# Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems
## 6 Results
Figure 10 represents the effect of the channel correlation over the capacity. The MMSE outperformed the ZF receiver. Therefore, the authors considered only the MMSE receiver. Particularly, this result deals with the comparison between the conventional MF and MMSE receivers with KZ/LLL  
Figure 6: CDF and PDF of channel capacity for MIMO system with LLLR-MMSE receiver. Figure 7: CDF and PDF of channel capacity for MIMO system with LLLR-MMSE receiver (with the variation in m). lattice reduction-aided receivers for a \(10\times 10\) MIMO system. Now, at 24 dB SNR and \(\rho_{rx}=0\), capacities corresponding to MF, MMSE, LLR-MMSE, and KZ-MMSE are 2.219, 10.4, 19.11, and 21.66 bits/S/Hz, respectively. It reveals the impact of the lattice reduction-aided receivers over the channel capacity of a MIMO system. Furthermore, at the same SNR level but with \(\rho_{rx}=0.2\), it is seen in the figure that the KZ receiver performs better than the other receivers. Figure 8: Orthogonal defect (OD) comparison. Figure 9: Orthogonal defect (OD) variation with the channel correlation (\(\rho_{rx}\)). In the comparison for KZ-MMSE and LLR-MMSE under SNR = 24 dB and \(\rho_{rx}=0\), we observed that KZ-MMSE receiver outperforms the LLR-MMSE receiver by 2.55 bits/S/Hz. However, with the increased channel correlation (\(\rho_{rx}=0.2\)), the capacity difference between the two receiver system is 9.1840 bits/S/Hz. This proves the effectiveness of the KZ-MMSE receivers over the correlated channel condition. Figure 11 describes the impact of the dimension of the MIMO system over capacity for different receiver algorithms, and for this particular result the authors consider 10 dB SNR and uncorrelated channel condition. The most critical point that one can observe is that, as the number of antennas increases, the performance of LLR-MMSE and KZ-MMSE tends to converge. However, the lattice reduction-aided receivers present a better performance in comparison to the conventional receiver system. Figure 11: Capacity variation with the antenna dimension. Figure 5.0 Effect of the channel correlation on the capacity. In this study, the authors compared the running times of ZF, MMSE, LLR-MMSE, and KZ-MMSE, as a part of complexity analysis. To obtain the above result, we took the median of the average runtime for 1000 random matrices. As we considered some random matrices, we observed specific long run times; thus, by taking the median, we smoothened the average time. Figure 12 shows the variation of average run time as a function of the numbers of antennas. The run time is dependent on the system configuration of the simulation platform. We used Lenevo laptop with a 2.4-GHz Intel (R)Core (TM) i3-311M CPU processor, with MATLAB running under Windows XP. As in the figure, with the lattice reduction-aided receiver, the complexity of the system is increased in comparison to the conventional receivers. Nevertheless, MIMO detector with LLR-aided technique is able to produce full diversity with reasonable complexity and that is much lower than the ML receiver. Figure 13 shows the comparison between the IF receiver [34; 38] with the LLR-MMSE and KZ-MMSE receivers. The IF receiver outperforms the ZF and MMSE receivers. However, the LLR-MMSE and KZ-MMSE receivers show better performance. Moreover, KZ-MMSE receiver outperforms the other receiver systems. The LLR-aided receivers improves the massive MIMO system performance significantly. In this study, we analyzed the performance of the same receiver system with multiple users in a single cell. The authors observed that, under spatially correlated channel condition, LLR-aided receiver systems have significantly improved performance when compared with than the conventional linear receiver system; it is most suitable for the usage in a multi-user environment. Figure 14 shows the impact of the number of users (3/6) and the number of antennas at the receiver side (2/4), over the channel capacity, with 12 antennas at the transmitter side. As depicted in Figures 3, 4 and 7, the ZF receiver outperforms the MMSE receiver.