## 6 Extension with Side-Lobes/Back-Lobes as Well as the Path Loss Effect
### Physical Channel Model  
We denote the node whose transmission causes the interference to other nodes as the _interfering_ node. The node whose reception is interfered by other _interfering_ nodes is denoted as _interfered_ node. We assume that the interfering node transmits with power \(P_{t}\). The received power at the interfered node at a distance \(d\) from the interfering node is denoted by \(P_{r}\), which can be calculated by  
\[P_{r}=CG_{t}G_{r}P_{t}\frac{1}{r^{\alpha}} \tag{8}\]  
where \(C\) is a constant, \(G_{t}\) and \(G_{r}\) denote the antenna gain of the interfering node and the antenna gain of the interfered node, respectively, and \(\alpha\) is the path loss factor usually ranging from 3 to 4 [28]. When an interfering node interferes with an interfered node, the received power at the interfered node \(P_{r}\) is required to be no less than a threshold \(P_{0}\), _i.e._, \(P_{r}\geq P_{0}\). Thus, to calculate \(P_{0}\), we  
\[P_{0}=CG_{t}G_{r}P_{t}\frac{1}{R_{i}^{\alpha}} \tag{9}\]  
where \(R_{i}\) is defined as the _interfering range_ in the physical channel model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Main Beamwidth \(\theta_{m}\)** & **Main Beam Gain \(g_{m}\)** & **Side-Lobe and Back-Lobe Gain \(g_{s}\)** \\ \hline \hline \(60^{\circ}\) & 10 (10 dBi) & \(0.18\) (\(-\)7.4 dBi) \\ \hline \(40^{\circ}\) & 25.12 (14 dBi) & \(0.17\) (\(-\)7.6 dBi) \\ \hline \(20^{\circ}\) & 100 (20 dBi) & \(0.22\) (\(-\)6.5 dBi) \\ \hline \(10^{\circ}\) & 398 (26 dBi) & \(0.398\) (\(-\)4.0 dBi) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Antenna gains (linear and logarithmic). Solving this equation, we have  
\[R_{i}=(\frac{CG_{t}G_{r}P_{t}}{P_{0}})^{\frac{1}{\alpha}} \tag{10}\]  
We next analyze the interfering range \(R_{i}\) according to the four different scenarios, which are summarized in Table 3.  
In particular, in Scenario I, two nodes \(X_{j}\) and \(X_{k}\) interfere with each other if and only if they fall into the interference range of each other and their main antenna beams are pointed toward each other, as shown in Figure 11(a). In this case, the interference range denoted by \(R_{i}(MM)\) can be calculated by  
\[R_{i}(MM)=(\frac{Cg_{m}g_{m}P_{t}}{P_{0}})^{\frac{1}{\alpha}} \tag{11}\]  
where we replace both \(G_{t}\) and \(G_{r}\) in Equation (10) by \(g_{m}\). In Scenario II, the main antenna beam of the interfering node \(X_{k}\) is pointed to the interfered node \(X_{j}\), which also falls into the interference range of \(X_{i}\). However, the main beam of the interfered node \(X_{j}\) is not necessarily pointed to the interfering node \(X_{k}\). Due to the existence of the side-lobes and the back-lobes, the reception of node \(X_{j}\) is interfered by node \(X_{k}\), as shown Figure 11(b). Thus, the interference range denoted by \(R_{i}(MS)\) can be calculated by  
\[R_{i}(MS)=(\frac{Cg_{m}g_{s}P_{t}}{P_{0}})^{\frac{1}{\alpha}} \tag{12}\]  
where we replace \(G_{t}\) and \(G_{r}\) in Equation (10) by \(g_{m}\) and \(g_{s}\), respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenarios** & **Interfering Node** & **Interfered Node** & **Interference Range \(R_{i}\)** \\ \hline I & Main beam & Main beam & \(R_{i}(MM)\) \\ \hline II & Main beam & Side-lobes and back-lobes & \(R_{i}(MS)\) \\ \hline III & Side-lobes and back-lobes & Main beam & \(R_{i}(SM)\) \\ \hline IV & Side-lobes and back-lobes & Side-lobes and back-lobes & \(R_{i}(SS)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Four scenarios. Figure 11: Four scenarios. (**a**) Scenario (I); (**b**) Scenario (II); (**c**) Scenario (III); (**d**) Scenario (IV). Similar to Scenario II, the interference range in Scenario III, which is denoted by \(R_{i}(MS)\), can be calculated by  
\[R_{i}(SM)=(\frac{Cg_{s}g_{m}P_{t}}{P_{0}})^{\frac{1}{\alpha}} \tag{13}\]  
where we replace \(G_{t}\) and \(G_{r}\) in Equation (10) by \(g_{s}\) and \(g_{m}\), respectively. It is obvious that \(R_{i}(MS)=R_{i}(SM)\). Thus, we regard \(R_{i}(MS)\) as \(R_{i}(SM)\) interchangeably throughout the remaining paper.