## 7 5G Radio Propagation Mechanisms
### The 5G Path loss Prediction Modelling Approaches
#### 7.3.1 Single-Frequency Models  
Close-In (CI) Model  
The CI model equation for electromagnetic wave propagation is given by [171] in Equation (5):  
\[PL^{CI}(f,l)=FSPL(f,1\text{ m})+10n\text{log}_{10}(l)+X_{\sigma}^{CI}[\text{ dB}]\text{for}l\geq 1\text{ m} \tag{5}\]  
where \(X_{\sigma}^{CI}\) denotes a Gaussian random variable described by the standard deviation \(\sigma^{CI}\)(dB), and the random variable has a mean value of zero (\(\mu=0\)). The free-space path loss is the loss with a reference distance of 1 m, as shown in Equation (6)  
\[(FSPL)(f,1\text{ m})=10\text{log}_{10}\bigg{(}\frac{4\pi f}{c}\bigg{)}^{2} \tag{6}\]  
where \(n\) is the path loss exponent (PLE). Equation (5) can be arranged using the MMSE-based optimization technique to determine the optimum \(n\) as given in (7):  
\[X_{\sigma}^{CI}=PL^{CI}(f,l)[\text{dB}]-FSPL(f,1\text{ m})-10n\text{log}_{10}(l); \text{for}l\geq 1\text{ m} \tag{7}\]  
Assuming that \(K=PL^{CI}(f,l)[\text{dB}]-FSPL(f,1\text{ m})\) and \(P=10\text{log}_{10}(l)\), Equation (5) can then be written as (8)  
\[\sigma^{CI}=\sqrt{\frac{\sum(X_{\sigma}^{CI})^{2}}{D}}=\sqrt{\frac{\sum(K-nP)^ {2}}{D}} \tag{8}\]  
By using the MMSE, the standard deviation of the shadow factor (SF) is calculated using (9):  
\[\sigma^{CI}=\sqrt{\frac{\sum(X_{\sigma}^{CI})^{2}}{D}}=\sqrt{\frac{\sum(K-nP)^ {2}}{D}} \tag{9}\]  
Figure 15: A graph showing the results of (**a**) average penetration loss and (**b**) average attenuation for ceiling, clear glass, drywall, and plywood at \(28,39,120\), and \(144\) GHz. The figure was reproduced from a paper [170] titled: Sub-Terahertz and mmWave penetration loss measurements for indoor environments; published in the 2021 IEEE International Conference on Communications Workshops (ICC Workshops). where \(D\) represents the Tx-Rx separation distances, which define the number of different measurement distances recorded. The SF can be minimized with the standard deviation \(\sigma^{CI}\) when the term \(\sum\bigl{(}K-nP\bigr{)}^{2}\) is reduced. To reduce \(\sum\bigl{(}K-nP\bigr{)}^{2}\), the derivative about \(n\) should be zero as in Equation (10)  
\[d\sum\frac{\bigl{(}K-nP\bigr{)}^{2}}{dn}=\sum 2P(nP-K)=2\sum L(nL-F)=2(n\sum L^{2}- \sum LF)=0 \tag{10}\]  
Thus, from Equation (10), Equation (11) is derived:  
\[n=\frac{\sum KP}{\sum P^{2}} \tag{11}\]  
Therefore, the minimum SF standard variation for the CI model is given by (12)  
\[\sigma_{min}^{CI\sqrt{\frac{\sum\bigl{(}K-nP\bigr{)}^{2}}{D}}} \tag{12}\]  
B. Floating-Intercept (FI) Model  
The FI approach is used by the Wireless World Initiative New Radio (WINNER) II and 3 GPP for 5G path loss prediction [171]. \[PL^{FI}(l)[\mathrm{dB}]=\alpha+10\cdot\beta\mathrm{log}_{10}(l)+X_{\sigma}^{ FI} \tag{13}\]  
where the FI in dB is denoted by \(\alpha\), and it is equivalent to FSPL, while the slope of the line is denoted by \(\beta\), which is similar to the path loss exponent (PLE). \(X_{\sigma}^{FI}(\mu,\ \sigma^{FI})\) denotes the Gaussian random variable with zero mean \((\mu=0)\) and \(\sigma^{FI}\) represents the standard deviation that defines large-scale signal fluctuations about the mean path loss over the Tx-Rx spacing. Equation (13) requires two parameters, unlike the CI model, which requires just the parameter \(n\).