# On the Spectral Efficiency for Distributed Massive MIMO Systems
## 2 Distributed MIMO Model  
The main goal to study the scenarios is the emulation of distributed MIMO systems by combining values from several communication channels acquired in different environments. Therefore, diverse channel matrices are combined in order to form a single matrix that symbolizes the D-MIMO channel matrix [6; 19]. Each propagation environment has a certain number of transmitting (TX) and receiving (RX) antennas. TXs simulate a transmitting BS, where each BS is placed in a different location, emitting to a user equipment (UE) located in a fixed place and composed by several RXs. Mathematically, the combination of channel matrices can be depicted in Equation (1). A deep and detailed explanation on channel matrix models can be found on [20; 21]. \[\mathbf{H}\ =\ \left[\mathbf{H}_{1}^{T},\mathbf{H}_{2}^{T},\ \ldots,\mathbf{H}_{N}^{T}\right] \ ^{T}, \tag{1}\]  
\(N\) is the total number of MIMO systems that form the D-MIMO. \(\mathbf{H}_{i}\) is the channel matrix from a single MIMO whose dimension is L\({}_{i}\)\(\times\) R. L\({}_{i}\) and R is the number of TX and RX in the BS and UE for the \(i\)-MIMO system, respectively. Finally, \(\mathbf{H}\) is the channel matrix of the D-MIMO with dimensions L \(\times\) R, where L is computed as \(\sum_{i=1}^{N}\) L\({}_{i}\). It is essential to emphasize that each \(\mathbf{H}_{i}\), related to different MIMO systems, represents propagation channels that may differ from each other due to the unique conditions of each scenario. Therefore, the power received by the UE is expected to be different from different schemes. In this paper, a modulation of \(\mathbf{H}_{i}\) is introduced to emulate the unbalance power that might occur in a D-MIMO system. In order to perform this unbalance, \(\mathbf{H}_{i}\) is multiplied by a balance factor (\(Att\)) that emulates losses in the transmission due to obstacles in the propagation path or shadow fading. \[\mathbf{H}_{i}^{Att}=Att_{i}\cdot\mathbf{H}_{i}\ ;\qquad Att_{i}\geq 0 \tag{2}\]  
By applying this balance factor to certain \(i\)-MIMO systems, the performance of D-MIMO systems can be analyzed when they suffer a strong power unbalance. The spectral efficiency of the system can be computed to determine the effect on the D-MIMO once the power balance is modified. This metric of the full communication channel is estimated using the following formulas [22]:  
\[\mathbf{H}_{norm}(f_{i})=\mathbf{H}(f_{i})\ \left[\frac{1}{N_{f}}\sum_{i=1}^{N_{f}} \left|\mathbf{H}(f_{i})\right|\right|^{2}\right]^{-\frac{1}{2}}, \tag{3}\]\[\eta(\text{bps}/\text{Hz})=log_{2}\bigg{|}\mathbf{I}_{L}+\frac{\text{SINR}}{\text{L}} \mathbf{H}\mathbf{H}^{H}\bigg{|}, \tag{4}\]  
where SINR is the signal-to-interference-plus-noise ratio rate, \(\mathbf{H}\) represents the channel matrix of the D-MIMO, \(\mathbf{I}\) is an identity matrix, and the superscript \(\ {}^{(H)}\) denotes the conjugate transpose operator. First of all, \(\mathbf{H}\) must be normalized (\(\mathbf{H}_{norm}\)) in order to take into account the frequency dependence of the communication channel in Equation (3) [23], where \(N_{f}\) is the number of frequency samples. Note that the number of frequency samples \(N_{f}\) must be chosen long enough so that no channel effect at a single frequency is unnoticed. The spectral efficiency in Equation (4) is calculated from the normalized channel frequency response \(\mathbf{H}_{norm}\). Throughout the article, additional modifications will be added to the channel frequency response shown in Equation (3) to demonstrate the importance of effects such as power unbalance in D-MIMO systems. A visual example of the composition of a D-MIMO system is illustrated in the Figure 2. This example shows the composition of a D-MIMO throughout several measurements acquired in a semi-anechoic and semi-reverberation chamber. As previously stated, \(n\)-MIMO systems are deployed, where the UE/RX is located in a fixed position. However, the BS/TX position is changed in order to move and point to different locations inside the chamber.