## 3 Link Level Simulation Model for Interference Evaluation in 5G UDN with LAB
### SIR Dependence on the Terrestrial Separation of Two gNB-UE Links in 5G UDN with LAB
#### 3.1.2 Mathematical Model for SIR Dependence on the Terrestrial Separation
Formulas (9)-(12) are also used to calculate AA steering vector \(\mathbf{w}_{a}\), when substituting the necessary gNB\(\rightarrow\)UE direction angles into them according to (6) and (7). Denote SOI gNB AA steering vector of coefficients \(\mathbf{w}_{as}=\mathbf{s}_{a}(\varphi_{eNB_{s}},\theta_{eNB_{s}})\), and SNOI gNB AA steering vector of coefficients \(\mathbf{w}_{as}=\mathbf{s}_{a}(\varphi_{eNB_{s}},\theta_{eNB_{s}})\). In this case, \(\left[\varphi_{gNB_{s}},\theta_{gNB_{s}}\right]\) are the angles of the direction of the neighboring gNB to its UE. The AA gain coefficient in a given direction \([\varphi,\theta]\), taking into account antenna array vector of steering coefficients \(\mathbf{w}_{a}\), is calculated as:  
\[g(\varphi,\theta)=|\mathbf{w}_{a}\mathbf{s}_{a}(\varphi,\theta)|^{2}. \tag{13}\]  
According to the formula (13), AA gain of the serving (SOI) gNB \(g_{s}\) and AA gain of the neighboring (SNOI) gNB \(g_{s}\) in the direction of UE is calculated, based on the angles \(\left[\varphi_{gNB_{s}},\theta_{gNB_{s}}\right]\) and \(\left[\varphi_{gNB_{i}},\theta_{gNB_{i}}\right]\), and vectors of steering coefficients \(\mathbf{w}_{as}\) and \(\mathbf{w}_{as}\) as:  
\[g_{s}(\varphi_{gNB_{s}},\theta_{gNB_{s}})=\left|\mathbf{w}_{as}\mathbf{s}_{a}\big{(} \varphi_{gNB_{s}},\theta_{gNB_{s}}\big{)}\right|^{2}; \tag{14}\]  
\[g_{i}(\varphi_{gNB_{i}},\theta_{gNB_{i}})=\left|\mathbf{w}_{as}\mathbf{s}_{a}\big{(} \varphi_{gNB_{i}},\theta_{gNB_{i}}\big{)}\right|^{2}. \tag{15}\]  
Power (in dB), received by the UE from gNB, given \(\mathbf{w}_{a}\) vector, is calculated as:  
\[P(g(\varphi,\theta),d)=10\log_{10}(g(\varphi,\theta))-20\log_{10}\bigg{(}\frac {4\pi d}{\lambda}\bigg{)}. \tag{16}\]  
where \(d\) is the distance between the UE and the gNB; \(\lambda\) is the wavelength. In formula (16), the second term describes FSPL. The value of SIR for the UE in dB is calculated as:  
\[SIR=P(g_{s},d_{s})-P(g_{i},d_{i}), \tag{17}\]  
where \(g_{s}\) and \(g_{i}\) denote AA gains of the serving (SOI) and neighboring (SNOI) gNBs in the direction of the UE, respectively; \(d_{s}\) and \(d_{i}\) are the distances from the UE to the serving and neighbor gNBs, respectively. If there is an AA at the UE, its gain is calculated in a similar manner to the gNB. The vector that specifies the direction UE\(\rightarrow\)gNB is calculated by Formula (1), but the resulting vector is taken with the opposite sign \(\mathbf{k}_{ue}=-\mathbf{k}\). Formulas (2)-(5) are used to translate the direction vector \(\mathbf{k}_{ue}\) into the UE AA local coordinate system. Formulas (6)-(8) are used to calculate the UE\(\rightarrow\)gNB direction angles. The calculation of the amplitude-phase distribution vector and the vector of UE antenna array steering coefficients is performed by (9)-(12). UE AA gain in the direction of gNB is calculated by (13). The power, received by the UE with the AA, is calculated by (16) with an additional term \(g_{UE}\), being the UE AA gain:  
\[P\big{(}g_{gNB},g_{UE},d\big{)}=10\log_{10}\big{(}g_{gNB}\big{)}+10\log_{10}(g _{UE})-20\log_{10}\bigg{(}\frac{4\pi d}{\lambda}\bigg{)}. \tag{18}\]  
The SIR value for the UE with AA is calculated by (17), taking into account (18). The results of SIR calculations are shown in Figures 1 and 2 for the case of omnidirectional UE and 4 \(\times\) 4 URA at gNB.