# Research on Hybrid Relay Protocol Design and Cross-Layer Performance Based on NOMA
## 2 System Model and Signal Processing
### Signal Processing  
This paper considers a hybrid relay system based on NOMA, assuming that multiple stations at the source end simultaneously send data to the relay node R. When the relay node R works in half-duplex mode, sending and receiving are completed in two orthogonal time slots, with no self-interference item, then the relay node R receives the following signals through the wireless interface:  
\[y_{\mathrm{R}}=\sum_{k=1}^{n}H_{\mathrm{SR_{k}}}\sqrt{a_{k}P_{\mathrm{S}}}x_{k }+u_{w}, \tag{1}\]  
where \(P_{\mathrm{S}}\) is the total transmitted power of S, \(x_{k}\), \(k\in(1,2,\ldots,n)\) represents the data information of user \(D_{k}\), and \(a_{k}\) represents the power distribution factor of user \(D_{k}\). Without loss of generality, let \(a_{1}>a_{2}>\ldots>a_{n}\) and \(a_{1}+a_{2}+\ldots+a_{n}=1\). \(H_{\mathrm{SR_{k}}}=h_{\mathrm{SR_{k}}}/\left(d_{\mathrm{SR_{k}}}\right)^{v}\), \(k\in(1,2,\ldots,n)\). Figure 1: NOMA-based two-hop hybrid communication system model. \((1,2,\ldots,n)\), \(d_{\text{SR}_{k}}\) represents the link distance between source node \(\text{S}_{k}\) and relay node R. \(\
u\) is the path fading factor of the wireless channel, which is usually taken as \(1.6\ \sim 6\)[16]. \(n_{w}\) represents white gaussian noise satisfying the normal distribution \(N(0,N_{w})\). \(h_{\text{SR}_{k}}\) represents the channel fading coefficient of the wireless channel S-R link. Relay node R first decodes \(x_{1}\) using ideal SIC technology, and after successfully decoding and deleting it, R continues to decode \(x_{2}\) until all signals are decoded, then the signal-to-noise ratio (SNR) of the signal \(x_{1}\) received by R to detect its own message is as follows [17]:  
\[\gamma_{\text{R},x_{k}}=\frac{a_{k}\Delta_{w}d_{\text{SR}_{k}}^{-2w}\big{|}h_{ \text{SR}_{k}}\big{|}^{2}}{\sum_{i=k+1}^{n}a_{i}\Delta_{w}d_{\text{SR}_{i}}^{-2 w}\big{|}h_{\text{SR}_{i}}\big{|}^{2}+1},1\leq k\leq n-1, \tag{2}\]  
where \(\Delta_{w}=P_{S}/\sigma_{w}^{2}\) is the average signal-to-noise ratio of the wireless channel. Assume that relay node R detects \(x_{n}\) through the perfect SIC and its SNR is as follows:  
\[\gamma_{\text{R},x_{n}}=a_{n}\Delta_{w}d_{\text{SR}_{n}}^{-2w}\big{|}h_{\text{ SR}_{n}}\big{|}^{2} \tag{3}\]  
When signals are decoded, R re-encodes and multiplexes them, and transmits the multiplexed signal to the users through the PLC interface. The signal is as follows:  
\[x_{\text{R}}=\sum_{k=1}^{n}\sqrt{a_{k}P_{\text{R}}}x_{k}, \tag{4}\]  
where \(P_{\text{R}}\) represents the total transmitting power of relay node R.  
Then the received signal at \(\text{D}_{j}\) is as follows:  
\[y_{\text{D}_{j}}=H_{\text{RD}_{j}}x_{\text{R}}+n_{p},j\in(1,2,\ldots,n), \tag{5}\]  
where \(H_{\text{RD}_{j}}=h_{\text{RD}_{j}}/\left(d_{\text{RD}_{j}}\right)^{a}\). \(d_{\text{RD}_{j}}\) represents the link distance between relay node R and user D\({}_{j}\). \(\alpha\) is the path fading factor of the power line channel, which usually takes the value \(1.3\sim 3.3\)[18]. \(h_{\text{RD}_{j}}\) represents the channel fading coefficient of the power line channel. \(n_{p}\) represents power line channel noise, which adopts Bernoulli-Gaussian model and consists of background noise and pulse noise. Its probability density function (PDF) has the following form [19]:  
\[f\big{(}n_{p}\big{)}=(1-p)N(0,N_{G})+pN(0,N_{G}+N_{I}), \tag{6}\]  
where \(N(0,N_{G})\) and \(N(0,N_{G}+N_{I})\) represent normal distribution, respectively, \(p\) represents the probability of pulse noise occurrence, \(N_{G}\) and \(N_{I}\) represent the power of background noise and pulse noise, respectively, and the average total noise power is \(N_{p}=N_{G}+pN_{I}\). In order to simplify the model, let \(m=N_{G_{j}}/N_{I_{j}},j\in(1,2,\ldots,n)\) represent the ratio of background noise and pulse noise. On the destination terminal, each user applies SIC to remove some interference between users. Each user decodes strong signals first and regards other signals as interference until its own signal is decoded.