# Optimal Resource Allocation Policies for Multi-User Backscatter Communication Systems
## 4 Optimal Resource Allocation of Passive Tags
For the semi-passive tags case, combing the priority of allocated reader transmission power and the proposed BCD-based optimization algorithm, we find the close-to-optimal solution. The simulation results confirm that the proposed policies can achieve larger total system goodput than conventional techniques. This work was supported by the National Natural Science Foundation of China (Nos. 61271335, 61671252, 61401069), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), the Key University Science Research Project of Jiangsu Province (No. 14 KJA510003), the Innovation Program of Graduate Education of Jiangsu Province, China (KYLX15-0828), the National Natural Science Foundation of China (Nos. 61471202, 61501251, 61571233), the Natural Science Foundation of Jiangsu Province (BK20140891). [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON] contributed analysis tools; [PERSON] and [PERSON] wrote the paper. The authors declare no conflict of interest. ## Appendix A

For the \(i\)-th sub-problem, if \(NP^{\text{ave}}\geq(N-i+1)P^{\text{max}}\), the maximum total goodput is given as \(\sum_{m=1}^{N-i+1}\hat{G}_{m}(P^{\text{max}})\). For the \(\hat{i}\)-th sub-problem, \(\hat{i}=i+1,\cdots,N\), \(NP^{\text{ave}}\geq(N-\hat{i}+1)P^{\text{max}}\) is still satisfied, and the maximum total goodput is given as \(\sum_{m=1}^{N-\hat{i}+1}\hat{G}_{m}(P^{\text{max}})\). It is obvious that \(\sum_{m=1}^{N-i+1}\hat{G}_{m}(P^{\text{max}})>\sum_{m=1}^{N-\hat{i}}\hat{G}_{m }(P^{\text{max}})>\cdots>\hat{G}_{1}(P^{\text{max}})\). Hence, we conclude that the solution for the \(\hat{i}\)-th sub-problem is not the optimal solution for Problem P1. Figure 4: Average total goodput vs. average reader transmission power with \(N=5\). ## Appendix B

The Lagrange function of Problem P2 is expressed as

\[L(\hat{\mathbf{P}})=\sum_{m=1}^{M}T\left[\frac{1}{2}+\frac{1}{2}\text{erf}\left( \sqrt{a_{m}P_{m}-b_{m}}\right)\right]-\lambda(\sum_{m=1}^{M}P_{m}-NP^{\text{ave}})\]

where \(\lambda\) denotes the Lagrange multiplier. Problem P2 can be solved by exploiting its Karush-Kuhn-Tucker (KKT) conditions [26], which are given by

\[\frac{\partial L}{\partial P_{m}^{*}} \begin{cases}<0,&P_{m}^{*}=P_{m}^{\text {th}},\\ =0,&P_{m}^{\text{th}}\leq P_{m}^{*}\leq P^{\text{max}},\quad\forall m,\\ >0,&P_{m}^{*}=P^{\text{max}},\end{cases}  \tag{1}\]

\[\lambda^{*}(\sum_{m=1}^{M}P_{m}^{*}-NP^{\text{ave}})=0, \tag{2}\]

where \(P_{m}^{*}\) and \(\lambda^{*}\) denote the optimal primal and dual solutions of Problem P2, respectively. It is easy to verify that \(\sum_{m=1}^{M}P_{m}^{*}=NP^{\text{ave}}\) holds for Problem P2. Hence, we can assume \(\lambda^{*}>0\) from Equation (2) without loss of generality. According to Equation (1), if \(P_{m}^{\text{th}}\leq P_{m}^{*}\leq P^{\text{max}}\), it follows that

\[\mathcal{G}(a_{m}P_{m}^{*}-b_{m},c_{m})=\lambda^{*},\quad\forall m, \tag{3}\]

where \(\mathcal{G}(\
u_{m},c_{m})\) with respect to \(\
u_{m}\) is defined in Theorem 2. Define \(\
u_{m}^{*}\) as the solution for the equation \(\mathcal{G}(\
u_{m},c_{m})=\lambda^{*}\). Since \(\mathcal{G}(\
u_{m},c_{m})\) is a monotonically decreasing function of \(\
u_{m}\), it is obtained that \(\
u_{m}^{*}\) is the unique solution. Based on \(\
u_{m}^{*}\), \(P_{m}^{*}=\frac{\
u_{m}^{*}+b_{m}}{a_{m}}\). Combining with \(P_{m}^{\text{th}}\leq P_{m}^{*}\leq P^{\text{max}}\), the optimal solution for Problem P2 is given as \(P_{m}^{*}=\min\{\max\{P_{m}^{\text{th}},\frac{\
u_{m}^{*}+b_{m}}{a_{m}}\},\text {p}^{\text{max}}\}\). ## Appendix C

Assume the optimal solution for Problem P4 is \(\{\hat{P}_{m},\hat{t}_{m}^{a},\hat{n}_{m}\}\) satisfying \(P^{c}\hat{t}_{m}^{a}<\eta\hat{P}_{m}h_{m}(T-\hat{t}_{m}^{a})+\eta(1-\hat{n}_{m} )\hat{P}_{m}h_{m}\hat{t}_{m}^{a}\), \(\forall m\). It is to prove that \(\{\hat{P}_{m},\hat{t}_{m}^{a},\hat{n}_{m}\}\) is not the optimal solution by contradiction.