# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 4 Simulation Results
The simulation results in Figure 15 indicate that the QOSTBC system outperforms the proposed work when perfect channel knowledge is available at the receiver, which is impractical. Although the EQOSTBC system is a feasible and practical version of the QOSTBC, due to the channel estimation block at the receiver, simulations using the least-squares channel estimation show that the EQOSTBC performance is degraded by more than 2.5 dB when compared with the QOSTBC [11]. Furthermore, even with a perfect channel estimator, it is computationally expensive to decode QOSTBC codes with maximum likelihood for more than four antennas, as addressed by [56]. Figure 11: Evolution of MSE values averaged over 10 realization sequences of the proposed MMPTRBF network decoder, using 4-QAM and \(E_{b}/N_{0}=12\) dB for (**a**) \(M_{T}=M_{R}=4\) antennas, (**b**) \(M_{T}=M_{R}=8\) antennas, (**c**) \(M_{T}=M_{R}=16\) antennas, (**d**) \(M_{T}=M_{R}=32\) antennas. Figure 12: Scatter plots for the 4-QAM symbols at the output of the MMPTRBF during the training period, for \(M_{T}=M_{R}=4\) antennas and \(E_{b}/N_{0}=12\) dB. (**a**) 1 epoch, (**b**) 5 epochs, (**c**) 10 epochs, (**d**) 15 epochs, (**e**) 20 epochs, (**f**) 25 epochs, (**g**) 30 epochs, and (**h**) 35 epochs. Figure 13: Scatter plots for the 16-QAM symbols at the output of the MMPTRBF during the training period, for \(M_{T}=M_{R}=4\) antennas and \(E_{b}/N_{0}=20\) dB. (**a**) 1 epoch, (**b**) 20 epochs, (**c**) 40 epochs, (**d**) 60 epochs, (**e**) 80 epochs, (**f**) 100 epochs, (**g**) 120 epochs, and (**h**) 140 epochs. Figure 16 and 17 show the BER\(\times E_{b}/N_{0}\) results of the QOSTBC, EQOSTBC, and MIMO-PTRBF systems operating with \(M_{T}=4\) and \(M_{T}=8\) with \(M_{R}=1\) antennas and 4-QAM modulation. Figures 16 and 17 highlight the diversity gain of the proposed system when compared with the EQOSTBC. Although the mathematical derivation of the proposed system diversity gain has not been obtained yet, simulations indicate a significant diversity gain [11]. Contrasting Figure 15 with Figure 16, one can see the increase in the diversity gain as the number of transmitting antennas increases. Figure 14: Scatter plots for the 64-QAM symbols at the output of the MMPTRBF in the training phase, for \(M_{T}=M_{R}=4\) antennas and \(E_{b}/N_{0}=26\) dB. (**a**) 1 epoch, (**b**) 1000 epochs, (**c**) 2000 epochs, (**d**) 3000 epochs, (**e**) 4000 epochs, (**f**) 5000 epochs, (**g**) 6000 epochs, and (**h**) 7000 epochs. Figure 15: Systems with \(M_{T}=M_{R}=4\) antennas for 4-QAM modulation. Figure 18 shows the BER vs. \(E_{b}/N_{0}\) results of the MMPTRBF systems operating with \(M_{T}=4\), 6, and 8 antennas for a 4-QAM modulation. Figure 18 also highlights the increase in the diversity gain as the number of transmitting antennas increases. For a \(\text{BER}=10^{-2}\), there is a gain of approximately 2 dB when doubling \(M_{T}\). To further investigate the effects of a larger number of transmitting and receiving antennas on the performances of BER, in Figure 19, we present the simulation results for a higher-order system with \(M_{T}=M_{R}=8\). It can be seen that the performance of the quasi-orthogonal code with channel estimation is worse for \(M_{T}=8\) than for \(M_{T}=4\) antennas, as shown in Table 7.0 It is shown in [53] that the performance of the linear estimator decreases proportionally with the number of transmitting antennas, which adds a constraint to the number of transmitting antennas for linearly decoded systems. Table 7.1 Systems with \(M_{T}=4\) and \(M_{R}=1\) antennas for 4-QAM modulation. Table 7.2 Systems with \(M_{T}=8\) and \(M_{R}=1\) antennas for 4-QAM modulation. Figure 20 presents the simulation results for \(M_{T}=M_{R}=4\) (\(N=100\)), \(M_{T}=M_{R}=8\) (\(N=150\)), \(M_{T}=M_{R}=16\) (\(N=200\)), and \(M_{T}=M_{R}=32\) (\(N=600\)) to examine the extent of the proposed work for massive MIMO operations. These results show the potential of the proposed work to efficiently operate with a massive number of transmitting and receiving antennas.