# Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems
## 6 Results  
This section represents the performance analysis of the massive MIMO system under realistic channel environment. To evaluate the performance, the authors used the MATLAB platform. The correlated Nakagami-m channel was taken to have a realistic environment. In addition, the authors also assumed equal power distribution for the transmitter section and full knowledge of channel state information (CSI) at the receiver side only. Figure 3 shows the performance comparison between the different receiver systems. The LLL algorithm was used to perform the lattice reduction. The performance was compared considering the MF receiver. As in Figure 3, with the increase in SNR, MMSE receiver approaches the ZF receiver. The same is true for the LLR-aided receiver system. In Figure 3, at an SNR 2 dB, the capacities corresponding to MF, MMSE, and LLR-MMSE receivers are 1.994, 3.559, and 4.472 bits/S/Hz, respectively. Now, at relatively high SNR (at 28 dB), the channel capacities for the same receivers are 2.492, 6.52, and 12.17 bits/S/Hz, respectively. Therefore, it is clear that there is a considerable capacity gain due to the lattice reduction-aided receiver system. This is because lattice reduction-aided receivers convert the ill-conditioned channel to a much better channel. Figure 4 represents the effect of the number of antennas over the channel capacities. The objective of this analysis is to figure out the capacity improvement due to LR-aided receivers. At a particular SNR level of 24 dB (as in Figure 4), the channel capacities for ZF receivers with \(6\times 6\), \(12\times 12\), and \(30\times 30\) antenna configurations are 8.093, 9.801, and 11.88 bits/S/Hz, respectively. Furthermore, with the same antenna configurations, for LLR-ZF receiver, the channel capacities are 16.19, 19.6, and 23.75 bits/S/Hz, respectively. Through this quantitative analysis, one can figure out that LLR-aided receivers improve the channel capacity tremendously. Similarly, if we compare the MMSE and LLR-MMSE receivers for these antenna configurations at same SNR level, we can visualize that LLR-MMSE is providing capacity increments of 7.608, 10.43, and 19.49 bits/S/Hz, respectively, concerning the MMSE receiver. Therefore, the lattice reduction-aided receivers effectively enhance the performance of the MIMO system, and, out of all the presented receiver systems, the LLLR-MMSE receiver provides the best solution for capacity enhancement. Figure 3: Capacity comparison with different receivers. The impact of the correlated channel over the capacity is presented in Figure 5. For this particular simulated result, we considered a \(30\times 30\) MIMO system with one receiver side correlated Nakagami-m channel (m = 1). Considering a SNR = 24 dB and correlation coefficient (\(\rho_{rx}\)) = 0.2, the channel capacities for ZF, MMSE, LLR-ZF, and LLR-MMSE are 4.326, 6.7263, 8.652, and 17.83 bits/S/Hz, respectively. Moreover, it establishes the utility of a lattice reduction-aided receiver system. Figure 6 shows the statistical analysis of the channel capacity for a MIMO system with LLR-MMSE receiver. Figure 7 presents the statistical analysis of the channel capacity for a MIMO system with the LLR-MMSE receiver with the variation m. There is a significant effect of m over the channel capacity and the higher is the value of m, the higher is the capacity of the MIMO system. Figure 4: Capacity comparison between different receiver with different antenna configurations. Figure 5: Impact of (\(\rho_{rx}\)) over the channel capacity. The variation of OD with respect to the dimension of the MIMO system is presented in Figure 8. It can be seen that the lattice reduction algorithms significantly improve the channel condition. Figure 9 describes the performance of the ZK and LLL reduction algorithm with the change in \(\rho_{rx}\). The KZ reduction algorithm has a better ability to reduce OD in comparison to the LLL algorithm. In addition, in identifies the effectiveness of the KZ algorithm when the channel correlation is high (\(\rho_{rx}\) = 0.5).