# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 5 Computational Complexities  
Table 2 presents the computational complexities of the OSTBC and EOSTBC with ML decoding, both with the additional complexity of channel estimation, and the proposed scheme with MIMO-PTRBF for training and decoding operation modes. \(M_{T}\) and \(M_{R}\) are the number of transmitting and receiving antennas, \(M_{s}\) is the number of transmitted symbols per MIMO block, \(P\) is the number of time samples per block of coded symbols, \(A\) is the constellation order (e.g., \(A=4\) in the case of 4-QAM), and \(N\) is the number of neurons used in the PTRBF neural network. Since the \(\exp(\cdot)\) function can be easily implemented in hardware by lookup tables, multiplication is the most costly operation. One may note that, for \(M_{T}\leq 8\), the complexity of the proposed algorithm is similar to the complexity of the OSTBC, for which no code exists for \(M_{T}>8\)[54]. The case of QOSTBC is similar, for which no simplified ML metric exists for \(M_{T}\
eq 4\)[11; 56]. Table 3 presents the computational complexities for the OSTBC, QOSTBC, and the proposed system, for \(M_{T}=M_{R}=4\). In Table 3, \(N=100\) neurons are used in the neural network, and maximum likelihood decoding is simulated with \(R=1\). Note that generic maximum likelihood decoding refers to the minimization of Equation (5) for a rate one (\(R=1\)) coding scheme (e.g., it could decode the QOSTBC for the case \(M_{T}=4\) at a higher computational cost), and it will be assumed as an upper bound for the computational \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decoder** & **Multiplications** & **Additions** & \(\mathbf{exp}(\cdot)\) \\ \hline OSTBC \({}^{\dagger}\) & \((2 PM_{R}+4)M_{s}A+ch\) & \((3 PM_{R}+3)M_{s}A+ch\) & _cl_ \\ OOSTBC \({}^{\dagger}\) & \(24M_{R}(M_{s}/2)A^{2}+ch\) & \(16(M_{R}-1)(M_{s}/2)A^{2}+ch\) & _cl_ \\ ML\({}^{\ast}\) with \(R=1\) & \(M_{R}M_{T}P(A^{M_{s}})+ch\) & \(M_{R}M_{T}(P-1)(A^{M_{s}})+ch\) & _cl_ \\ MIMO-PTRBF (train) & \(12 NM_{S}+12N+8 NM_{R}P+2M_{s}\) & \(6 NM_{R}P+12 NM_{s}-2N\) & \(2N\) \\ MIMO-PTRBF (decoding) & \(2 NM_{R}P+2N+4 NM_{s}\) & \(4 NM_{R}P+4 NM_{s}-2N-2M_{s}\) & \(2N\) \\ \hline \hline \end{tabular} \({}^{\dagger}\)_ch_ refers to the additional complexity of channel estimation.  
\end{table} 
Table 2: Computational complexities. Figure 24: MMPTRBF with \(M_{T}=M_{R}=4\) and 8 antennas operating at the same bitrate with 64-QAM modulation. complexities of the other quasi-orthogonal systems. Furthermore, appropriate modulation schemes are used to provide the desired transmission rate for the evaluated systems; i.e., 4-QAM for rate one code (\(R=1\)) and 16-QAM for half-rate code (\(R=1/2\)) [11]. Table 4 displays the computational complexities for \(M_{T}=M_{R}=8\), when the PTRBF is equipped with \(N=150\) neurons. QOSTBC is defined as not applicable since no simplified ML decoding metric has been presented in the literature. Thus, we need to rely on the usual ML metric to perform decoding, which implies the limitation of using QOSTBC combined with ML for a higher number of antennas in practical approaches. Table 5 displays the computational complexities for \(M_{T}=M_{R}=32\), when the PTRBF is equipped with \(N=600\) neurons. As the OSTBC coding matrix is limited to eight antennas (see [54]), it is not applicable for \(M_{T}=M_{R}=32\). As already mentioned, since there is no simplified ML metric to perform ML decoding with QOSTBC, it results in an explosion of computational complexity for \(M_{T}=M_{R}=32\). On the other hand, the proposed approach can expand the number of antennas, maintaining a reasonable compromise between computational complexity and BER, as discussed in Section 4. The decoding computational complexities, shown in Figure 25, are addressed in terms of real-valued multiplications per MIMO symbol, as a function of \(M_{T}=M_{R}\) antennas.