# On Multi-Hop Decode-and-Forward Cooperative Relaying for Industrial Wireless Sensor Networks
## Appendix B Channel Capacity Performance Analysis  
### Derivation of the Average Capacity in Proposition 4  
To derive the closed-form expression for average capacity, we first obtain the following alternative expression of the logarithm term in (23) by expanding it to Taylor series ([51], p. 68):  
\[\ln\Bigl{(}1+\frac{\gamma_{\epsilon}}{\beta_{n}}\Bigr{)} =\sum_{q=1}^{\infty}\frac{2}{2q-1}\biggl{(}1-\frac{2\beta_{n}}{ \gamma_{\epsilon}+2\beta_{n}}\biggr{)}^{2q-1} \tag{101}\] \[=\sum_{q=1}^{\infty}\sum_{k=0}^{2q-1}\frac{2}{2q-1}\binom{2q-1}{k }\cdot\biggl{(}-\frac{2\beta_{n}}{\gamma_{\epsilon}+2\beta_{n}}\biggr{)}^{k}. \tag{102}\]  
Substituting (102) and (23) into (27), we can obtain the following expression for the end-to-end average capacity \(C_{avg}\) of the multi-hop network:  
\[C_{avg} =\sum_{n=0}^{\infty}\frac{\alpha_{n}}{\ln(2)}\cdot\int_{0}^{ \infty}\sum_{q=1}^{\infty}\sum_{k=0}^{2q-1}\frac{2}{2q-1}\cdot\binom{2q-1}{k} \cdot\biggl{(}-\frac{2\beta_{n}}{\gamma_{\epsilon}+2\beta_{n}}\biggr{)}^{k} \cdot f_{\gamma_{\epsilon}}(\gamma_{\epsilon})\,d\gamma_{\epsilon} \tag{103}\] \[=\sum_{n=0}^{\infty}\sum_{q=1}^{\infty}\sum_{k=0}^{2q-1}\frac{ \exp(-A)\cdot A^{n}}{\ln(2)\,n!}\cdot\frac{2\cdot(-1)^{k}}{2q-1}\cdot\binom{2q -1}{k}\cdot\mathrm{E}\Big{[}\Bigl{(}\frac{2\beta_{n}}{\gamma_{\epsilon}+2 \beta_{n}}\Bigr{)}^{k}\Big{]}. \tag{104}\]  
It is obvious from (104) that to calculate the average capacity, we need to derive the moments of the ratio \(\frac{2\beta_{n}}{\gamma_{\epsilon}+2\beta_{n}}\). By definition, the \(k\)-th moment \(\mathrm{E}[\bigl{(}\frac{2\beta_{n}}{\gamma_{\epsilon}+2\beta_{n}}\bigr{)}^{k}]\) is formulated as  
\[\mathrm{E}\Big{[}\Bigl{(}\frac{2\beta_{n}}{\gamma_{\epsilon}+2\beta_{n}} \Bigr{)}^{k}\Big{]}=\int_{0}^{\infty}\Bigl{(}\frac{2\beta_{n}}{\gamma+2\beta_{ n}}\Bigr{)}^{k}\cdot f_{\gamma_{\epsilon}}(\gamma)\,d\gamma=\frac{L}{\Gamma(m)} \cdot\Bigl{(}\frac{mN_{0}}{\Omega E_{b}P_{Rx,\ell+1}}\Bigr{)}^{m}\cdot I_{a}, \tag{105}\]  
where the integral \(I_{a}\) is given as  
\[I_{a}=\int_{0}^{\infty}\gamma^{m-1}\cdot\Bigl{(}\frac{2\beta_{n}}{\gamma+2 \beta_{n}}\Bigr{)}^{k}\cdot\exp\Bigl{(}-\frac{mN_{0}\gamma}{\Omega E_{b}P_{Rx, \ell+1}}\Bigr{)}\cdot\Bigl{[}\widehat{\Gamma}\Bigl{(}m,\frac{mN_{0}\gamma}{ \Omega E_{b}P_{Rx,\ell+1}}\Bigr{)}\Bigr{]}^{L-1}d\gamma. \tag{106}\]  
To obtain the closed-form expression for integral \(I_{a}\) in (105), we follow the similar rationale in [52], namely, representing the Gamma function as a sum of terms. The normalized upper incomplete Gamma function can be rewritten as an finite sum of terms ([46], p. 899), i.e.,  
\[\widehat{\Gamma}\Bigl{(}m,\frac{mN_{0}\gamma}{\Omega E_{b}P_{Rx,\ell+1}}\Bigr{)} =\exp\Bigl{(}\frac{-mN_{0}\gamma}{\Omega E_{b}P_{Rx,\ell+1}}\Bigr{)}\cdot \Bigl{[}\sum_{\
u=0}^{m-1}\frac{1}{\
u!}\Bigl{(}\frac{mN_{0}}{\Omega E_{b}P_{Rx, \ell+1}}\gamma\Bigr{)}^{\
u}\Bigr{]}. \tag{107}\]  
Next, we define \(\varphi=(mN_{0}\gamma)/\left(\Omega E_{b}P_{Rx,\ell+1}\right)\) and \(\psi_{\
u}=1/(\
u!)\). According to the polynomial theory, the finite series in (107) corresponds to a polynomial \(Q_{1}\) in \(\varphi\) with degree \((m-1)\) and coefficient \(\psi_{v}\) ([46], p. 17). Then, the \((L-1)\)-th power of \(Q_{1}\) is equivalent to another polynomial \(Q_{2}\) with degree \((L-1)(m-1)\) as follows  
\[\Big{[}\sum_{v=0}^{m-1}\psi_{v}\cdot\varphi^{v}\Big{]}^{L-1}=\sum_{v=0}^{(L-1)(m -1)}\omega_{v}\cdot\varphi^{v}, \tag{113}\]  
where the coefficients \(\omega_{v}\) are expressed as  
\[\omega_{0}=1,\quad\omega_{1}=L-1,\quad\omega_{(L-1)(m-1)}=\frac{1}{[(m-1)!]^{ L-1}}, \tag{114}\]  
and  
\[\omega_{v}=\frac{1}{v}\sum_{\lambda=1}^{\Lambda}\frac{\lambda L-v}{\lambda!} \omega_{v-\lambda},\quad v=2,3,\ldots,(L-1)(m-1)-1 \tag{115}\]  
are computed recursively with \(\Lambda=\min\{v,m-1\}\).