# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 4 Simulation Results  
Using the formerly mentioned OSTBC [54] and QOSTBC [56] coding schemes, several setups are compared with the proposed approach to validate and assess their performance in massive MIMO-OFDM. OSTBC and QOSTBC are simulated with the maximum likelihood (ML) decoding with perfect channel knowledge. This configuration achieves the maximum diversity gain \(G_{d}=M_{T}M_{R}\) at the cost of half of the theoretical bandwidth efficiency, since \(R=1/2\) in this case. Considering a practical QOSTBC application, we also implement the EQOSTBC with the least-squares (ML-LS) channel estimation [11]. Figure 7 illustrates the simulated system. In Figure 7, the binary input data are created by a pseudo-random generator with uniform distribution. The bit-stream is then modulated according to the \(M\)-QAM or \(M\)-PSK modulation scheme used in the simulation. Subsequently, using the coding scheme proposed in Section 3, the modulated symbols are encoded in the STBC block. In the IDFT block, the STBC symbols are frequency-multiplexed for OFDM transmission. At the receiver side, after the transmitted signal passes through the channel, the DFT is applied to demultiplex the STBC symbols. In the decoder, the proposed ANN-based technique presented in Section 3 and the ML algorithm (either with perfect channel knowledge at the receiver or with channel estimation by LS) are employed to assess the system performance. In the sequel, the decoder output symbols are demodulated, and BER is computed. For the sake of comparison, the BER as a function of \(E_{b}/N_{0}\) (energy per bit to noise power spectral density ratio) is used in the simulations. By adjusting the transmitting power for each antenna, the received signals are normalized by \(M_{T}\) transmitting antenna, by the receiver array gain \(M_{R}\), and by the code rate gain \(R\), implying  
\[SNR_{(dB)}=E_{b}/N_{0(dB)}+10\log_{10}\biggl{(}\frac{b}{RM_{T}M_{R}}\biggr{)}, \tag{26}\]  
in which \(b\) is the number of bits per QAM symbol. In Figure 8, aiming to validate the simulator shown in Figure 7, we compare the obtained results with the theoretical performance of OSTBC for 4 th, 8 th, 16 th, and 64 th diversity orders using 4-QAM modulation for a Rayleigh channel with AWGN. For all OSTBC diversity orders, theoretical and simulated results were approximately the same, validating the framework. In addition, Figure 9 presents the reference results of [56] with \(M_{T}=4\) antennas and \(M_{R}=1\) antenna for 16-QAM OSTBC and 4-QAM QOSTBC and the obtained results for the same scenarios. The simulated results are in line with theoretical results, which also corroborates the framework's reliability. Figure 7: Block model of the simulated systems. With the simulation framework validated, we can compare the proposed coding algorithm with results from the literature. Firstly, Figure 10 shows the results of the proposed coding algorithm and theoretical results for 2 nd, 3 rd, 4 th, 5 th, 8 th, and 10 th order diversity. As addressed by [56], quasi-orthogonal transmitting schemes with four antennas achieve at least half of the theoretical diversity (\(D_{o}=\frac{M_{\mathrm{T}}}{2}=2\)) of the orthogonal four antenna scheme (\(D_{o}=M_{\mathrm{T}}=4\)). This can be seen in the solid blue curve with squares of Figure 10, which is located between the theoretical second and third-order curves. Using simulations, we extend the concept introduced by [56] for QOSTBC with \(8\times 1\) and \(16\times 1\) antennas. In order to simulate these scenarios, we employ the proposed coding algorithm presented in Equation (8). As expected, the solid green curve with diamonds and the solid orange curve with circles are between the theoretical 4 th and 5 th order and the 8 th and 10 th order, respectively. Then, utilizing this analysis, we validate the proposed code algorithm and show that it is a suitable approach for generating QOSTBC matrices for at least 16 antennas.