# Optimal Resource Allocation Policies for Multi-User Backscatter Communication Systems
## 4 Optimal Resource Allocation of Passive Tags
Theorem 4.2: _If \(NP^{\text{ave}}<MP^{\max}\), then the optimal power allocation for Problem P2 is given as follows_  
\[P_{m}^{*}=\min\{\max\{P_{m}^{\text{th}},\frac{\
u_{m}^{*}+b_{m}}{a_{m}}\},P^{ \max}\},\quad\forall m,\]  
_where \(\
u_{m}^{*}\) is the unique solution for the equation \(\mathcal{G}(\
u_{m},c_{m})-\lambda^{*}=0\) with respect to \(\
u_{m}\), \(\mathcal{G}(\
u_{m},c_{m})=\frac{1}{2\sqrt{\pi}}c_{m}e^{-\
u_{m}}\
u_{m}^{- \frac{1}{2}}\), \(c_{m}=a_{m}T\), \(\lambda^{*}\) is the optimal Lagrange multiplier, and \(\sum_{m=1}^{M}P_{m}^{*}=NP^{\text{ave}}\)._  
Proof: See Appendix B. The optimal Lagrange multiplier \(\lambda^{*}\) can be obtained by the bisection method. Based on above discussions, the optimal power allocation policy can be derived using Algorithm 1. The asymptotic complexity of Algorithm 1 is analyzed as follows. The main complexity of Problem P2 comes from deriving Theorem 4.2. Suppose \(\Delta\) iterations are needed for the bisection method used in Theorem 4.2 to guarantee the converge. The complexity of deriving Theorem 4.2 is given as \(\mathcal{O}(\Delta M)\). We assume that \(NP^{\text{ave}}\geq(N-i+1)P^{\max}\) if and only if \(i>K\) (\(K\leq N\)) and consider the worst-case; i.e., the first \(K\) sub-problems are all feasible, the total complexity of Algorithm 1 is thus \(\mathcal{O}(K\Delta M)\). ```

**Algorithm 1** Algorithm for Problem P1

**Remark 1** (Activate or Not): _It can be derived that the decision on whether a tag can be activated is determined by the allocated reader transmission power (or the receive power at the tag). If the allocated power \(P_{i}\) is not smaller than the threshold \(P_{i}^{\text{th}}\), the \(i\)-th tag is activated; otherwise, this tag stays in the non-activated state. From Algorithm 1, if the \(i\)-th sub-problem corresponds to the maximum total goodput, only the first \(N-i+1\) tags can be activated._

Generally, for a BackCom system with passive tags, the tags corresponding to worse channel conditions cannot be activated because the average and peak of reader transmission power are limited. To make the system work in lower reader transmission power, a BackCom system with semi-passive tags is considered in the following sections. ## 5 Problem Formulation of Semi-Passive Tags

In this section, the total goodput maximization problem of a BackCom system with \(N\) semi-passive tags is studied. Since each semi-passive tag has a battery, it can first enter the sleep state to store energy and then operate in the active state. Therefore, the activation threshold for the \(i\)-th semi-passive tag can be reduced compared to the case of passive tags. This threshold is given as follows:

\[P_{i}^{\text{th}}=\frac{[\text{erfc}^{-1}(2\varepsilon)]^{2}}{a_{i}},\quad \forall i.\]

Following the case of passive tags, Equation (1) is also suitable here, while \(t_{i}^{a}\) needs to further optimized. Hence, Equation (2) is rewritten as

\[G_{i}(P_{i},t_{i}^{a},n_{i})= \begin{cases}t_{i}^{a}\left[\frac{1}{2}+\frac{1}{ 2}\text{erf}(\sqrt{a_{i}n_{i}P_{i}})\right],&P_{i}\geq P_{i}^{\text{th}}\\ 0,&P_{i}<P_{i}^{\text{th}}.\end{cases} \]

If the \(i\)-th semi-passive tag can be activated, the reflection ratio satisfies \(n_{i}^{\text{min}}\leq n_{i}\leq 1\). To power up the \(i\)-th semi-passive tag, the energy harvested during both the sleep and active states should be sufficient to cover the circuit energy consumption, resulting in the following circuit power constraint:

\[P^{c}t_{i}^{a}\leq\eta P_{i}h_{i}(T-t_{i}^{a})+\eta(1-n_{i})P_{i}h_{i}t_{i}^{a },\quad\forall i.\]

Based on the above discussions, the optimization problem of maximizing the total goodput is given as follows\[