# Optimal Resource Allocation Policies for Multi-User Backscatter Communication Systems
## 4 Optimal Resource Allocation of Passive Tags
It is not difficult to find \(P_{m}^{*}\), \(t_{m}^{a*}\), and \(n_{m}^{*}\), satisfying that \(t_{m}^{a*}=\frac{\eta P_{m}^{*}h_{m}T}{\mathbb{R}+\eta P_{m}^{*}h_{m}\hat{n}_{m }^{a}}\) with \(n_{m}^{*}=\hat{n}_{m}\), and \(P_{m}^{*}=\hat{P}_{m}\). Since \(t_{m}^{a}\left[\frac{1}{2}+\frac{1}{2}\text{erf}(\sqrt{a_{m}P_{m}n_{m}}) \right]\) is an increasing function of \(t_{m}^{a}\) and \(n_{m}\), respectively, it is evident that \(\sum_{m=1}^{M}t_{m}^{a*}\left[\frac{1}{2}+\frac{1}{2}\text{erf}(\sqrt{a_{m}P_ {m}^{*}n_{m}^{a}})\right]>\sum_{m=1}^{M}\hat{t}_{m}^{a}\left[\frac{1}{2}+\frac {1}{2}\text{erf}(\sqrt{a_{m}\hat{P}_{m}\hat{n}_{m}})\right]\). Therefore, the assumption is contradiction, completing the proof. ## References

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