# Hybrid Active-Passive Reconfigurable Intelligent Surface for Cooperative Transmission Systems
## 2 System Model
### Active Mode  
During active mode, Switch 1 is \"on\", Switch 2 is \"off\", and all REs of HAPR are active elements. The transmit signal at S is \(x\), with \(E\left[\left|x\right|^{2}\right]=1\). In the meantime, the transmit power of S is defined as \(P_{S}\). The transmit signal from S that passes through the HAPR-assisted link reaches D, and the received signal at D is represented as follows:  
\[\begin{split} y_{a}&=\sum_{n=1}^{N}h_{D}e^{i\phi_{n }}\Big{(}\sqrt{P_{S}}a_{n}e^{i\phi_{n}}h_{S}e^{i\phi_{n}}x+a_{n}e^{i\phi_{n}}n_ {R}\Big{)}+z_{D}\\ &=\sqrt{P_{S}}\sum_{n=1}^{N}a_{n}h_{S}h_{D}e^{i(\theta_{n}+\varphi _{n}+\phi_{n})}x+\sum_{n=1}^{N}a_{n}h_{D}e^{i(\theta_{n}+\phi_{n})}n_{R}+z_{D }\end{split} \tag{1}\]  
where \(z_{D}\sim\mathcal{CN}\big{(}0,\sigma_{D}^{2}\big{)}\) stands for the additive white Gaussian noise (AWGN) at D, and \(n_{R}\sim\mathcal{CN}\big{(}0,\sigma_{R}^{2}\big{)}\) is the noise caused by the active RE. Hence, SNR at D could be expressed as  
\[\gamma_{a}=\frac{P_{S}\Big{|}\sum\limits_{n=1}^{N}a_{n}h_{S}h_{D}e^{i(\theta_{ n}+\varphi_{n}+\phi_{n})}\Big{|}^{2}}{\sum\limits_{n=1}^{N}\left|a_{n}h_{D}e^{i( \theta_{n}+\phi_{n})}\right|^{2}\sigma_{R}^{2}+\sigma_{D}^{2}} \tag{2}\]  
The first part is focused on the optimal phase design. Given that HAPR can dynamically change the incident signal's phase for maximum SNR, from [14] we learned that the optimal phase design is as follows,  
\[\theta_{n}=-(\varphi_{n}+\phi_{n}) \tag{3}\]  
Referring to [19], the high altitude of S and HAPR lead to the presence of line-of-sight (LOS) links, and each RE has the same large-scale path loss, so the equal-gain amplification strategy is adopted in this paper as follows,  
\[a_{1}=a_{2}...=a_{N}=a \tag{4}\]  
The reflection coefficient \(a\) is the second optimization problem we needed to pay attention to. The power consumption expression of the reflective amplifier in each active element was similar to that of the classical power amplifier,  
\[P_{RIS}=P_{C}+P_{DC}+\eta P_{oa} \tag{5}\]  
where \(P_{C}\), \(P_{DC}\), and \(P_{oa}\) are the power consumption of switching and control circuit, DC biased power consumption of each active element, and the output power of the reflective  
Figure 1: Hybrid active–passive RIS for cooperative communication system.  
amplifier, respectively. As the active RIS amplification signal works in the linear region, the output power is associated with the incident power [19], which can be  
\[P_{\alpha a}=a^{2}P_{ia}=a^{2}\Big{(}P_{S}h_{S}^{2}+\sigma_{R}^{2}\Big{)} \tag{6}\]  
In addition, \(\eta\triangleq\
u^{-1}\) is the reciprocal of the amplification efficiency. Considering the practical situation, RE has a pre-fixed power budget, which means that its amplification power is limited. At the same time, under active code, the reflection coefficient \(a\in(1,a_{\max})\). Therefore, the optimization problem of the reflection coefficient is expressed as  
\[\begin{split}\underset{a}{\max}& a\\ s.t.&\eta P_{\alpha a}\leq P_{RIS}-P_{\mathrm{C}}-P_{DC} \\ a&\leq a_{\max}\end{split} \tag{7}\]  
The optimal reflection coefficient could be obtained as follows,  
\[a=\min\left(\sqrt{\frac{P_{A}}{P_{S}h_{S}^{2}+\sigma_{R}^{2}}},a_{\max}\right), \tag{8}\]  
where \(P_{A}=\
u(P_{RIS}-P_{\mathrm{C}}-P_{\mathrm{DC}})\). Because of realistic factors, the reflection coefficient is hardly equal to \(a_{\max}\), so we adopt the more general case of \(a=\sqrt{\frac{P_{A}}{P_{S}h_{S}^{2}+\sigma_{R}^{2}}}\) in the follow-up research.  
By taking \(a=\sqrt{\frac{P_{A}}{P_{S}h_{S}^{2}+\sigma_{R}^{2}}}\) and (3) into (2), the SNR in active mode is obtained as follows,  
\[\gamma_{a}=\frac{N^{2}P_{S}P_{A}h_{S}^{2}h_{D}^{2}}{P_{S}\sigma_{D}^{2}h_{S}^{ 2}+NP_{A}\sigma_{R}^{2}h_{D}^{2}+\sigma_{R}^{2}\sigma_{D}^{2}} \tag{9}\]  
Under the condition of a high SNR, the typical calculation expression of the SNR at D is approximately:  
\[\gamma_{a}=\frac{N^{2}P_{S}P_{A}h_{S}^{2}h_{D}^{2}}{P_{S}\sigma_{D}^{2}h_{S}^{ 2}+NP_{A}\sigma_{R}^{2}h_{D}^{2}} \tag{10}\]