# Research on Hybrid Relay Protocol Design and Cross-Layer Performance Based on NOMA
## 3 Cross-Layer Performance Analysis of Two-Hop Relay System Based on NOMA
### Physical Layer Outage Probability  
This paper considers that both wireless and power line channels are affected by multiplicative fading and additive noise. Wireless channel fading satisfies the Nakagami-m distribution, and power line channel involves LogN distribution fading and Bernoulli-Gaussian pulse noise. Let \(h_{\text{SR}_{k}}\) represent the channel fading factor of the wireless channel, which satisfies the Nakagami distribution:  
\[f_{h_{\text{SR}_{k}}}(x)=\frac{2}{\Gamma\big{(}m_{w_{k}}\big{)}}\bigg{(}\frac{ m_{w_{k}}}{\Omega_{w_{k}}}\bigg{)}^{m_{w_{k}}}x^{2m_{w_{k}}-1}e^{(-\frac{m_{w_{k}} \!^{2}}{\Omega_{w_{k}}})},k\in(1,2), \tag{9}\]  
where \(m_{w_{k}}\) is the Nakagami distribution parameter, and \(m_{w_{k}}\geq 0.5\); \(\Gamma(x)\) is the gamma function; \(\Omega_{w_{k}}\) represents the mean of the fading amplitude, that is, \(\Omega_{w_{k}}=\text{E}\big{(}\big{|}h_{\text{SR}_{k}}\big{|}^{2}\big{)}\). In order to ensure that the fading does not change the average power of the received signal, the normalization is carried out and let \(\Omega_{w_{k}}=1\). \(h_{\text{SR}_{k}}\) is known to obey the Nakagami distribution, then \(\big{|}h_{\text{SR}_{k}}\big{|}^{2}\) satisfies the Gamma distribution \(G(a_{w},\beta_{w})\), and according to the nature of the Gamma function, \(\big{|}h_{\text{SR}_{k}}\big{|}^{2}\sim G\big{(}m_{w_{k}},\Omega_{w_{k}}/m_{w_ {k}}\big{)}\), then its PDF and cumulative distribution function (CDF) are shown as follows:  
\[f_{\big{|}h_{\text{SR}_{k}}\big{|}^{2}}(x)=\bigg{(}\frac{m_{w_{k}}}{\Omega_{ w_{k}}}\bigg{)}^{m_{w_{k}}}\frac{x^{m_{w_{k}}-1}}{\Gamma\big{(}m_{w_{k}} \big{)}}x, \tag{10}\]  
\[F_{\big{|}h_{\text{SR}_{k}}\big{|}^{2}}(x)=1-e^{(-\frac{m_{w_{k}}\!^{2}}{\Omega _{w_{k}}})}\sum_{m=0}^{m_{w_{k}}-1}\frac{\big{(}\big{(}m_{w_{k}}/\Omega_{w_{k}} \big{)}x\big{)}^{m}}{m!}. \tag{11}\]Let \(h_{\text{RD}_{j}}\) represents the channel fading factor of the power line channel, satisfying the lognormal distribution:  
\[f_{h_{\text{RD}_{j}}}(x)=\frac{1}{\sqrt{2\pi}\sigma_{j}x}\exp\left(-\frac{\left( \ln x-\mu_{j}\right)^{2}}{2\sigma_{j}^{2}}\right),j\in(1,2), \tag{12}\]  
where \(\mu_{j}\) and \(\sigma_{j}\) are the mean and standard deviation of variables \(\ln h_{RD_{j}}\), respectively. Normalizing the fading energy, that is, \(E(h_{\text{RD}_{j}}^{2})=\exp(2\mu_{\text{RD}_{j}}+2\sigma_{\text{RD}_{j}}^{2 })=1\), yields \(\mu_{\text{RD}_{j}}=-\sigma_{\text{RD}_{j}}^{2}\). By the nature of the logarithmic normal distribution, if \(x\sim\log\text{N}(\mu,\sigma^{2})\), then \(x^{2}\sim\log\text{N}(2\mu,4\sigma^{2})\). So \(h_{\text{RD}_{j}}^{2}\sim\log\text{N}(2\mu_{j},4\sigma_{j}^{2})\), its PDF and CDF are as follows:  
\[f_{\left|h_{\text{RD}_{j}}\right|^{2}}(x)=\frac{1}{2\sigma_{j}\sqrt{2\pi}\cdot x }e^{-\frac{\left(\ln(x)-2\mu_{j}\right)^{2}}{24\sigma_{j}^{2}}}, \tag{13}\]  
\[F_{\left|h_{\text{RD}_{j}}\right|^{2}}(x)=Q\left(\frac{2\mu_{j}-\ln(x)}{4 \sigma_{j}^{2}}\right). \tag{14}\]  
The outage probability can be defined as the probability that the SNR falls below a certain threshold.