# A Relay Optimization Method for NOMA-Based Power Line Communication Systems
## 2 System Model  
### Channel and Noise Model  
The power line communication system consists of a signal source (S) and \(n\) users (U), where users communicate through the power line. Users are sorted by their distance from the signal source as \(U_{i}\), where \(i=1,2,\ldots,n\). For example, a simplified structure with one source and five users is illustrated in Figure 2.  
Users are arranged in the order of distance from the source as \(U_{1}\) to \(U_{5}\). The channel gains between \(S-U_{1}\),\(\ldots\), \(S-U_{5}\), \(U_{1}-U_{3}\), \(U_{1}-U_{4}\), \(U_{2}-U_{4}\), and \(U_{3}-U_{4}\) are denoted as \(h_{su1}\),\(\ldots\), \(h_{u3u4}\), respectively. These gains are modeled as independent normal distribution random variables. The log-normal fading model is widely used in PLC systems as it describes network topology mismatch and multipath effects [19]. The probability density function (PDF) of these channel gains is given by [20; 21; 22; 23]:  
\[f(h_{i})=\frac{1}{\sqrt{2\pi}\sigma_{i}h_{i}}\text{exp}\left[-\frac{(\log h_{i }-\mu_{i})^{2}}{2\sigma_{i}^{2}}\right] \tag{1}\]  
where \(i\in\){\(S-U_{1}\), \(S-U_{2}\), \(\ldots\), \(S-U_{5}\), \(U_{1}-U_{3}\), \(U_{1}-U_{4}\), \(U_{2}-U_{4}\), \(U_{3}-U_{4}\)}, \(\mu_{i}\) and \(\sigma_{i}^{2}\) are the mean and standard deviation of the \(10 log_{10}(hi)\), respectively. \(\mu_{i}\) represents the mean logarithmic channel gain, while \(\sigma_{i}^{2}\) quantifies channel variability due to impedance mismatches. For instance, a higher \(\sigma_{i}^{2}\) indicates severe multipath fluctuations, common in branched powerline networks. This model effectively captures the characteristics of signal attenuation and multipath propagation in PLC environments. Furthermore, the channel attenuation due to distance and frequency dependence is denoted as \(A_{i}(d_{i},f)\), where f represents the communication frequency and \(d_{i}\) represents the distance between nodes \(i\in\){\(S-U_{1}\), \(S-U_{2}\), \(\ldots\), \(S-U_{5}\), \(U_{1}-U_{3}\), \(U_{1}-U_{4}\), \(U_{2}-U_{4}\), \(U_{3}-U_{4}\)} [24; 25; 26; 27]. Specifically, it is given by:  
\[A_{i}(f,d_{i})=e^{-(b_{i}+b_{i}f^{k})d_{i}} \tag{2}\]  
Figure 1: Full-text structure flow chart. Figure 2: Example of PLC structure for users. where \(k\) is the exponent of the attenuation coefficient, and \(b_{o}\) and \(b_{1}\) are constants obtained through measurements. Channel noise mainly includes two types: background noise and impulsive noise. Since the power spectral density of background noise is relatively low and changes slowly over time, it manifests as a slowly varying random process in the time domain. Therefore, it can be represented by a stationary random process [10; 28; 29], described using an autoregressive (AR) model. Given that the spectral characteristics of background noise in power line channels are similar to those of a stationary random process, and its variance is finite, the simplest stationary random process, the AR model, can be used for its modeling [27]. A time series signal can be represented by a \(p\)-order difference equation, as shown below:  
\[x(n)+a_{1}x(n-1)+\ldots+a_{p}x(n-p) =w(n)+b_{1}w(n-1)+\ldots+b_{q}w(n-q) \tag{3}\]  
where \(w(n)\) is zero-mean white noise with variance \(\sigma^{2}\). The AR model sets all coefficients \(b_{l}(i=1,2,\ldots,q)\) to zero:  
\[x(n)+a_{1}x(n-1)+\ldots+a_{p}x(n-p)=w(n) \tag{4}\]  
Its system function is given by:  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1+\sum_{i=1}^{p}a_{i}z^{-i}} \tag{5}\]  
From the above equation, it is known that the AR model parameters of the background noise in power line channels include the variance \(\sigma^{2}\) and \(p\) coefficients \(a_{k}(k=1,2,\ldots,p)\). Once these \(p+1\) coefficients are determined, the AR model can be established. The AR model approximates background noise as a linear combination of past samples plus white noise, widely used for stationary processes. However, impulsive noise does not exhibit the stationary characteristics of random signals.