# Backscatter Assisted NOMA-PLNC Based Wireless Networks
## 3 Power Optimization in Downlink NOMA  
Using optimal power allocation, the sum capacity of the proposed network is enhanced at the downlink. The objective function is formed by maximizing the sum capacity while maintaining the constraint for the total transmit power of the AP and the minimum data rate requirement for each node. Mathematically, the problem can be formulated as follows  
\[\max_{p_{1},p_{2}} \sum_{m=1}^{4}\log(1+\gamma_{m}^{d})\] \[\text{s.t.} \sum_{m=1}^{4}\log(1+\gamma_{m}^{d})-\text{C}_{min}\geq 0 \tag{9}\] \[p_{1}+p_{2}\leq p_{max},\]where \(p_{max}\) is the maximum transmit power of the AP, \(C_{min}\) is the minimum required data rate for each node, \(\gamma_{1}^{d}\) and \(\gamma_{2}^{d}\) are the downlink signal to noise ratio (SNR) of the near node pair, \(\gamma_{3}^{d}\) and \(\gamma_{4}^{d}\) are the downlink signal to interference plus noise ratio (SINR) of the far node pair. The SINR of the far node to decode the near node can be written as  
\[\gamma_{m,n\to f}^{d}=\frac{|g_{m}|^{2}p_{2}}{|g_{m}|^{2}p_{1}+ \sigma^{2}};\quad m\in(1,2), \tag{10}\]  
After removing the far node signal, the SNR of the near node can be computed. \[\gamma_{m}^{d}=\frac{p_{1}|g_{m}|^{2}}{\sigma^{2}};\quad m\in(1,2), \tag{11}\]  
Similarly, SINR expression for the far node pair can be expressed as  
\[\gamma_{m}^{d}=\frac{|g_{m}|^{2}p_{2}+\beta|e_{m}|^{2}|b_{m}|^{2}p_{2}}{|g_{m }|^{2}p_{1}+\beta|e_{m}|^{2}|b_{m}|^{2}p_{1}+\sigma^{2}};\quad m\in(3,4), \tag{12}\]  
It is noted that the far node achieves the diversity from the ABS signal. Furthermore, it can be designed such that it can decode the ABS signal after removing its own message. Thus, far node can also be used as an IoT reader [26]. The SNR of the far node to decode the ABS signal is given as  
\[\gamma_{m,x|x_{b}}^{d}=\frac{\beta|e_{m}|^{2}|b_{m}|^{2}p_{2}}{ \sigma^{2}};\quad m\in(3,4), \tag{13}\]  
Since the constraints are linear and the objective function is concave [5], the equivalent problem becomes the convex maximization, it can be solved using Karush Kuhn Tucker (KKT) conditions [31]. Using Lagrangian multiplier's, (9) can be rewritten as  
\[ \begin{split}&\mathcal{L}(p,\lambda,\mu)=\sum_{m=1}^{4}\log(1+ \gamma_{m}^{d})\\ &-\sum_{m=1}^{4}\lambda_{m}\Big{(}C_{min}-\log(1+\gamma_{m}^{d}) \Big{)}\\ &\qquad\qquad-\mu((p_{1}+p_{2})-p_{max}),\end{split}  \tag{14}\]  
where \(\lambda_{i}\geq 0,i\in(1,2,3,4)\) are the [PERSON]'s multipliers for the minimum rate constraints of near node and far node respectively and \(\mu\geq 0\) is the Lagrangian's multiplier of the maximum transmit power. The problem can be modeled as dual objective function such that it maximizes the sum capacity while minimize the constraints. Mathematically, the first objective function to maximize the sum capacity can be written as  
\[a(\lambda,\mu)=\max_{p_{1},p_{2}}\quad\mathcal{L}(p,\lambda,\mu), \tag{15}\]  
Similarly, the dual objective function can be expressed as  
\[\min_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\mu}\quad a(\lambda,\mu), \tag{16}\]  
\[s.t.\quad\sum_{m=1}^{4}\lambda_{m}\geq 0,\mu\geq 0, \tag{17}\]  
Using gradient descent algorithm, the optimal power \(p_{1}\) and \(p_{2}\) can be estimated. By substituting \(\gamma_{m}^{d}\) in (14) and differentiating with respect to \(p_{1}\) and \(p_{2}\), the optimal powers \(p_{1}\) and \(p_{2}\) can be computed and either \(p_{1}\) or \(p_{2}\) is used for optimal power allocation.