# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 3 Proposed Approach
### Complex MIMO-PTRBF Neural Network for Massive MIMO Decoding  
In the proposed system, the maximum likelihood decoder is replaced by a neural network, the MIMO-PTRBF, to decode the received symbols, as shown in Figure 4. The MIMO-PTRBF has a supervised learning stage, in which a training sequence is used to fit the hyper-parameters of the neural network. A pseudo-random generator creates this training sequence, which is known both at the transmitter and receiver sides. When the neural network output achieves the desired MSE, it switches from the learning stage to the decoding stage. At this time, the information data are then effectively transmitted over the system, and the BER is computed. These two stages are implemented as in Figure 4, with the input switch of the MIMO STBC Encoder block and the output switch of the Neural Network Decoder block. The switches have two states represented by (a) and (b), which shift between the training and decoding stages [11]. As in the maximum likelihood detector, the input signal to the MIMO-PTRBF algorithm is the set of received vectors \(\mathbf{r}\), as shown in Figure 4. The MIMO-PTRBF architecture, with \(N\) neurons, has three free parameters: the matrix of synaptic weights \(\mathbf{W}\in\mathbb{C}^{M_{s}\times N}\), the matrix of center vectors \(\mathbf{\Gamma}\in\mathbb{C}^{M_{R}P\times N}\), and the vector of variances \(\boldsymbol{\sigma^{2}}\in\mathbb{C}^{N\times 1}\). The MIMO-PTRBF is an extension of the PTRBF for multiple outputs. The key difference between both architectures is the multiple-output layer, which fits each output individually. Figure 5 shows a closer view of the receiver side using the MIMO-PTRBF neural network for decoding [11]. The output vector is thus given by  
\[\mathfrak{s}[u]=\mathbf{W}[u]\boldsymbol{\phi}[u]. \tag{14}\]  
Following the complex-valued radial basis function presented in [41], the \(n\)th neuron output of the MIMO-PTRBF (\(\phi_{n}\)), for the \(p\)th output vector of \(\mathbf{r}\), is [11]  
Figure 4: Complete vision of the proposed MIMO-OFDM model [11]. Figure 5: Closer view of the system with neural network decoding [11]. \[\phi_{n}=\exp\bigg{(}-\frac{||\text{Re}\{\mathbf{r}\}-\text{Re}\{\mathbf{\gamma}_{n}\} ||_{2}^{2}}{\text{Re}\{\sigma_{n}^{2}\}}\bigg{)}+j\exp\bigg{(}-\frac{||\text{Im} \{\mathbf{r}\}-\text{Im}\{\mathbf{\gamma}_{n}\}||_{2}^{2}}{\text{Im}\{\sigma_{n}^{2 }\}}\bigg{)}, \tag{15}\]  
where \(||\cdot||_{2}\) is the operator which returns the Euclidean norm of its argument, and \(\text{Re}\{\cdot\}\) and \(\text{Im}\{\cdot\}\) are the respective real and imaginary parts of their arguments. Additionally, as shown in Figure 6, the output of the neurons can be represented by the vector \(\mathbf{\phi}=[\phi_{1}\quad\phi_{2}\ \cdots\ \phi_{N}]^{T}\in\mathbb{C}^{N\times 1}\). This kernel partitioning into real and imaginary components has an important role in avoiding any phase invariance at the output of the neurons [11; 35; 41]. Thus, by means of the steepest descent algorithm, the update of the MIMO-PTRBF free parameters is given by  
\[w_{m_{s},n}[u+1] =w_{m_{s},n}[u]-\eta_{w}\
abla^{w}[u],\] \[\mathbf{\gamma}_{n}[u+1] =\mathbf{\gamma}_{n}[u]-\eta_{\gamma}\
abla^{\gamma}J[u], \tag{16}\] \[\sigma_{n}^{2}[u+1] =\sigma_{n}^{2}[u]-\eta_{\sigma}\
abla^{\sigma}J[u],\]  
in which \(\eta_{w}\), \(\eta_{\gamma}\), and \(\eta_{\sigma}\) are the adaptive steps of \(w_{m_{s},n}\), \(\mathbf{\gamma}_{n}\), and \(\sigma_{n}^{2}\), respectively. Furthermore, \(\
abla^{w}\), \(\
abla^{\gamma}\), and \(\
abla^{\sigma}\) are the complex gradient operators of \(w_{m_{s},n}\), \(\mathbf{\gamma}_{n}\), and \(\sigma_{n}^{2}\), respectively.