## 3 Method: Massive MIMO Network Design
### Problem Description
#### 3.1.1 Tri-Objective Fitness Function
For each grid point i, the electric field \(E_{i,A}\) due to A in 5G is modeled as follows [31; 33]:  
\[E_{i,A}=10^{\frac{EIRP_{\text{th}}+10\text{log}_{10}(DC_{\text{TDD}}^{\text{DL }}+DC_{\text{SPA}})-\Delta 3.15+20\text{-log}_{10}(f)-PL_{d}}{20}} \tag{9}\]  
where \(f\) is the operating frequency in MHz; \(PL_{d}\) is the path loss in dB; \(\text{EIRP}_{\text{th}}\) is the theoretical maximum radiated power in dBm; \(\text{DC}_{\text{TDD}}^{\text{DL}}\) is the TDD duty cycle in the downlink (\(\text{DC}_{\text{TDD}}^{\text{DL}}=0.75\)) capturing the fact that the BS does not transmit continuously and \(\text{DC}_{\text{SPA}}\) (=0.15) is the spatial duty cycle accounting for the fraction of the BS transmit power towards the direction of the user [9]. The total EMF exposure \(\text{E}_{i}\) at the considered location \(i\) (grid point), due to the transmission of all \(N\) base stations in the area, each with \(M\) antenna elements, is given by the below expression:  
\[E_{i}=\sqrt{\sum_{A=0}^{N-M}E_{i,A}^{2}} \tag{10}\]  
With \(E_{i,A}\) is the EMF exposure at a location \(i\) due to a single antenna element, calculated via Equation (9). 1. [label=3.] 2. Uplink exposure fitness function The UL EMF exposure fitness function is given by the below formula:  
\[f_{3}^{3}=\min_{\{x,y\}}\quad\left(1-\frac{SAR_{ull}\left(x_{kn}y_{k}\right)}{ SAR_{\text{max}}}\right) \tag{11}\]  
With \(SAR_{ull}\left(\right)\) is the uplink whole-body specific absorption rate in (W/Kg) due to the transmissions of the users covered by the designed network. \(SAR_{ul}\) is obtained as a weighted average of the median and the 95 th percentile values over the 40 simulations and over all users:  
\[SAR_{ul}=\psi_{1}\cdot SAR_{ul50}+\psi_{2}\cdot SAR_{ul95} \tag{12}\]  
\(SAR_{ulmax}\) is the uplink whole-body specific absorption rate in (\(\frac{\text{W}}{\text{Kg}}\)) due to the transmission of the users at maximum transmit power (23 dBm). \(SAR_{ulmax}\) is obtained similarly to [31] in terms of weighted average as follows:  
\[SAR_{ulmax}=\psi_{1}\cdot SAR_{ulmax50}+\psi_{2}\cdot SAR_{ulmax95} \tag{13}\]  
The whole-body \(SAR_{ul}\) in (\(\frac{\text{W}}{\text{Kg}}\)) due to a user's uplink traffic is obtained as per the formula below:  
\[SAR_{ul}=P_{act}\cdot DC_{TDD}^{UL}\cdot SAR_{ref}^{UL} \tag{14}\]  
Figure 2:0  
where \(DC_{TDD}^{UL}\) is the uplink duty cycle of the TDD network mode of operation. This parameter accounts for the relative transmission time with the TDD mode. Here, we assume a value of 0.25 as proposed in [9]. \(SAR_{ref}^{UL}\) is the uplink reference whole-body specific absorption rate in (\(\frac{\text{W}}{\text{Kg}}\)) for 1 W of transmitted power (\(SAR_{ref}^{UL}\) = 0.0052 (\(\frac{\text{W}}{\text{Kg}}\))). This value has been obtained by simulations using the numerical tool (Sim4 life) with 1 W of transmitting power, using a dipole at 3.7 GHz, positioned next to the right ear. \(P_{act}\) is the actual transmit power of the user's device in (W). The value of \(P_{act}\) ranges from \(-26\) to 23 dBm. \(PL\) is the propagation path loss experienced by the user (in dB) and \(rxsens\) is the sensitivity of the base station (in (dBm)) for maintaining the user's call defined as follows:  
Figure 2:1  
With NF is the noise figure in (dB) (NF = 7 dB [34]), SNR is the signal to noise ratio in (dB) and TN is the thermal noise in (dB) defined as follows:  
Figure 2:2  
where Kg is the Boltzman constant (\(1.38\times 10^{-23}\) (J/K)), T is the reference temperature in Kelvin (T = 290 K) and B is the bandwidth.