# On Multi-Hop Decode-and-Forward Cooperative Relaying for Industrial Wireless Sensor Networks
## 3 Bit Error Rate Performance Analysis
### The End-to-End Average BER  
**Proposition 1**.: _The average BER \(\mathcal{P}_{\ell}\) of the M-ary phase shift keying (M-PSK) for the \(\ell\)-th hop over the Nakagami-m fading channel in the presence of impulsive noise is given by_  
\[\mathcal{P}_{\ell} =\sum_{n=0}^{\infty}\frac{\exp(-A)\cdot A^{n}}{n!}\cdot\frac{(m \cdot N_{0})^{2m}}{(\Omega E_{b}P_{Rx,\ell+1})^{2m}}\cdot\frac{1}{\zeta_{M}} \sum_{i=1}^{max(\frac{M}{4},1)}\left[1-\sqrt{\frac{2A(1+\rho)mN_{0}\cdot\eta_{ i}^{2}}{\pi(n+A\rho)\Omega E_{b}\cdot P_{Rx,l+1}}}\right.\] \[\quad\cdot\left(\frac{\Omega E_{b}P_{Rx,\ell+1}}{m\cdot N_{0}} \right)^{(m+\frac{1}{2})}\cdot{}_{2}F_{1}\left(1,m+\frac{1}{2};\frac{3}{2}; \frac{A(1+\rho)\Omega E_{b}P_{Rx,\ell+1}\cdot\eta_{i}^{2}}{2 mN_{0}(n+A\rho) +\Omega E_{b}P_{Rx,\ell+1}A(1+\rho)}\right)\] \[\quad\cdot\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\cdot\frac{1}{ \left(\frac{mN_{0}}{\Gamma E_{b}P_{Rx,\ell+1}}+\frac{A(1+\rho)\eta_{i}^{2}}{2 (n+A\rho)}\right)^{m+\frac{1}{2}}}\Bigg{]}, \tag{18}\]  
_where \(\zeta_{M}=\max(\log_{2}\mathrm{M},2)\), \(\eta_{i}=\sin(\frac{(2-1)\cdot\pi}{M})\) and \({}_{2}F_{1}(a,b;c;x)\) represents the Gauss hypergeometric function ([46], p. 1005) defined by_  
\[{}_{2}F_{1}(a,b;c;x)=\sum_{p=0}^{\infty}\frac{(a)_{p}(b)_{p}}{(c)_{p}}\frac{x^ {p}}{p!}, \tag{19}\]  
_with \((g)_{p}=g(g+1)\cdots(g+p-1)\) denoting the ascending factorial._  
Proof.: See Appendix A.1.  
**Remark**.: _The average BER \(\mathcal{P}_{\ell}\) of equiprobable binary phase shift keying (BPSK) modulated symbols for the \(\ell\)-th hop over the Nakagami-m fading channel in the presence of impulsive noise can be readily obtained from (18) as follows:_  
\[\mathcal{P}_{\ell} =\sum_{n=0}^{\infty}\frac{\exp(-A)\cdot A^{n}}{2n!}\cdot\frac{(m \cdot N_{0})^{2m}}{(\Omega E_{b}P_{Rx,\ell+1})^{2m}}\cdot\left[1-\sqrt{\frac{ 2A(1+\rho)mN_{0}}{\pi(n+A\rho)\Omega E_{b}P_{Rx,\ell+1}}}\cdot\frac{\Gamma(m+ \frac{1}{2})}{\Gamma(m)}\right.\] \[\quad\cdot\left(\frac{\Omega E_{b}P_{Rx,\ell+1}}{m\cdot N_{0}} \right)^{(m+\frac{1}{2})}{}_{2}F_{1}\left(1,m+\frac{1}{2};\frac{3}{2};\frac{A( 1+\rho)\Omega E_{b}P_{Rx,\ell+1}}{2 mN_{0}(n+A\rho)+\Omega E_{b}P_{Rx,\ell+1}A (1+\rho)}\right)\] \[\quad\cdot\frac{1}{\left(\frac{mN_{0}}{\Gamma E_{b}P_{Rx,\ell+1} }+\frac{A(1+\rho)}{2(n+A\rho)}\right)^{m+\frac{1}{2}}}\Bigg{]}. \tag{20}\]  
**Proposition 2**.: _If the average BER per hop is the same for all hops (namely, \(\mathcal{P}_{\ell}=\mathcal{P}_{0}\), \(\ell=1,\cdots,L\)), then the end-to-end average BER \(\mathcal{P}_{L}^{E}\) of \(L\)-hop the wireless DF relaying system can be expressed as:_  
\[\mathcal{P}_{L}^{E}=\frac{1}{2}\Big{[}1-(1-2\mathcal{P}_{0})^{L}\Big{]}. \tag{21}\]  
Proof.: See Appendix A.2.  
Figure 4 shows the relationship between the end-to-end average BER of the \(L\)-hop relay system and the single-hop average BER in log-log representation assuming the identical statistical behavior for all single hops. From Figure 4, we can observe a nearly linear relationship between the two BERs when the single-hop average BER is below some threshold. This linearity can be theoretically proved by noticing that the linear term will dominate for low values of \(\mathcal{P}_{0}\) after extending the power series in (21) ([46], p. 25).