# Iterative Analog-Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems
## 5 Performance Results
From the Figures 3 and 4 we can see that the theoretical curves almost overlap with the simulated ones, which means that our semi-analytical approach is quite accurate. As also observed in Figures 3 and 4, the performance improves for both cases as the number of iterations increases. We may also see from these two figures that the performance gap from iteration 1 to iteration 2 is higher than from iteration 3 to iteration 4. This change occurs because most of the residual multi-user and intersymbol interferences are mitigated from the first to the second iteration. From the third to the fourth iteration, there is also a benefit from residual interference removal; however, the gain is smaller because most of the interference was previously removed. We can also verify that the gap between NCSI (Figure 3) and PCSI (Figure 4) precoders at a target BER of is 5.3 dB for the fourth iteration, i.e., with only the knowledge of average AoD information at the UTs with the proposed efficient hybrid equalizers, the average BER performance has a significant improvement. Let us compare the average BER performance between the proposed Algorithm 1 and the fully digital iterative multi-user equalizer. Figures 3 and 4 show that, for iteration 4, the proposed Algorithm 1 almost achieves the fully digital bound, i.e., the proposed receive and transmit structures are sufficiently efficient to overtake the constraints imposed by the hybrid architectures. In Figures 5 and 6, we compare the performance of the proposed iterative hybrid multi-user equalizer, against the performance of two-step approach proposed in [31], and the scheme proposed in [14], where the analog part is computed by applying the Gram-Schmidt (GS) method, while in the digital part a MMSE equalizer is used, referred here as GS/MMSE approach. Starting by analyzing the first iteration, we can see that the BER performance of the two-step approach and Algorithm 1 are similar, since for iteration 1, both algorithms assume \(\boldsymbol{\Psi}^{(0)}=0_{U}\), and then they are equivalent. Focusing now on the fourth iteration, the performance of two-step approach is far from the performance of Algorithm 1. This happens because only the digital part is iteratively, while the proposed algorithm the analog and digital parts are both computed in each iteration, and thus it is more efficient to remove the interferences. From the figure, we can also observe that the performance of the GS/MMSE approach is approximately the same as the one obtained by our schemes for the first iteration. This can be explained by the fact that in our schemes the digital part, for the first iteration, falls in the MMSE equalizer. Now, let us analyze the results presented in Figures 7 and 8 using 16-QAM and considering PCSI precoder. Figure 7 presents the results for Algorithm 1 for 1 up to 6 iterations. As for the QPSK, the  
Figure 4:0 Performance comparison between the Algorithm 1 and the two-step approach for the PCSI precoder with ULA configuration, path loss, and shadowing and QPSK modulation. performance improves, as the number of iterations increases. Comparing these results with the ones obtained in Figure 4 for QPSK we observe a performance penalty since the 16-QAM is more prone to errors and therefore more iterations are required to remove the residual multi-user and intersymbol interferences. However, similarly to the QPSK case, Algorithm 1 almost achieves the fully digital bound for 16-QAM, requiring only a few iterations. Figure 8 presents the results for both Algorithm 1, the two-step approach considering first and fourth iterations and the GS/MMSE for the first iteration. Comparing these results with the ones presented in Figure 6 for QPSK the same conclusions can be reached. In Figures 9 and 10 we present the results for the same schemes of the Figures 3 and 4 but now considering UPA configuration. From them, we can see that the theoretical curves almost overlap with the simulated ones, which means that our semi-analytical approach is also quite accurate for the UPA configuration.