# Capacity Bounds for Dense Massive MIMO in a Line-of-Sight Propagation Environment
## 4 Channel Sum-Capacity in LoS and Favorable Propagation Conditions
The equality in the second line of Equation (5) holds if and only if \(\mathbf{G}^{H}\mathbf{G}\) is a diagonal matrix (i.e., the channel matrix \(\mathbf{G}\) has mutually orthogonal columns) that must satisfy [35],  
\[\mathbf{g}_{i}^{H}\mathbf{g}_{j}= \begin{cases}0,&i,j=1,\ldots,K,\text{ for }i \
eq j\\ \|\mathbf{g}_{k}\|^{2}=M\beta_{k},&k=1,\ldots,K,\text{ for }i=j,\end{cases}  \tag{6}\]  
which is the case when the channel exhibits favorable propagation [36]. The equality \(\|\mathbf{g}_{k}\|^{2}=M\beta_{k}\) is detailed in Appendix A. Given the assumption of channels exhibiting asymptotically favorable propagation, we know from [1] that the spatial signature vectors, denoted by \(e^{i\phi_{k}}\mathbf{h}_{k}\), become asymptotically mutually orthogonal and consequently, it can be shown that  
\[\frac{\mathbf{B}^{H}\mathbf{H}^{H}\mathbf{H}\mathbf{B}}{M}\overset{ u_{k+\infty}}{\approx}\mathbf{I}_{K}, \tag{7}\]  
and therefore,  
\[\frac{\mathbf{G}^{H}\mathbf{G}}{M}=\mathbf{D}^{1/2}\frac{\mathbf{B}^{H} \mathbf{H}^{H}\mathbf{H}\mathbf{B}}{M}\mathbf{D}^{1/2}\overset{M\to\infty}{ \approx}\mathbf{D}, \tag{8}\]  
hence, the results presented in this work hold for ULAs and URAs, as in the asymptotic regime of Equation (8) will always tend to the matrix containing the large-scale fading coefficients. Now using \(\|\mathbf{g}_{k}\|^{2}=M\beta_{k}\) and \(\beta_{k}=\frac{\eta}{d_{k}^{2}}\) in the last line of Equation (5), the sum-capacity upper bound can be rewritten as  
\[C =\sum_{k=1}^{K}\mathbb{E}\left[\log_{2}\left(1+\frac{\rho M\eta}{ d_{k}^{2}}\right)\right] \tag{9}\] \[=K\mathbb{E}\left[\log_{2}\left(1+\frac{\rho M\eta}{d_{k}^{2}} \right)\right],\]  
where the last equality is due to the fact that \(d_{k}\) is an i.i.d. random variable for all \(k\). Next, considering \(z=\frac{1}{d_{k}^{2}}\) as a random variable denoted by \(Z\) with PDF given by  
\[f_{Z}(z)= \begin{cases}\frac{1}{2(R-R_{\min})z\sqrt{z}},&\frac{1}{R^{2}}\leq z \leq\frac{1}{R_{\min}^{2}},\\ 0,&\text{otherwise},\end{cases}  \tag{10}\]then, Equation (9) can be re-expressed as  
\[ \begin{split} C&=K\mathbb{E}\left[\log_{2}\left(1+\frac{ \rho M\eta}{d_{k}^{2}}\right)\right]\\ &=K\mathbb{E}\left[\log_{2}\left(1+\rho M\eta z\right)\right]\\ &=\frac{K}{2(R-R_{min})}\int_{1/R^{2}}^{1/R_{min}^{2}}\log_{2}(1+ \rho M\eta z)\frac{1}{z\sqrt{z}}dz,\end{split}  \tag{11}\]  
where the proof of Equation (10) is given in Appendix B. Next, solving Equation (11) with an integral solver [37], we find an exact closed-form expression given in Equation (12) for the capacity when the channel offers favorable propagation. \[C=\tfrac{K}{(R-R_{min})}\frac{2\sqrt{\rho M\eta}\left(\tan^{-1}\left(\sqrt{ \frac{R_{min}^{2}}{R_{min}^{2}}}\right)-\tan^{-1}\left(\sqrt{\frac{R_{min}^{2} }{R^{2}}}\right)\right)+R\log\left(\frac{\rho M\eta}{R^{2}}+1\right)-R_{min} \log\left(\frac{\rho M\eta}{R_{min}^{2}}+1\right)}{\log(2)}. \tag{12}\]  
**Remark 1**. : _After analyzing Equation (12), we see that if we make the transmit power of each device equal to \(P/M^{\alpha}\), where \(\alpha>1\), then the sum-capacity, \(C\), will go to zero as \(M\rightarrow\infty\). When \(\alpha<1\) the sum-capacity grows without bound as \(M\rightarrow\infty\). This means that \(1/M\) (i.e., \(\alpha=1\)) is the fastest rate at which we can decrease the transmit power of each device and still have a fixed capacity as \(M\rightarrow\infty\)._  
Remark 1 clearly shows that as \(M\) grows without bound, the transmit power of each device can be reduced proportionally to \(1/M\) and that the spectral efficiency increases by a factor of \(K\), meaning that the BS can simultaneously serve \(K\) devices over the same time-frequency resources. This reduction of the transmit power per device is very important to power-constrained devices such as IoT devices.