# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 4 Simulation Results
In addition, taking as reference the BER \(=10^{-3}\), one can notice that the gain increment is reduced when doubling the number of antennas as \(M_{T}\) increases. For instance, increasing from \(M_{T}=M_{R}=4\) to \(M_{T}=M_{R}=8\) yields a gain of 1.22 dB, while increasing from \(M_{T}=M_{R}=16\) to \(M_{T}=M_{R}=32\) yields a gain of only 0.2 dB. It is important to highlight that, in contrast to the maximum likelihood detection, the proposed MMPTRBF decoding is able to operate with more than 16 antennas due to its reduced computational complexity, as discussed below in Section 5. Table 7.3 Systems with \(M_{T}=M_{R}=4\), \(M_{T}=M_{R}=8\) antennas for 4-QAM modulation. Table 7.4 MMPTRBF with \(M_{T}=4\), \(M_{T}=8\), and \(M_{R}=1\) antennas for 4-QAM modulation. Figure 21 shows the BER vs. \(E_{b}/N_{0}\) results of the ML-QQSTBC and ML-LS-QQSTBC with 16-PSK and MIMO-PTRBF with 16-QAM to further examine the extent of the proposed work for higher-order modulations with \(M_{T}=M_{R}=4\) antennas. This result shows that the proposed work operates efficiently with a higher modulation order of 16-QAM. It is important to note, however, that different modulation formats are used in this scenario because the maximum likelihood QOSTBC decoding is not capable of dealing with quadrature amplitude modulation, as addressed in [56]. For this reason, in order to keep 16-order modulation, a 16-PSK modulation format for QOSTBC is used. Figure 21 shows that the proposed work outperforms the QOSTBC using maximum likelihood (which is the optimal decoder for 16-PSK) by about 2 dB and outperforms the channel estimated scenario by more than 4 dB. Although this robust result seems to show a great advantage of using the proposed approach, we should be careful as it is not quite fair to compare 16-QAM and 16-PSK formats under the proposed nonlinear scenario. Table 7.5 Systems with \(M_{T}=M_{R}=4\) antennas operating at the same bitrate with 16-QAM (MMPTRBF) and 16-PSK (ML-QQSTBC and ML-LS-QQSTBC). Table 7.6 MMPTRBF with \(M_{T}=M_{R}=4\), \(M_{T}=M_{R}=8\), \(M_{T}=M_{R}=16\), \(M_{T}=M_{R}=32\), antennas for 4-QAM modulation. Figure 22 shows the BER vs. \(E_{b}/N_{0}\) results of the MIMO-PTRBF systems operating with \(M_{T}=M_{R}=4\) and \(M_{T}=M_{R}=8\) for 16-QAM modulation. Figure 22 highlights the increase in the diversity gain as a result of the increase in the number of transmitting antennas. Although the MMPTRBF with \(M_{T}=M_{R}=8\) has the worst results for low values of \(E_{b}/N_{0}\), for values above \(E_{b}/N_{0}=10\) dB, the performance with \(M_{T}=M_{R}=8\) surpasses the \(M_{T}=M_{R}=4\) results by about 1.6 dB for \(\text{BER}=10^{-3}\). Figure 23 shows the BER vs. \(E_{b}/N_{0}\) results of the MIMO-PTRBF with 64-QAM modulation to further investigate the proposed work for higher-order modulations with \(M_{T}=M_{R}=4\) antennas. Figure 23 shows the potential of the proposed approach of working with high-order modulation and highlights the gains over the QOSTBC with 64-PSK and perfect channel estimation, by about 5 dB, and for the channel estimated scenario by about 7 dB. It is important to emphasize, once again, that it is not quite fair to compare 64-QAM and 64-PSK formats under the proposed nonlinear scenario. Table 7.7 MMPTRBF with \(M_{T}=M_{R}=4\) and \(M_{T}=M_{R}=8\) antennas operating at the same bitrate with 16-QAM modulation. Figure 24 shows the BER vs. \(E_{b}/N_{0}\) results of the MIMO-PTRBF systems operating with \(M_{T}=M_{R}=4\) and \(M_{T}=M_{R}=8\) for 64-QAM modulation. Figure 24 highlights the increase in the diversity gain as a result of the increase in transmitting antennas. When doubling the number of antennas from \(M_{T}=M_{R}=4\) to \(M_{T}=M_{R}=8\), the gain is about 0.87 dB for \(\text{BER}=10^{-3}\).