# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 4 Simulation Results
The PAPR issue is usually more critical in the uplink since the efficiency of the power amplifier is critical due to the limited battery power in a mobile terminal. For this purpose, from now on, the results assume mild amplifier nonlinearities, represented by a first-grade power amplifier or an appropriate back-off operating point [11]. Based on [41], the nonlinearity vector \(\boldsymbol{\rho}=[\rho_{1}\quad\rho_{2}\quad\rho_{3}]^{T}=[0.9\quad 0.1 \quad 0.05]^{T}\) implies 90%, 10%, and 5% first, second, and third-order coefficients, respectively. With the model properly validated and the specified 5G channel model, we are now able to analyze the proposed complex-valued ANN-based decoder for MIMO-OFDM systems. First, we present the MSE convergence curves during the learning process. The MSE curves are averaged over 10 subsequent simulation traces, and a 4-QAM modulation with \(E_{b}/N_{0}=12\) dB is employed. Figure 11a-d show the MSE evolution for \(M_{T}=M_{R}=4\), \(M_{T}=M_{R}=8\), \(M_{T}=M_{R}=16\), and \(M_{T}=M_{R}=32\) antennas, respectively. The red bottom and top curves in Figure 11 refer to the MSE standard deviations over the 10 subsequent simulation traces, and the green curves refer to the mean values. Although the steady-state MSEs decrease slightly as the number of antennas increases, one may notice that the decays of the standard deviations are more conspicuous as \(M_{T}=M_{R}\) increases. This is due to the MIMO characteristics that mitigate the channel effects by sending several samples of the same signal to the receiver. Thus, sudden channel variations are smoothed, suggesting that the PTRBF learning process presents a robust and cohesive behavior. The 4-QAM scatter plots presented in Figure 12 show the convergence of the proposed neural network decoder for the first 35 training epochs. For this sequence of scatter plots, each training epoch corresponds to one OFDM symbol; i.e., 256 4-QAM symbols. As shown in Figure 12, the proposed algorithm has a fast convergence rate since only 10 training epochs are sufficient to separate the 4-QAM constellation symbols efficiently [11]. The 16-QAM scatter plots presented in Figure 13 show the convergence of the proposed neural network decoder for the first 140 training epochs, spaced in intervals of 20 OFDM symbols. In this case, the number of training epochs necessary for algorithm convergence is greater than for the 4-QAM case, given the intrinsic complexity of the higher-order constellation. Nevertheless, with only 20 training epochs, it is already possible to visually identify the 16-QAM constellation symbols and, with 80 training epochs, to see the correctly grouped symbols in the scatter plot [11]. The 64-QAM scatter plots presented in Figure 14 show the convergence of the proposed neural network decoder for the first 7000 training epochs, spaced in intervals of 1000 OFDM symbols. In this case, the number of training epochs necessary for algorithm convergence is greater than for the former 16/4-QAM cases, in view of the intrinsic complexity of the much higher-order 64-QAM constellation. Nevertheless, with 3000 training epochs, it is already possible to visually identify the 64-QAM constellation symbols and, with 5000 training epochs, to see the correctly grouped symbols in the scatter plot [11]. Although an abrupt increase in training epochs occurs, when compared with 4-QAM in Figure 12, it represents a time interval of only 80 ms (5000 OFDM symbols with 240 kHz sub-carrier spacing). After analyzing the MSE curves and constellations of the MMPTRBF, we can further investigate the BER vs. \(E_{b}/N_{0}\) of the proposed approach. Figure 15 shows the BER vs. \(E_{b}/N_{0}\) results of the QOSTBC, EQOSTBC, and MIMO-PTRBF systems operating with \(M_{T}=M_{R}=4\) antennas and 4-QAM modulation. The 4-QAM AWGN curve defines a lower bound for all MIMO systems that use 4-QAM modulation.