# On Multi-Hop Decode-and-Forward Cooperative Relaying for Industrial Wireless Sensor Networks
## 2 System and Channel Model
### Channel Model  
The channel fading amplitudes \(h_{\ell}\) (\(\ell=1,\ldots,L\)) in (1) are modelled as independent and identically distributed (i.i.d.) Nakagami-\(m\) RVs. The PDF \(f_{h_{\ell}}(h)\) of the RVs \(h_{\ell}\) (\(\ell=1,\ldots,L\)) is expressed as  
\[f_{h_{\ell}}(h)=\frac{2m^{m}h^{2m-1}}{\Omega^{m}\Gamma(m)}\cdot\exp\Bigl{(}- \frac{mh^{2}}{\Omega}\Bigr{)},\hskip 14.226378 pth\geq 0 \tag{5}\]  
where \(\Gamma(\cdot)\) represents the complete Gamma function defined as \(\Gamma(\tau)=\int_{0}^{\infty}t^{\tau-1}\cdot\exp(-t)\,dt\), \(\Omega=\mathrm{E}[|h_{\ell}|^{2}]\) is the expectation of \(|h_{\ell}|^{2}\); and \(m\) is the Nakagami-\(m\) fading parameter, which determines the severity of fading channels and ranges from 0.5 to \(\infty\). The additive noise \(z_{\ell}\) (\(\ell=1,\ldots,L\)) in (1) are i.i.d. RVs modeled by the MCA model. The PDF \(f_{z_{\ell}}(z)\) of the RVs \(z_{\ell}\) is given by [41]  
\[f_{z_{\ell}}(z) =\sum_{n=0}^{\infty}\frac{\alpha_{n}}{\sqrt{2\pi}\sigma_{n}} \cdot\exp\Bigl{(}\frac{-z^{2}}{2\sigma_{n}^{2}}\Bigr{)} \tag{6}\] \[=\sum_{n=0}^{\infty}\alpha_{n}\cdot\mathcal{N}(z;0,\sigma_{n}), \tag{7}\]  
where \(\mathcal{N}(z;0,\sigma_{n})\) denotes a zero-mean Gaussian PDF with variance \(\sigma_{n}^{2}\); and the parameter \(\alpha_{n}\) is the Poisson-distributed probability expressed by  
\[\alpha_{n}=\frac{\exp(-A)\cdot A^{n}}{n!}. \tag{8}\]  
The variance \(\sigma_{n}^{2}\) in (6) and (7) is defined with the auxiliary RV \(\beta_{n}\) as follows:  
\[\sigma_{n}^{2} =(\sigma_{g}^{2}+\sigma_{i}^{2})\cdot\beta_{n} \tag{9}\] \[\beta_{n} =\frac{(n+A\rho)}{A\cdot(1+\rho)}, \tag{10}\]  
where \(\rho\) represents the ratio of the Gaussian noise power \(\sigma_{g}^{2}\) to the impulsive noise power \(\sigma_{i}^{2}\). The parameter \(A\) is called the impulsive index and defined as the product of average rate of impulsive noise and mean duration of the impulsive interference. The impulsive index determines impulsiveness the noise: a smaller value of the impulsive index implies a higher level of impulsive interference. It is known that as \(A\) approaches a value around 10 or larger, the MCA distribution is very close to a Gaussian PDF; while for \(A\) and \(\rho\) lower than 1, the PDF gets very heavy tails and the interference can be seen as very impulsive [26]. The extraction of MCA model parameters from real measurements can be done with the algorithms developed in (e.g., [42; 43]). It should be noted that the MCA model includes both the impact of impulsive interference as well as thermal noise in the communication systems [41]. Figure 2 shows the difference between pure AWGN and impulsive noise generated with the MCA model and Figure 3 illustrates the effects of the parameters \(A\) and \(\rho\) used in the MCA model. As can be seen from (7), the MCA model can be interpreted as a Gaussian-mixture model [44]. Therefore, it is straightforward to show that the mean of the noise \(u_{z_{\ell}}=E[z_{\ell}]=0\); and the average power \(N_{0}\) of the noise, i.e., the variance of \(z_{\ell}\), is calculated as  
\[N_{0} =E[z_{\ell}^{2}]=\sum_{n=0}^{\infty}\frac{\exp(-A)\cdot A^{n}}{n! \sqrt{2\pi\sigma_{n}}}\int_{-\infty}^{\infty}z^{2}\cdot\exp\!\left(-\frac{z^{2 }}{2\sigma_{n}^{2}}\right)dz \tag{11}\] \[=\frac{\exp(-A)\cdot\sigma_{g}^{2}}{\rho}\cdot\sum_{n=0}^{\infty }\frac{A^{n}}{n!}\cdot\left(\frac{n}{A}+\rho\right)=\sigma_{g}^{2}+\sigma_{i} ^{2}. \tag{12}\]  
Alternatively, the generation of MCA noise can be interpreted as a stationary random process: for each channel realization, the probability of experiencing an interference noise with distribution \(\mathcal{N}(z;0,\sigma_{n})\)\((n=0,\ldots,\infty)\) is \(\alpha_{n}\) given in (8). Despite the PDF in (7) has an infinite number of terms, we can safely restrict our analysis to the first \(N\) terms without significant loss by noticing that as \(n\) increases, the probability \(\alpha_{n}\) approaches zero and is thus negligible.