# Capacity Bounds for Dense Massive MIMO in a Line-of-Sight Propagation Environment
## 4 Channel Sum-Capacity in LoS and Favorable Propagation Conditions  
The distance between the device's antennas and the BS's antennas is not static as the devices are always moving within the cell (e.g., drones, cars, etc.). Therefore, the distance has to be treated as a random variable, resulting in a different channel realization for every time instant. As it is known [1], the instantaneous sum-capacity is not a meaningful performance metric under such random conditions. Here, we are interested in the capacity performance averaged over all different positions a device might be within the cell. Therefore, in order to assess the performance in this scenario, we need to employ the notion of ergodic capacity, which results in the following uplink sum link capacity [32]  
\[ \begin{split} C&=\mathbb{E}\left[\log_{2}|\mathbf{ I}_{M}+\rho\mathbf{G}\mathbf{G}^{H}|\right]\\ &=\mathbb{E}\left[\log_{2}|\mathbf{I}_{K}+\rho\mathbf{G}^{H} \mathbf{G}|\right],\end{split}  \tag{3}\]  
where \(\rho\) is the average SNR, also known as the average transmitted power of each device, and the expectation is taken over the joint distribution of all possible positions of the devices. As the additive white noise is assumed to have mean and variance equal to 0 and 1 respectively, therefore, \(\rho\) has, consequently, the interpretation of normalized transmit SNR and is therefore dimensionless [33]. It is important to highlight that the expectation in Equation (3) is taken in relation with devices' channels, more specifically in relation to their positions within the cell (i.e., distances and angles of arrival), which are considered as random variables. The second line of Equation (3) is found by applying the [PERSON]'s determinant theorem. Furthermore, we assume that the base station has perfect knowledge of the channel matrix \(\mathbf{G}\). The rationale behind the assumption of perfect knowledge of the channel matrix is that the results obtained with this assumption are readily comprehended and they bound the performance of massive MIMO systems. Finding the exact ergodic capacity given by Equation (3) is quite a complex task that involves finding the distribution of the eigenvalues of \(\mathbf{G}^{H}\mathbf{G}\)[34]. In this work, as we will show next, we are concerned with finding the sum-capacity considering a massive MIMO scenario and asymptotically favorable propagation [1]. In asymptotically favorable propagation scenarios, the channel vectors of different users become mutually orthogonal as the number of antennas, \(M\), increases, i.e.,  
\[\frac{\mathbf{g}_{i}^{H}\mathbf{g}_{j}}{M}\overset{M\rightarrow\infty}{ \rightarrow}\ \ 0,\ \ \forall\ i\
eq j. \tag{4}\]  
The environment is said to offer asymptotically favorable propagation when Equation (4) is satisfied. The mutual orthogonality offered by environments exhibiting the asymptotic favorable propagation condition is the most beneficial situation from the perspective of sum-capacity maximization. Then, the sum-capacity in Equation (3) can be re-expressed as [1]  
Figure 1: Illustration of the adopted system model. \[C =\mathbb{E}\left[\log_{2}|\mathbf{I}_{K}+\rho\mathbf{G}^{H}\mathbf{G }|\right] \tag{5}\] \[\leq\mathbb{E}\left[\log_{2}\left(\prod_{k=1}^{K}\left[\mathbf{I}_ {K}+\rho\mathbf{G}^{H}\mathbf{G}\right]_{kk}\right)\right]\] \[=\mathbb{E}\left[\sum_{k=1}^{K}\log_{2}\left(\left[\mathbf{I}_{K }+\rho\mathbf{G}^{H}\mathbf{G}\right]_{kk}\right)\right]\] \[=\mathbb{E}\left[\sum_{k=1}^{K}\log_{2}\left(1+\rho\|\mathbf{g}_ {k}\|^{2}\right)\right].\]  
The inequality in the second line of Equation (5) is found applying the Hadamard inequality and assuming that \(\|\mathbf{g}_{k}\|^{2},\forall k\) is known, where \(\mathbf{g}_{k},\forall k\), are the columns of the channel matrix \(\mathbf{G}\). As will be shown later, this bound will be proven to be very tight.