receivers is much more efficient in reducing the orthogonal defect under the correlated channel condition in comparison to LLL algorithms. This efficiency results in a robust conditioned channel and significantly impacts the achievable capacity, and, as a consequence, KZ reduction-aided receivers outperforms the LLL reduction-aided receivers. Under correlated channel condition (\(\rho_{rx}\) = 0.2) and high SNR condition (SNR = 30 dB), the KZ-MMSE receiver provides an increment in the capacity of 17.73, 12.209, and 9.835 Bits/S/Hz for MF, MMSE, and LLR-MMSE receivers, respectively. Therefore, this presents KZ reduction-aided receivers as a default choice for a massive MIMO system.  
As future work, the authors will analyze the achievable capacity for millimeter-wave massive MIMO systems under the realistic highly dynamic system. This study also has limitations since the authors did not explore the impact of the Doppler shift, and this analysis will be highly beneficial for a high-speed vehicular communication system. As an extension of the presented study, the authors plan to focus on the performance analysis of the lattice reduction-aided receivers under the influence of multi-user interference (MUI), particularly at a cell edge environment.  
All authors were responsible for mathematical design; simulation and analyzing data; and writing the original manuscript. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Receiver** & \multicolumn{4}{c}{**Capacity (Bits/S/Hz)**} \\ \hline  & \multicolumn{2}{c}{\(\rho_{rx}\) = 0} & \multicolumn{2}{c}{\(\rho_{rx}\) = 0.2} \\ \hline
**System** & **SNR = 3 dB** & **SNR = 30 dB** & **SNR = 3 dB** & **SNR = 30 dB** \\ \hline MF & 2.17 & 2.22 & 1.531 & 1.533 \\ \hline MMSE & 3.709 & 11.66 & 2.453 & 7.051 \\ \hline LLR-MMSE & 5.519 & 21.73 & 2.455 & 9.425 \\ \hline KZ-MMSE & 12.57 & 23.23 & 5.264 & 19.26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Achievable capacity comparison between different Receivers.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{4}{c}{**Capacity (Bits/S/Hz)**} \\ \hline
**Antenna** & \multicolumn{2}{c}{\(4\times 4\)} & \multicolumn{2}{c}{\(10\times 10\)} & \multicolumn{2}{c}{\(20\times 20\)} \\ \hline ME & 2.189 & 2.273 & 2.293 \\ \hline MMSE & 5.897 & 6.779 & 10.25 \\ \hline LLR-MMSE & 10.16 & 12.45 & 18.52 \\ \hline KZ-MMSE & 15.14 & 17.42 & 20.29 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Achievable capacity comparison with the variation in number of antennas.