# User Association Performance Trade-Offs in Integrated RF/mmWave/THz Communications
## 3 System Model
### SINR Calculation  
SINR is estimated using frequency-specific path loss and noise models. For the RF band, the path loss is modeled using a logarithmic distance-based model (COST 231) [38]. For mmWave bands, additional penetration losses are considered based on common building materials measured as in TR38.901 [39]. For THz bands, molecular absorption losses are characterized using coefficients from propagation measurements in varying atmospheric compositions [40; 41].  
The SINR model for RF communication can be given as follows [42]:  
\[SINR^{RF}=\frac{P_{T_{0}}G_{T}G_{R}Ad_{0}^{-\alpha}h_{0}}{N_{0}+A\sum_{i=1}^{N }P_{T_{i}}G_{i}d_{i}^{-\alpha}h_{i}} \tag{2}\]  
where \(A=\frac{e^{2}}{16\pi^{2}f^{2}}\), \(f\) is frequency, \(c\) is light speed, \(G_{T}\) and \(G_{R}\) are the directional gains of the transmitting and receiving beams, \(\alpha\) is the attenuation coefficient, \(d\) is the distance from the transmitter to the receiver, \(P_{T_{0}}\) is the transmitted power, \(N_{0}\) is the Johnson-Nyquist noise, \(N\) is the number of interferes, \(G_{i}\) is the interfering link beam gain that is reduced if beams do not align well, and \(h_{0}\) is the small-scale fading gain.  
SINR for mmW LOS BSs communication is calculated as [43; 44]:  
\[SINR_{LOS}^{mmW}=\frac{P_{T_{0}}G_{T}G_{R}A_{0}d_{0}^{-\alpha}B_{mm}}{N_{0}+ \sum_{i=1}^{N}P_{T_{i}}G_{i}A_{i}d_{i}^{-\alpha}} \tag{3}\]  
SINR for mmW NLOS BSs communication is calculated as:  
\[SINR_{NLOS}^{mmW}=\frac{P_{T_{0}}G_{T}G_{R}A(d_{10}+d_{20})^{-\alpha}B_{mm}}{ N_{0}+A\sum_{i=1}^{N_{mm}}\sum_{i=1}^{N}P_{T_{i}}G_{i}d_{i}^{-\alpha}} \tag{4}\]  
The path loss received from NLOS components communication considers the presence of scatters between the transmitter and the receiver [43], where \(d_{1}\) is the distance from the transmitter to the scatterer, \(d_{2}\) is the distance from the scatterer to the receiver, \(N_{rays}\) is the number of NLOS rays, and \(B_{mm}\) represents the blockage probability, which is calculated analytically as:  
\[B_{mm}=e^{-\frac{2\pi\lambda_{b}\rho}{\gamma}} \tag{5}\]  
Here, \(\lambda_{b}\) is the density of blockages, \(\rho\) is the typical width of blockages, and \(\gamma\) is the viewing angle. The blockage calculation considers beam directions--a blockage may only partially obstruct a directional link.  
SINR for THz LOS communication is calculated as [43; 44]:  
\[SINR_{LOS}^{THz}=\frac{P_{T_{0}}G_{T}G_{R}Ad_{0}^{-\alpha}e^{-Kd_{0}}B_{mm}}{ N_{0}+A\sum_{i=1}^{N}P_{T_{i}}G_{i}d_{i}^{-\alpha}} \tag{6}\]where \(K\) is the absorption coefficient of the medium, and \(N_{0}\) is the Johnson-Nyquist noise plus the molecular noise equal to \(N_{0}=K_{B}T+P_{T_{0}}A_{0}d_{0}^{-2}(1-e^{-Kd_{0}})\), considering \(K_{B}\) as the Boltzman constant and T as the temperature.  
SINR for Tera Hertz NLOS communication is calculated as:  
\[SINR_{NLOS}^{THz}=\frac{P_{T_{0}}G_{T}G_{R}A(d_{1_{0}}+d_{2_{0}})^{-n}e^{-K(d _{1_{0}}+d_{2_{0}})}R(f)B_{mm}}{N_{0}+A\sum_{i=1}^{N_{m}}\sum_{i=1}^{N}P_{T_{i} }G_{i}(d_{1_{0}}+d_{2_{i}})^{-\alpha}} \tag{7}\]  
where \(R(f)\) is the reflection coefficient of a rough surface equal to the Rayleigh roughness factor times the smooth surface reflection coefficient derived from the Fresnel equations. This is given by the expression [45]:  
\[R(f)=e^{-\frac{2\cos(\theta_{1})}{\sqrt{\mu_{r}\epsilon_{r}}-1}}\times e^{- \frac{8\pi f\sigma\cos(\theta_{1})}{c\times d_{2}}} \tag{8}\]  
where \(\theta_{1}\) is the angle of incidence, and \(\sigma\) is the rough surface height standard deviation, commonly assumed to be Gaussian distributed.  
Based on the equations provided, the total SINR would be a weighted sum of the LOS and NLOS SINR:  
\[SINR_{total}=w_{LOS}*SINR_{LOS}+w_{NLOS}*SINR_{NLOS} \tag{9}\]  
where \(w_{LOS}\) and \(w_{NLOS}\) are the weights for the LOS and NLOS SINR, representing their relative importance or reliability \(w_{LOS}+w_{NLOS}=1\).  
LOS signal tends to be more reliable, so typically \(w_{LOS}>w_{NLOS}\).