# Energy Efficient Access Point Placement for Distributed Massive MIMO
## 3 Different Channel Models
### Euclidean Distance-Based Model  
Let us denote the channel gain from AP location \(l\) to user location \(i\) as \(\beta_{il}\). Considering the generic statistical model commonly applied in existing studies [14] and the radiation pattern of the array in the AP (the user has a single isotropic antenna), we calculate \(\beta_{il}\) as  
\[\beta_{il}=G_{\Upsilon\theta_{il}}\frac{\zeta}{d_{il}^{2}}10^{\chi_{il}}\| \mathbf{f}_{\delta_{il}}\|^{2}, \tag{10}\]  
where \(\theta_{il}\) is the angle of user location \(i\) regarding the broadside direction of the array at AP location \(l\) and \(G_{\Upsilon\theta_{il}}\) is the antenna gain from the AP at location \(l\) in the direction of user location \(i\). Parameter \(\zeta\) is the free space loss factor [39] determined by the wavelength. The power loss due to the propagation distance is represented by \(\frac{1}{d_{il}^{2}}\), where two is the pathloss exponent and \(d_{il}\) is the Euclidean distance between user location \(i\) and AP location \(l\). We model the shadowing as \(10^{\chi_{il}}\), where \(\chi_{il}\) is a zero mean Gaussian random variable with a given variance. Each entry of the small-fading vector \(\mathbf{f}_{Sil}\) is an independent and identically distributed Gaussian random variable with zero mean and unit variance. Considering Equation (10), the average channel gain is defined as  
\[\widetilde{\beta}_{il}=\mathbb{E}\{\beta_{il}\}=G_{\Upsilon\theta_{il}}\frac{ \zeta}{d_{il}^{2}}\mathbb{E}\{10^{\chi_{il}}\}\mathbb{E}\Big{\{}\|\mathbf{f} _{\delta_{il}}\|^{2}\Big{\}} \tag{11}\]  
The distributions of shadowing and small-scale fading in the model are commonly assumed to be independent of AP location \(l\) and user location \(i\)[14]. The constant \(\zeta\) is also independent of the user location and the AP location. Therefore, we can state that the constant \(\zeta^{\prime}\) equals  
\[\zeta^{\prime}=\zeta\mathbb{E}\{10^{\chi_{il}}\}\mathbb{E}\Big{\{}\|\mathbf{ f}_{\delta_{il}}\|^{2}\Big{\}}, \tag{12}\]and reformulate Equation (11) as  
\[\widetilde{\beta}_{il}=\zeta^{\prime}\frac{G\tau_{\theta_{il}}}{d_{il}^{2}}. \tag{13}\]  
Regarding Constraints (8), the value for \(\zeta^{\prime}\) has no impact on the optimal AP placement. This is consistent with the conclusion made in [14]. We only use Equation (13) for calculating the AP placement and then evaluate the required transmit power with an RTS. Therefore, we simply set \(\zeta^{\prime}\) to one in our experiments.  
As indicated by Equation (13), the average channel gain received at a user location \(i\) from an AP location \(l\) is only characterized by the Euclidean distance between both locations. As such, it does not consider the local environment information, for example buildings, causing obstruction or not. Consequently, the estimated \(\theta_{il}\) may not correspond to the direction of the highest power. This is illustrated in Figure 5, using the same map as in Figure 2. The two highlighted dots in Figure 5 represent a user location and an AP location. The blue arrow represents the broadside direction of the array at the AP location. The green straight line indicates the Euclidean distance between the user location and the AP location. The orange polyline in Figure 5 illustrates one of the possible propagation paths from the AP location to the user location. As shown in Figure 5, the Euclidean distance-based model ignores the building between the user location and the AP location. For both the angle of departure and length, the green line significantly deviates from the orange polyline. The channel gain may be severely overestimated if we estimate it based on the green line.