# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## Appendix A  
**Definition A1**. : _A non-monotone and differentiable transformation \(\mathbf{g}:\mathbb{R}^{N}\mapsto\mathbb{R}^{N}\), composed of M-real solutions, yields a multivariate change of probability density function:_  
\[f_{\mathbf{Y}}(\mathbf{y}) = \sum_{m=1}^{M}f_{\mathbf{X}}(\mathbf{x})\Big{|}J\Big{(}\mathbf{ g}_{m}^{-1}(\mathbf{y})\Big{)}\Big{|},\]  
_in which \(\mathbf{x},\mathbf{y}\in\mathbb{R}^{N}\) are random vectors, \(J(\cdot)\) is the determinant of the Jacobi operator, and \(\mathbf{g}_{m}^{-1}\) is the inverse transformation of \(\mathbf{g}\), regarding the \(m\)th real solution (see [65], p. 322 and [66], pp. 69-70)._  
**Theorem A1**. : _If \(\mathbf{x}\in\mathbb{R}^{N}\) is an input random vector of a kernel function \(g(\mathbf{x})=y\triangleq\exp\left\{-\frac{\|\mathbf{x}-\mathbf{\gamma}\|_{2}^{2 }}{z}\right\}\), with constant \(\mathbf{\gamma}\in\mathbb{R}^{N}\) and \(z\in\mathbb{R}\); then, \(g\) performs a trans-dimensional transformation of probability density function from \(f_{\mathbf{X}}(\mathbf{x})\) to \(f_{Y}(y)\)._Proof. : Let the Gaussian kernel function \(g(\mathbf{x},\boldsymbol{\gamma},z)\triangleq\exp\left\{-\frac{||\mathbf{x}- \boldsymbol{\gamma}||_{2}^{2}}{z}\right\}\), with constant center vector \(\boldsymbol{\gamma}\) and variance \(z\), be denoted by \(y=g(\mathbf{x})\), since it is only dependent on the random vector \(\mathbf{x}\in\mathbb{R}^{N}\). Furthermore, let \(\mathbf{y}\triangleq\left[g(\mathbf{x})\quad y_{2}\,\cdots y_{N}\right]^{ \mathsf{T}}\) be the auxiliary expanded vector of mapping. With Definition A1, we have the following transformation of the probability density function (PDF):  
\[f_{\mathbf{Y}}(\mathbf{y})=\sum_{m=1}^{M}f_{\mathbf{X}}(\mathbf{x})\big{|}I \Big{(}\mathbf{g}_{m}^{-1}(\mathbf{y})\Big{)}\big{|}. \tag{10}\]  
However, we need to marginalize the extra dimensions of \(\mathbf{y}\) in (10) to obtain the PDF of the desired random variable \(f_{Y}(y)\):  
\[f_{Y}(y)=\sum_{m=1}^{M}\int f_{\mathbf{X}}(\mathbf{x})\big{|}I\Big{(}\mathbf{g }_{m}^{-1}(\mathbf{y})\Big{)}\big{|}dy_{2}\cdots dy_{N}. \tag{11}\]  
Then, finding some non-monotone function \(h(y)=h(g(\mathbf{x}))=f_{\mathbf{X}}(\mathbf{x})\), in which \(h(y)\) is not a function of the expanded auxiliary terms \(y_{2}\), \(y_{3}\), \(\cdots\), \(y_{N}\), we can rewrite (11) as  
\[f_{Y}(y)=\sum_{m=1}^{M}h_{m}(y)\int\big{|}I\Big{(}\mathbf{g}_{m}^{-1}(\mathbf{ y})\Big{)}\big{|}dy_{2}\cdots dy_{N}, \tag{12}\]  
where the right-hand term under the integral is called integral Jacobian, and it is responsible for the volume correction [67] of the trans-dimensional transformation. In addition, as the Gaussian kernel is a non-monotone function with two real solutions (due to the Euclidean norm), (12) results in  
\[f_{Y}(y)=h_{1}(y)\int\big{|}J\Big{(}\mathbf{g}_{1}^{-1}(\mathbf{y})\Big{)} \big{|}dy_{2}\cdots dy_{N}+h_{2}(y)\int\big{|}J\Big{(}\mathbf{g}_{2}^{-1}( \mathbf{y})\Big{)}\big{|}dy_{2}\cdots dy_{N}, \tag{13}\]  
which is the trans-dimensional transformation of PDF performed by a Gaussian kernel function. For any complex-valued vectors of same size \(\mathbf{x},\mathbf{y}\in\mathbb{C}^{N}\) and a scalar \(z\in\mathbb{C}\), a kernel function \(f(\mathbf{x},\mathbf{y},z)\triangleq\exp\left\{-\frac{||\Re(\mathbf{x})-\Re( \mathbf{y})||_{2}^{2}}{\Re(z)}\right\}+j\exp\left\{-\frac{||\Im(\mathbf{x})- \Im(\mathbf{y})||_{2}^{2}}{\Im(z)}\right\}\) have real and imaginary boundaries between \(0\) and \(1\), if \(\Re(z)>0\) and \(\Im(z)>0\). Proof. : The proof is straightforward using Theorem A1 of [35], where the real-valued analyses must be independently performed for the real and imaginary components of \(\mathbf{x}\), \(\mathbf{y}\), and z.