# Energy Efficiency and Coverage Trade-Off in 5G for Eco-Friendly and Sustainable Cellular Networks
## 3 System Model and Problem Formulation
### Problem Formulation and Mathematical Modeling  
#### 3.2.1 Propagation Channel Model  
The properties of the received signal are affected by three attributes, small-scale fading-multipath, large-scale path fading-shadowing, and propagation path loss. Hence, an all-inclusive received power (\(P_{rx}\)) propagation model can be written as follows [35]:  
\[P_{r}=P_{tx}+G-P_{Loss}-\sigma, \tag{1}\]  
where \(P_{tx}\), \(G\), \(P_{Loss}\), \(\sigma\), denote device transmitted power, total antenna gains, path loss, shadow fading margin, respectively.  
LTE Path Loss Model  
The path loss model utilized herein is the 3 GPP Urban Macro (Uma-NLOS propagation model for the channels between BSs and UEs [36]. The model consists of: the frequency (\(f\)), BS antenna height (\(h_{BS}\)), UE antenna height (\(h_{UE}\)), average building height (\(h_{bl}\)), street width (\(w_{st}\)), and radius (\(R\)) in meters. The function is given as follows:  
\[\begin{split}& P_{L}=161.04-7.1\log_{10}(w_{st})+7.5\log_{10}(h_{bl}) -\left(24.37-23.7\left(\frac{h}{h_{BS}}\right)^{2}\right)\times\log_{10}(h_{BS} )\\ &+\left(43.42-3.1\log_{10}(h_{BS})\right)\times\left(\log_{10}(R) -3\right)\\ &+20\log_{10}(f)-\left(3.2\left(\log_{10}(11.75h_{UE})\right)^{2 }-4.97\right)\end{split} \tag{2}\]  
II. 5G Path Loss Model  
The modified Stanford University Interim (SUI) path loss model (\(P_{L\_mod\_SUI}\)) for a frequency of 28 GHz for Non-Line of Sight (NLOS) is proposed, as given in Equation (3). This mathematical model (the modified SUI path loss model) is constructed based on extensive empirical measurements [37].  
\[P_{L\_mod\_SUI}(d)=\alpha_{NLOS}\times\left(P_{L\_SUI}(d)-P_{L\_SUI}(d_{o}) \right)+P_{L}(d_{o})+X_{\sigma}, \tag{3}\]  
where \(\alpha_{NLOS}\) is the mean slope correction factor, obtained directly from the NLOS empirical results; \(P_{L\_SUI}(d)\) is an original SUI model at a distance \(d\); \(P_{L\_SUI}(d_{o})\) is an original SUI model at a reference distance \(d_{o}\); \(P_{L}(d_{o})\) represents free space path loss (dB) in close-in reference distance \(d_{o}\); and \(X_{\sigma}\) is a typical lognormal random shadowing with variable mean of (0 dB) and standard deviation (\(\sigma\)). The original SUI model \(P_{L\_SUI}(d)\) for a frequency above 2 GHz is [37]  
\[P_{L\_SUI}(d)=P_{L}(d_{o})+10n\log_{10}(d/d_{o})+X_{fc}+X_{RX}+X_{\sigma}, \tag{4}\]  
where  
\[P_{L}(d_{o})\ =\ 20\log_{10}\left(\frac{4\Pi d_{o}}{\lambda}\right) \tag{5}\]  
\[n=a-b\cdot h_{TX}+\frac{C}{h_{TX}} \tag{6}\]  
\[X_{fc}=6\log_{10}\left(\frac{f_{MHz}}{2000}\right)X \tag{7}\]  
\[X_{RX}=-10.8\log_{10}\left(\frac{h_{RX}}{2}\right). \tag{8}\]  
\(\lambda\) denotes carrier wavelength (m); \(X_{fc}\) and \(X_{RX}\) signify the correction factors for frequency and receiver heights, respectively; \(f_{MHz}\) denotes carrier frequency (MHz); and \(h_{TX}\) and \(h_{RX}\) denote the transmitter and receiver antenna heights in meters, respectively. The parameters \(a\), \(b\), and \(c\) are constants that are service area. The SUI terrain type A is considered, with parameters given as \(a\) = 4.6, \(b\) = 0.0075, and \(c\) = 12.6 [37].  
Cellular cell coverage area is defined as the fraction of the area within a cell receiving signal power above a given minimum, \(P_{\min}\). In reality, signal degradation factors of random shadowing and path loss will lead to some cell portions having a received power below \(P_{\min}\). Given by [38],  
\[P_{\min}=N_{o}BW+N_{f}+SINR+IM, \tag{9}\]  
where \(N_{o}BW\) connotes thermal noise level assigned for a specified noise bandwidth; \(N_{f}\) is the receiver noise figure; while \(IM\) connotes implementation margin.