# Two-Step Multiuser Equalization for Hybrid mmWave Massive MIMO GFDM Systems
## 5 Performance Results  
This section shows the performance of mmWave mMIMO hybrid analog-digital system using GFDM modulation. The results are evaluated through the BER (\(E_{b}/N_{0}\)) as a performance metric, where \(E_{b}\) is the average bit energy and \(N_{0}\) is one-sided noise power spectral density (PSD). We assume that the average \(E_{b}/N_{0}=\sigma_{u}^{2}/(2\sigma_{n}^{2})=\sigma_{n}^{-2}/2\) is the same for all users, where \(\sigma_{1}^{2}=\ldots,=\sigma_{U}^{2}=1\). In this work, the used simulation parameters are depicted in Table 1.  
Figure 3 shows the comparison between the proposed hybrid analog-digital scheme and the full digital one, for two types of precoders: (a) random-based precoder and (b) AoD-based precoder. Taking into consideration that the number of transmitting antennas is \(N_{tx}=8\), and the number of receiving antennas is \(N_{rx}=32\) for different number of RF chains, where \(N_{rx}^{RF}=4,\ 8,\ 16,\ 32\). For the random precoding (case (a), we can observe that the best performance among the addressed hybrid cases was when \(N_{rx}^{RF}=16\), with a gap of only around \(0.5\) dB at a BER target of \(10^{-3}\) compared with the full digital curve. Furthermore, it can be noticed that the gap from \(N_{rx}^{RF}=16\) to \(N_{rx}^{RF}=8\) is much lower than the gap between \(N_{rx}^{RF}=16\) and \(N_{rx}^{RF}=4\), especially at the high SNR region. Better performance results can be depicted in case b, where AoD precoding is applied because partial CSI is available at the transmitter side. For both cases, the performance tends to the one obtained with full digital approach when the number of RF chains increases as expected. For more illustration, Figure 4 clearly presents the enhancement in the performance of the two discussed cases, for \(N_{rx}^{RF}=8\), where the case of partial CSI gives better results \(4\) dB for BER \(=10^{-3}\) compared with the random precoding approach, which is expected because the information about the channels at the terminals is quite small. \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Parameters** & **Value** \\ \hline Modulation type & & GFDM \\ Number of subcarriers & & \(K=64\) \\ Number of timeslots & & \(M=15\) \\ Oversampling factor & & \(L=1\) \\ Samples per symbol & & \(N=64\) \\ Number of users & & \(U=4\) \\ Transmitting antennas & & \(N_{\rm tr}=8\) or 16 \\ Receiving antennas & & \(N_{\rm nr}=32,~{}64\) or 128 \\  & Array configuration & ULA \\  & Time in samples where the channel rays has a & \(D=NM/4=240\) \\  & Carrier frequency & \(f_{c}=72\,\rm GHz\) \\  & Antenna element spacing & \(\delta=\lambda/2\) \\ Channel model parameters & Number of clusters & \(N_{\rm d}=4\) \\  & Number of rays per cluster & \(N_{\rm nr}=5\) \\  & Angle spread for the transmitter and receiver & \(\sigma_{\sigma_{\rm tr}^{\rm nr}}=\sigma_{\sigma_{\rm tr}^{\rm nr}}=10o\) \\  & Decay from the first to the last channel cluster & \(\beta_{0,N_{\rm d}-1}=10\,\rm dB\) \\  & Decay from the first to the last ray of each & \(\beta_{\rm g,N_{\rm nr}-1}=10\,\rm dB\), \\  & channel cluster & \(q=0,\ldots,N_{\rm d}-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. GFDM, generalized frequency division multiplexing; ULA, uniform linear array. Figure 3: Performance comparison of the proposed hybrid precoders with the full digital system for \(N_{\rm rx}=8\) and \(N_{\rm rx}=32\): (**a**) random-based precoder, (**b**) angle of departure (AoD)-based precoder. Similarly, Figure 5 also compares the BER of the hybrid and the full digital schemes, for two types of precoders: (a) random-based precoder and (b) AoD-based precoder, where \(N_{tx}=16\), \(N_{rx}=64\) and \(N_{rx}^{RF}=4,\ 8,\ 12,\ 16,\ 64\). Even with the increasing number of transmitting and receiving antennas, we can see from both cases, as the number of RF chains increases, the diversity order increases the performance toward the full digital curve.