## 6 Path Loss Channel Model
The following is a summary of the entire scenario:  
\[\min\left\{\sum_{i=1}^{x}P_{j,n}^{i}\right\},\ s.t.B_{S}\sum_{i=1}^{x}\log_{2} \left[1+SNR\big{(}d_{j,n},f_{i}\big{)}\right]\quad\geq R_{0}\ and\ P_{j,n}^{i}\geq 0 \ \forall\ i,j,n \tag{9}\]  
\(SNR\) (\(d_{j,n},f_{i}\)) is derived from Equation (8), and '\(x\)' is the number of sub-channels utilized by node '\(j\)' to send data to node '\(n\)'. It is worth noting that it is just like the convex function that is used for the objective function. The function of \(B_{S}\sum_{i=1}^{x}\log_{2}\left[1+SNR\big{(}d_{j,n},f_{i}\big{)}\right]\) is in terms of \(P_{j}^{i}\). All these details are described in Equation (9) to minimize the cost function [39]. We differentiated Equation (10) with some derivatives by substitution, in which we used '\(\lambda\)' as a long-range multiplier. \[L\left\{\left(P_{j,n}^{i}\right)_{i=1}^{x},\lambda\right\}=\sum_{i=1}^{x}P_{j,n}^{i}-\lambda\left\{B_{S}\sum_{i=1}^{x}\log_{2}\left[1+\frac{\left|a_{j,n}^{ i}\right|^{2}P_{j,n}^{i}}{A\big{(}d_{j,n},f_{i}\big{)}N(f_{i})B_{S}}\right]-R_{0}\right\} \tag{10}\]  
\[\frac{\delta L}{\delta P_{j,n}^{m}}=\frac{1-\frac{\lambda}{ln2}\left[\frac{ \left|a_{j,n}^{m}\right|^{2}}{A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})}\right]}{1 +\left[\frac{\left|a_{j,n}^{m}\right|^{2}P_{j,n}^{m}}{A\big{(}d_{j,n},f_{m} \big{)}N(f_{m})B_{S}}\right]}=0 \tag{11}\]  
Here we can elaborate Equation (11) for \(P^{m},_{j,n}\), as:  
\[ \begin{split} 1-\frac{\lambda}{ln2}\left[\frac{\left|a_{j,n}^{m} \right|^{2}}{A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})}\right]&=1+ \left[\frac{\left|a_{j,n}^{m}\right|^{2}P_{j,n}^{m}}{A\big{(}d_{j,n},f_{m}\big{)} N(f_{m})B_{S}}\right]\\ \frac{ln2A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})-\lambda\left|a_{j,n }^{m}\right|^{2}}{ln2A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})}&= \frac{A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})B_{S}-\left|a_{j,n}^{m}\right|^{2}P _{j,n}^{m}}{A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})B_{S}}\end{split}  \tag{12}\]  
Since in cross multiplication '\(P_{j,n}^{m}\)' will disappear and become a free variable, and we can assign any value to it. \[ \begin{split} B_{S}\left[ln2A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})- \lambda\left|a_{j,n}^{m}\right|^{2}\right]=ln2\left[A\big{(}d_{j,n},f_{m} \big{)}N(f_{m})B_{S}-P_{j,n}^{m}\right]\\ P_{j,n}^{m}=B_{S}\frac{\left|a_{j,n}^{m}\right|^{2}-ln2A\big{(}d_{ j,n}f_{m}\big{)}N(f_{m})}{ln2\left|a_{j,n}^{m}\right|^{2}}\end{split}  \tag{13}\]  
We obtain:  
\[P_{j,n}^{m}=B_{S}\left[\frac{\lambda}{ln2}-\frac{A\big{(}d_{j,n},f_{m}\big{)}N (f_{m})}{\left|a_{j,n}^{m}\right|^{2}}\right]^{+},\ \forall\ m=1,2,\ldots,x \tag{14}\]where '+' means to have a source of projections taking non-negative numbers. \[\frac{\delta L}{\delta\lambda}=0\ \Rightarrow\ B_{\mathrm{s}}\sum_{i=1}^{x}\log_{2} \left[1+\frac{\left|a_{j,n}^{i}\right|^{2}P_{j,n}^{i}}{A\big{(}d_{j,n},f_{i} \big{)}N(f_{i})B_{\mathrm{s}}}\right]=R_{0} \tag{15}\]  
We obtain from Equation (14), borrow the Equation (15) values, and perform simple procedures. \[\lambda=\left[\frac{part_{1}}{part_{2}}\right](ln2);\ where\ part_{1}=2^{\frac{R_{0}}{2 bits}}\ and\ part_{2}=\left[\prod_{i=1}^{x}\frac{\left|a_{j,n}^{i}\right|^{2}}{A \big{(}d_{j,n},f_{i}\big{)}N(f_{i})}\right]^{1/x} \tag{16}\]  
Solving for the minimum power \(P_{j,n}^{m}\) we obtain: \(P_{j,n}^{m}=B_{\mathrm{s}}\Big{(}\frac{part_{1}}{part_{2}}-\frac{part_{3}}{part _{4}}\Big{)}^{+}\); \(\ part_{3}=A\big{(}d_{j,n},f_{m}\big{)}N(f_{m})\ and\ part_{4}=\left|a_{j,n}^{i} \right|^{2}\). Therefore, the number of sub-channel '\(x\)' is used by node '\(j\)' to transmit data to node '\(n\)'.