# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## Appendix A
If \(\mathbf{x}\in\mathbb{C}^{N}\) is a complex-valued input random vector of a complex-valued kernel function \(g(\mathbf{x})=y\triangleq\exp\left\{-\frac{||\Re(\mathbf{x})-\Re(\mathbf{y})|| _{2}^{2}}{\Re(z)}\right\}+j\exp\left\{-\frac{||\Im(\mathbf{x})-\Im(\mathbf{y} )||_{2}^{2}}{\Im(z)}\right\}\), with constant \(\boldsymbol{\gamma}\in\mathbb{C}^{N}\) and \(z\in\mathbb{C}\), then \(g\) performs a complex-valued trans-dimensional transformation of probability density function from \(f_{\Re(\mathbf{x})}(\Re(\mathbf{x}))+f_{\Im(\mathbf{X})}(\Im(\mathbf{X}))\) to \(f_{\Re(\mathbf{Y})}(\Re(\mathbf{y}))+f_{\Im(\mathbf{Y})}(\Im(\mathbf{y}))\), with independent real and imaginary components. Proof. : The proof is straightforward using Theorem A1, where the real-valued analyses must be independently performed for the real and imaginary components of \(\mathbf{x}\) and \(y\). If \(\mathbf{x}\in\mathbb{C}^{N}\) is a stationary complex-valued input random vector of a PT-RBF, then the matrix of synaptic weights \(\mathbf{W}[k]\in\mathbb{C}^{M\times N}\) converges in the mean to the optimum matrix of synaptic weights \(\mathbf{W}_{o}\) when \(k\rightarrow\infty\), if the matrix of center vectors and the vector of variances are constants. Proof. : Let the PT-RBF output \(\mathbf{y}[k]=\mathbf{W}[k]\boldsymbol{\phi}[k]\in\mathbb{C}^{M}\), where \(\boldsymbol{\phi}[k]\in\mathbb{C}^{N}\) is the vector of the neuron output. Furthermore, let the update function of the matrix of synaptic weights \(\mathbf{W}[k+1]=\mathbf{W}[k]+\eta_{w}\mathbf{e}[k]\boldsymbol{\phi}^{\mathrm{ H}}[k]\in\mathbb{C}^{M\times N}\), where \(\eta_{w}\) is the adaptive step and \(\mathbf{e}[k]\in\mathbb{C}^{M}\) is the vector of errors. We can assume without loss in generality that the desired response is  
\[\mathbf{d}[k]=\mathbf{W}_{o}\boldsymbol{\phi}[k]+\mathbf{q}[k]\in\mathbb{C}^{ M}, \tag{100}\]  
where \(\mathbf{q}[k]\in\mathbb{C}^{M}\) is a complex-valued white Gaussian noise vector, with zero mean and variance \(\sigma_{q}^{2}\), which is uncorrelated with \(\boldsymbol{\phi}[k]\). Then, substituting (100) into the error equation \(\mathbf{e}[k]=\mathbf{d}[k]-\mathbf{y}[k]\), we have  
\[\mathbf{e}[k]=\mathbf{W}_{o}\boldsymbol{\phi}[k]+\mathbf{q}[k]-\mathbf{W}[k] \boldsymbol{\phi}[k]. \tag{101}\]  
Thus, replacing (101) into the update function of the matrix of synaptic weights:  
\[\mathbf{W}[k+1]=\mathbf{W}[k]+\eta_{w}\mathbf{W}_{o}\boldsymbol{\phi}[k] \boldsymbol{\phi}^{\mathrm{H}}[k]+\eta_{w}\mathbf{q}[k]\boldsymbol{\phi}^{ \mathrm{H}}[k]-\eta_{w}\mathbf{W}[k]\boldsymbol{\phi}[k]\boldsymbol{\phi}^{ \mathrm{H}}[k]. \tag{102}\]  
Subtracting \(\mathbf{W}_{o}\) from both sides of (102), the synaptic weights error matrix \(\mathbf{V}[k]=\mathbf{W}[k]-\mathbf{W}_{o}\) can be given by  
\[\mathbf{V}[k+1]=\mathbf{V}[k]-\eta_{w}\mathbf{V}[k]\boldsymbol{\phi}[k] \boldsymbol{\phi}^{\mathrm{H}}[k]+\eta_{w}\mathbf{q}[k]\boldsymbol{\phi}^{ \mathrm{H}}[k]. \tag{103}\]  
Applying the expectation operator to both sides of (103):  
\[\mathrm{E}(\mathbf{V}[k+1])=\mathrm{E}\Big{(}\mathbf{V}[k]-\eta_{w}\mathbf{V} [k]\boldsymbol{\phi}[k]\boldsymbol{\phi}^{\mathrm{H}}[k]+\eta_{w}\mathbf{q}[k] \boldsymbol{\phi}^{\mathrm{H}}[k]\Big{)}, \tag{104}\]  
and employing the independence assumptions of \(\mathbf{W}[k]\perp\!\!\!\perp\!\!\!\perp\boldsymbol{\phi}[k]\) and \(\mathbf{q}[k]\perp\!\!\!\perp\boldsymbol{\phi}[k]\):  
\[f_{\mathbf{Y}}(\mathbf{y})=\sum_{m=1}^{M}f_{\mathbf{X}}(\mathbf{x})\big{|}I \Big{(}\mathbf{g}_{m}^{-1}(\mathbf{y})\Big{)}\big{|}. \tag{10}\]0  
which results in:  
\[f_{\mathbf{Y}}(\mathbf{y})=\sum_{m=1}^{M}f_{\mathbf{X}}(\mathbf{x})\big{|}I \Big{(}\mathbf{g}_{m}^{-1}(\mathbf{y})\Big{)}\big{|}. \tag{10}\]1  
where \(\mathbf{I}\) is the identity matrix and \(\mathbf{R}_{\boldsymbol{\phi}\boldsymbol{\phi}}\) is the correlation matrix of the neuron outputs. As \(\mathbf{R}_{\boldsymbol{\phi}\boldsymbol{\phi}}\) is Hermitian and positive semidefinite (see [68], pp.