# Capacity Bounds for Dense Massive MIMO in a Line-of-Sight Propagation Environment
## Appendix D Approximated Distribution of the Total Power Gain of the Channel Matrix  
**Lemma A1**. : _Let \(\beta_{k}=\eta/d_{k}^{2}\), where \(d_{k},\forall k\) are i.i.d. random variables following the uniform distribution with PDF given by \(1/(R-R_{min})\), therefore, as the number of served devices increases, the PDF of \(Z=\sum_{k=1}^{K}M\beta_{k}\) can be approximated by a Gamma distribution with parameters \(\kappa=\frac{3 KRR_{min}}{(R-R_{min})^{2}}\) and \(\theta=\frac{M\eta(R-R_{min})^{2}}{3R^{2}R_{min}^{2}}\), i.e., \(\Gamma(\kappa,\theta)\)._  
Proof. : This is empirically proven by comparing the normalized histogram of \(Z\) against the theoretical PDF of a Gamma random variable with the parameters defined earlier. The parameters \(\kappa\) and \(\theta\) are found by using the mean and variance statistics of \(Z\), where \(\kappa=\frac{\mu_{Z}^{2}}{\sigma_{Z}^{2}}\) and \(\theta=\frac{\sigma_{Z}^{2}}{\mu_{Z}}\). These relations are derived based on the definitions of mean and variance of a Gamma random variable. The mean of \(Z\) is given by \[\mu_{Z} =M\eta\sum_{k=1}^{K}\mathbb{E}\left[\frac{1}{d_{k}^{2}}\right] \tag{11}\] \[=MK\eta\mathbb{E}\left[\frac{1}{d_{k}^{2}}\right]\] \[=\frac{MK\eta}{RR_{min}},\]  
where we used the assumption that \(d_{k},\forall k\) are i.i.d. random variables and the results in Appendix C. Next, the variance of \(Z\) is given by  
\[\sigma_{Z}^{2}=\mathbb{E}[Z^{2}]-\mu_{Z}^{2}, \tag{12}\]  
where  
\[\mathbb{E}[Z^{2}] =M^{2}K\{\mathbb{E}[\beta^{2}]+(K-1)\mathbb{E}[\beta_{k}]^{2}\} \tag{13}\] \[=M^{2}\eta^{2}K\bigg{\{}\mathbb{E}\left[\frac{1}{d_{k}^{4}} \right]+(K-1)\mathbb{E}\left[\frac{1}{d_{k}^{2}}\right]^{2}\bigg{\}}\] \[=M^{2}\eta^{2}K\left[\frac{R^{2}+RR_{min}+R_{min}^{2}}{3R^{3}R_{ min}^{3}}-\frac{(K-1)}{R^{2}R_{min}^{2}}\right],\]  
where \(\mathbb{E}\left[\frac{1}{d_{k}^{4}}\right]\) is calculated in Appendix C. Therefore, plugging Equation (13) into Equation (12) we have  
\[\sigma_{Z}^{2}=\frac{K\eta^{2}M^{2}(R-R_{min})^{2}}{3R^{3}R_{min}^{3}}. \tag{14}\]  
The proof is concluded by replacing Equations (11) and (14) into the definitions of \(\kappa\) and \(\theta\). Appendix E Approximated Distribution of the Instantaneous Sum-Capacity in the High-SNR Regime and Favorable Propagation  
In this appendix we demonstrate how the parameters \(\kappa\) and \(\theta\) are derived. These parameters, which are used to approximate the instantaneous sum-capacity PDF in the high-SNR regime and favorable propagation, are found by using the mean and variance statistics of the random variable given by Equation (30), where \(\kappa=\frac{\mu_{\text{clust.}}^{2}}{\sigma_{\text{clust. }}^{2}}\) and \(\theta=\frac{\sigma_{\text{clust.}}^{2}}{\mu_{\text{clust.}}}\). These relations are derived based on the definitions of mean and variance of a Gamma random variable. The mean of Equation (30) is given by \[ \begin{split}\mu_{\text{C}_{\text{inst.}}}&=\mathbb{E}[ \text{C}_{\text{inst.}}]\\ &=\mathbb{E}\left[\sum_{k=1}^{K}\log_{2}\left(\frac{\rho M\eta}{d_ {k}^{2}}\right)\right]\\ &=\sum_{k=1}^{K}\mathbb{E}\left[\log_{2}\left(\frac{\rho M\eta}{d_ {k}^{2}}\right)\right]\\ &=K\mathbb{E}\left[\log_{2}\left(\frac{\rho M\eta}{d_{k}^{2}} \right)\right]\\ &=K\frac{\log(\rho M\eta)+\frac{2(R-R_{min}-R\log(R)+R_{min} \log(R_{min}))}{R-R_{min}}}{\log(2)},\end{split}  \tag{15}\]  
where we used the assumption that \(d_{k},\forall k\) are i.i.d. random variables and the result in Appendix C. Next, the variance of Equation (30) is given by  
\[\sigma_{\text{C}_{\text{inst.}}}^{2}=\mathbb{E}[\text{C}_{\text{inst.}}^{2}]- \mathbb{E}[\text{C}_{\text{inst.}}]^{2}, \tag{16}\]  
where \(\mathbb{E}[\text{C}_{\text{inst. }}^{2}]\) is given by Equation (17) with \(\mathbb{E}\left[\log_{2}^{2}\left(\frac{\rho M\eta}{d_{k}^{2}}\right)\right]\) being defined by Equation (18) with \(a=\log(\rho M\eta)\), \(b=R-R_{min}\), \(c=\log(R)\), \(d=\log(R_{min})\). Therefore, plugging Equations (15) and (17) into Equation (16) we have Equation (19).