# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 3 Proposed Approach
### Complex MIMO-PTRBF Neural Network for Massive MIMO Decoding
Thus, with \(\mathbf{r}\) and \(\mathbf{s}\), the MIMO-PTRBF algorithm can be used to estimate the output vector \(\mathbf{\hat{s}}\) at the \(u\)th training epoch by the minimization of the following cost function:  
\[J[u]=\frac{1}{2}||\mathbf{s}[u]-\mathbf{\hat{s}}[u]||_{2}^{2}, \tag{17}\]  
where \(\mathbf{s}\) and \(\mathbf{\hat{s}}\) are the training sequence and the output vector, respectively. Applying the complex gradient operators (\(\
abla^{w}\), \(\
abla^{\gamma}\), and \(\
abla^{\sigma}\)) to (17) yields  
Figure 4.0 Figure 4.1 Figure 4.2  
in which \(e_{m_{s}}[u]=s_{m_{s}}[u]-\hat{s}_{m_{s}}[u]\) is the instantaneous error for the output \(\hat{s}_{m_{s}}\) at the \(u\)th training epoch. Then, substituting Equation (18) in (16) yields  
Figure 4.3 MIMO phase transmittance radial basis function neural network architecture [11]. Figure 4.4 Figure 4.5 Figure 4.6  
in which \([\cdot]^{*}\) denotes the complex conjugate operator and \(\xi_{n}[u]\) is the \(n\)th synaptic transmittance, given by  
Figure 4.7  
Furthermore, \(\mathbf{\alpha}_{n}[u]\in\mathbb{C}^{\mathbb{R}}\) is the \(m\)th vector of the matrix of weighted centers (\(\mathbf{A}[u]\in\mathbb{C}^{\mathbb{N}\times\mathbb{R}}\)):  
Figure 4.8  
In a similar way, \(\beta_{n}[u]\in\mathbb{C}\) is the \(n\)th element of the vector of weighted kernel (\(\mathbf{\beta}[u]\in\mathbb{C}^{\mathbb{N}}\)):  
Figure 4.9  
Generalizing Equation (19) to matrix structures results in  
\[\mathfrak{s}[u]=\mathbf{W}[u]\boldsymbol{\phi}[u]. \tag{14}\]0 \[\mathfrak{s}[u]=\mathbf{W}[u]\boldsymbol{\phi}[u]. \tag{14}\]1 \[\mathfrak{s}[u]=\mathbf{W}[u]\boldsymbol{\phi}[u]. \tag{14}\]2  
where \(\mathbf{A}[u]\) and \(\mathbf{x}[u]\) are auxiliary variables used to reduce the computational complexity, \(\mathbf{A}[u]\) and \(\mathbf{x}[u]\) are given by  
\[\mathfrak{s}[u]=\mathbf{W}[u]\boldsymbol{\phi}[u]. \tag{14}\]3 \[\mathfrak{s}[u]=\mathbf{W}[u]\boldsymbol{\phi}[u]. \tag{14}\]4  
in which \(\mathbf{\Xi}[u]\) is the diagonal matrix of synaptic transmittance:  
\[\mathbf{\Xi}[u]= \begin{bmatrix}\xi_{1}[u]&0&\cdots&0\\ 0&\xi_{2}[u]&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\xi_{N}[u]\end{bmatrix} \in\mathbb{C}^{\mathbb{N}\times\mathbb{N}}. \tag{25}\]  
Each training update is given by Equation (23); however, for \(u=0\), the MIMO-PTRBF free parameters are initialized following some criterion defined by the user (e.g., based on the probability distribution of the input data). Although (23) minimizes the error between the output vector \(\mathbf{\hat{s}}\) and the reference vector \(\mathbf{s}\), as the neurons are dependent on exponential functions, a risk of instability is assumed if the exponential argument is positive. In order to circumvent this issue, based on Theorem A1 of [35], the real and imaginary parts of each scalar component of the vector of variances are lower-bounded by the limit \(\mu>0\), which, consequently, bounds the real and imaginary parts of the neurons output from \(0\) to \(1\)[11]. In addition, taking into account Theorem A2 (see Appendix A), the adaptive step of the matrix of synaptic weights is limited by \(\eta_{w}<1/N\) for all simulations, to guarantee convergence in the mean. In addition, in the Appendix, Corollaries A1 and A2 are of utmost importance to prove Theorem A1. In addition, Definition A1 is used to prove Corollary A2.