## 2 System Models
### Csin Model  
According to the collected data from field measurements over powerline channels, researchers have reported that the dominant noise source in NB-PLC systems can be modeled as a cyclostationary process, which possess impulsive property and with a period equal to half the period of mains voltage [7; 31; 32; 33; 34]. The cyclostationary process is refereed to as CSIN and its sample function can be modeled by a zero-mean function and time-varying variance function \(\sigma_{\eta}^{2}[n]\) Gaussian random process. Note that this variance function exhibits periodicity with period \(N_{\eta}\), i.e., \(\sigma_{\eta}^{2}[n]=\sigma_{\eta}^{2}[n+kN_{\eta}]\) with \(k\in\mathbb{Z}\). In the current study, we used the piecewise spectral cyclostationary Gaussian model (PSCGM) as our CSIN model, based on IEEE 1901.2 standard [13]. Figure 2 presents the block diagram of PSCGM used to generate CSIN. The multiplexer (MUX) was used to select one branch from the \(M\) regions to the output. In most cases, \(M\) falls between 2 and 4, and here we assume there are \(M=3\) regions of temporal and spectral shaping functions. The CSIN process \(\eta[n]\) can be expressed as:  
\[ \begin{array}{rl}\eta[n]&=\sum_{t}h[n,\tau]x_{in}[\tau]\\ &=\sum_{i=1}^{M}\mathbb{I}_{n\in\mathbb{R}_{i}}\sum_{\tau}t_{i}[\tau]h_{i}[\tau ]x_{in}[n-\tau]\end{array}  \tag{3}\]  
where \(x_{in}[n]\) is the AWGN input excitation signal; \(t_{i}[n]\) and \(h_{i}[n]\) respectively indicate the temporal and spectral shaping functions for the \(i\)-th region of a period; \(\mathbb{R}_{i}\) denotes the set of discrete time indices for the \(i\)-th region of a period; \(H_{i}(z)\) represents the Z-transform of \(h_{i}(n)\); \(\mathbb{I}_{n}\) denotes the indicator function;  
Figure 2: Block diagram of the piecewise spectral cyclostationary Gaussian model (PSCGM) for generating cyclostationary impulsive noise (CSIN). \(0\leq n<N_{\eta}\); and \(N_{\eta}=0.5f_{s}\cdot T_{\mbox{AC}}\) is the period of the CSIN, where \(T_{\mbox{AC}}\) denotes the period of main voltage and \(f_{s}\) is the sampling rate of the received signal. For a random process \(\eta[n]\), the corresponding instantaneous autocorrelation function with lag \(m\) can be defined as  
\[R_{\eta}[n,n+m]=\mathbb{E}\left\{\eta^{*}\left[n\right]\cdot\eta \left[n+m\right]\right\}, \tag{4}\]  
where \(\mathbb{E}\left\{\cdot\right\}\) denotes the expectation operation and the superscript \(*\) represents complex conjugate operation of a complex number. Furthermore, the periodicity of CSIN results in the periodicity of its instantaneous autocorrelation function. Therefore, we can rewrite (4) in its Fourier series (FS) synthesis form as follows:  
\[R_{\eta}[n,n+m]=\sum_{\alpha_{k}}R_{\eta}[m;\alpha_{k}]e^{j2\pi \alpha_{k}n}, \tag{5}\]  
where \(\alpha_{k}\) is referred to as the \(k\)-th cyclic frequency connected with the FS coefficients \(R_{\eta}[m;\alpha_{k}]\) (cyclic auto-correlation function), which be expressed as follows [35]:  
\[R_{\eta}[m;\alpha_{k}]=\lim_{N_{L}\rightarrow\infty}\frac{1}{N_{L}}\sum_{n=-N _{L}/2}^{N_{L}/2}R_{\eta}[n,n+m]e^{-j2\pi\alpha_{k}n}, \tag{6}\]  
where \(N_{L}\) is the observation window size. Note also that the cyclic spectrum density (CSD) function and the cyclic auto-correlation function are Fourier transform pair. Hence, the CSD function can be expressed as follows:  
\[S_{\eta}(f;\alpha_{k})=\sum_{m=-\infty}^{\infty}R_{\eta}[m;\alpha_{k}]e^{-j2 \pi fm}. \tag{7}\]  
We also adopted a parameter referred to as the Gaussian-to-impulsive-noise ratio (GINR) to evaluate the impact of the CSIN on BER performance in NB-PLC systems as follows:  
\[\Gamma:=\sigma_{\
u}^{2}/\sigma_{\overline{\eta}}^{2}, \tag{8}\]  
where \(\sigma_{\
u}^{2}\) denotes the variance of the background zero-mean AWGN \(v[n]\) and \(\sigma_{\overline{\eta}}^{2}\) represents the averaged variance of the CSIN \(\eta[n]\) in one period. The value of GINR indicates the strength of impulsive noise. The value of \(\Gamma\) is inversely proportional to the magnitude of the impulsive noise.