# Multiuser Equalizer for Hybrid Massive MIMO mmWave CE-OFDM Systems
## 4 Performance Results  
The main performance results of the proposed hybrid analog-digital multiuser equalizer, designed for CE-OFDM based systems, are shown in this section. The performance metric is the average BER as a function of \(E_{b}/N_{0}\), where \(E_{b}\) is the average bit energy, and \(N_{0}\) is the one-sided noise power spectral density. We assume that the signal transmitted power of all users is the same, such that, \(\sigma_{s}^{2}=1\), and then \(E_{b}/N_{0}=\sigma_{s}^{2}/(2\sigma_{n}^{2})=\sigma_{n}^{-2}/2\). The quadrature phase shift keying (QPSK) modulation is adopted. For each UT, the wideband mmWave channel model defined in Equation (3) is considered and the parameters presented in Table 1. At the receiver side, it is assumed a perfect synchronization and a perfect CSI knowledge. The main fixed simulation parameters are also presented in Table 1, and it is assumed the worst case, i.e., \(U=N_{rx}^{RF}\), since for \(U\ll N_{rx}^{RF}\). The performance results were obtained by varying the modulation index, \(h\), the number of users and of RF chains, of receive antennas, \(N_{rx}\). The specific values for these parameters are indicated in the figures. First, let us evaluate the impact of modulation index on the performance of proposed hybrid system. As we can see in Figure 3, several values for \(2\pi h\) were tested, and the results for 1 st and 4 th iterations were compared. Cleary, there is an optimal value, which is approximately \(2\pi h=0.9\). A deeper discussion about the impact of the modulation index in the system performance can be found in [19]. Hereinafter, the results are obtained for \(2\pi h=0.9\). In Figure 4, we compare the simulation results with the semi-analytic BER approximation in (14). We can see that the results are very close, then we conclude that \(\mathbf{c}_{k}\) is approximately Gaussian distributed, and we confirm that the assumption made in Equation (12) was correct. We may also see that the performance gap from iteration 1 to iteration 2 is higher than from iteration 2 to iteration 4. This happens because most of the residual ISI and multiuser interference are removed from the 1 st to the 2 nd iteration. From the 2 nd to the 4 th iteration, the gain is smaller because most of the interference was already removed. In Figures 5 and 6, a comparison between the proposed iterative hybrid equalizer and the iterative fully digital one, for \(N_{rx}=32\) and \(N_{rx}=64\) was performed, respectively. All expressions of the hybrid case are valid for the full-digital case, assuming the number of RF chains are equal to the number of receive antennas. Therefore, for the fully digital case, only Algorithm 1 is used, assuming \(\mathbf{W}_{a}=\mathbf{I}_{N_{rx}}\). For \(N_{rx}=32\) (Figure 5), the gain of the proposed hybrid equalizer, from the 1 st to 4 th iteration, for a \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency & 72 GHz \\ Antenna element spacing & Half-wavelength \\ Array configuration & Uniform Linear Array (ULA) \\ Decay from the first to the last channel cluster \((N_{cl}\beta_{q_{1},q_{2}})\) & 10 dB \\ Decay from the first to the last ray of each channel cluster & 10 dB \\ \((\beta_{q_{1},l_{2}})\) & 4 \\ \(N_{cl}\) & 5 \\ \(N_{nx}\) & 10 degrees \\ \(N_{\infty}\) & 2 \\ \(N_{c}\) & 512 \\ \(D\) & 128 \\ \(N_{tx}\) & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 4: Performance of the proposed iterative hybrid equalizer and semi-analytic bit-error-rate (BER) approximation, with \(N_{rx}^{RF}=U=8\) and \(N_{rx}=32\). Figure 3: Performance comparison of the proposed iterative hybrid equalizer with \(N_{rx}^{RF}=U=8\), and \(N_{rx}=32\) for different modulation indexes. BER target of \(10^{-3}\), is 7.2 dB, and the penalties from hybrid to fully digital case are 7.1 dB and 2.7 dB, for iterations 1 and 4, respectively.