## 2 System Model Network
### Information Transmission (IT)  
In this model, the IT phase is divided into two equal-length subintervals with the length _T_/2. In the first interval, we can calculate the received signal as  
\[y_{r}=\sqrt{1-\rho}h(x_{s}+\mu_{s})+n_{r}=\sqrt{1-\rho}hx_{s}+\sqrt{1-\rho}h\mu _{s}+n_{r} \tag{5}\]  
where \(x_{r}\) is the transmitted signal, which satisfies \(\mathrm{E}\left(\left|x_{r}\right|^{2}\right)=P_{r}\), \(\mu_{r}\) denotes the distortion error caused by hardware impairment at R, which is modeled as a zero-mean Gaussian random variable with variance \(P_{r}\sigma_{2}{}^{2}\) and \(\mathrm{E}\left(\left|\mu_{r}\right|^{2}\right)=P_{r}\sigma_{2}{}^{2}\), and \(n_{r}\) is the AWGN noise at R node. Here, we use the amplify-and-forward (AF) protocol for our model. Then, the received signal at R is amplified by a factor \(\beta\), which is given by Equation (6)  
\[\beta=\frac{x_{r}}{y_{r}}=\sqrt{\frac{P_{r}}{(1-\rho)|h|^{2}P_{s}+(1-\rho)|h|^{ 2}P_{s}\sigma^{2}+N_{o}}} \tag{6}\]  
In the remaining _T_/2 interval time, R transfers the information to D. Hence, the received signal at D node is formulated by  
Figure 2: Power splitting (PS) protocol. The cumulative density function (CDF) of RV \(\varphi_{1}\) can be written as  
\[F_{\varphi_{1}}(x)=1-e^{-\lambda_{b}x} \tag{14}\]  
Similarly, the PDF of RV \(\varphi_{2}\) can be obtained as in [26], giving  
\[f_{\varphi_{2}}(x)=\frac{(K+1)e^{-K}}{\lambda_{g}}e^{-\frac{(K+1)x}{\lambda_{g }}I_{0}}\Bigg{(}2\sqrt{\frac{K(K+1)x}{\lambda_{g}}}\Bigg{)} \tag{15}\]  
where \(\lambda_{g}\) is the mean value of RV \(\varphi_{2}\), K denotes the Rician K-factor, and \(I_{0}(\bullet)\) is the zero-th order modified Bessel function of the first kind [26]. Then the Equation (14) can be reformulated as the following  
\[f_{\varphi_{2}}(x)=a\underset{l=0}{\overset{\infty}{\underset{(l!)^{2}}{}}} \frac{(bK)^{l}}{(l!)^{2}}x^{l}e^{-bx} \tag{16}\]  
where we denote \(a=\frac{(K+1)e^{-K}}{\lambda_{g}},b=\frac{K+1}{\lambda_{g}}\) and \(I_{0}(x)=\underset{l=0}{\overset{\infty}{\underset{2^{2}}{}}}\frac{x^{2l}}{2^ {2}(l!)^{2}}\)[24]. The cumulative density function (CDF) of RV \(\varphi_{2}\) can be computed like in [27]  
\[F_{\varphi_{2}}(\varsigma)=\underset{0}{\overset{\zeta}{\underset{0}{}}}f_{ \varphi_{2}}(x)dx=1-\frac{a}{b}\underset{l=0}{\overset{\infty}{\underset{1}{ }}}\sum_{m=0}^{\underset{l}{\overset{K}{\underset{l}{}}}}\frac{K^{l}b^{m}}{ l!m!}\varsigma^{m}e^{-b\varsigma} \tag{17}\]  
After that, the OP of the proposed system can be computed as  
\[P_{out}=F_{\gamma\omega_{2}}(\gamma)=\Pr(\gamma_{\epsilon 2e}<\gamma) \tag{18}\]  
If we denote \(\gamma=2^{R}-1\) to be the lower threshold for SNR at both R and D, and R is fixed transmission rate at S, then Equation (18) can be reformulated as the following  
\[P_{out}=\Pr\Bigg{\{} \begin{array}{l}\varphi_{1}\varphi_{2}\gamma_{0}[1- \rho-\gamma\sigma_{1}^{2}-\gamma(1-\rho)\sigma_{2}^{2}\\ -\gamma(1-\rho)\sigma_{1}^{2}\sigma_{2}^{2}]<\varphi_{2}(\gamma+\gamma\sigma_ {2}^{2})+\gamma\kappa+\gamma\kappa\sigma_{1}^{2}\end{array} \Bigg{\}} \tag{19}\]  
Here we denote \(c_{1}=\gamma+\gamma\sigma_{2}^{2}\), \(c_{2}=\gamma\kappa+\gamma\kappa\sigma_{1}^{2}\), and \(c_{3}=\gamma_{0}[1-\rho-\gamma\sigma_{1}^{2}-\gamma(1-\rho)\sigma_{2}^{2}- \gamma(1-\rho)\sigma_{1}^{2}\sigma_{2}^{2}]\). We assume that \(c_{3}\) is positive, because if \(c_{3}\) is negative, the OP of the system is always equal to 1.  
a.