# Distributed-Satellite-Clusters-Based Spectrum Sensing with Two-Stage Phase Alignment
## Appendix A  
Recall that \(Z=\sum_{i=1}^{L}h_{i}\), yeilding \(f_{Z}\triangleq f_{h_{1}}*f_{h_{2}}*\cdots*f_{h_{L}}\), where \((*)\) denotes the statistical convolution between independent distributions. First, we study the case where \(Z=h_{1}+h_{2}\), and the respective PDF of \(Z\) is obtained as  
\[f_{2}(Z)=\int_{0}^{Z}f_{h_{2}}(Z-h_{1})f_{h_{1}}(h_{1})dh_{1} \tag{11}\]  
After some algebraic manipulations, \(f_{2}(Z)\) can be rewritten as  
\[\begin{split} f_{2}(Z)=&(2a)^{2}\sum_{k_{1}=0}^{ \infty}\frac{(m_{s})_{k_{1}}\delta^{k_{1}}}{(1)_{k_{1}}k_{1}!}\sum_{k_{2}=0}^{ \infty}\frac{(m_{s})_{k_{2}}\delta^{k_{2}}}{(1)_{k_{2}}k_{2}!}\\ &\underbrace{\int_{0}^{Z}\exp\big{(}-\beta((Z-x)^{2}+x^{2})\big{)} (Z-x)^{2k_{2}+1}x^{2k_{1}+1}dx}_{J1}\end{split} \tag{12}\]  
By substituting \(x-\frac{Z}{2}=\frac{Z}{2}\varrho\) and using the binomial theorem, \(J1\) can be expressed as  
\[\begin{split} J1=&\bigg{(}\frac{Z}{2}\bigg{)}^{2k_ {1}+2k_{2}+3}e^{-\beta\frac{Z^{2}}{2}}\sum_{t_{1}}^{2k_{1}+1}\binom{t_{1}}{2k _{1}+1}\sum_{t_{2}}^{2k_{2}+1}\binom{t_{2}}{2k_{2}+1}(-1)^{t_{2}}\\ &\int_{0}^{1}\varrho^{t_{1}+t_{2}}(1+(-1)^{t_{1}+t_{2}})e^{- \beta\frac{Z^{2}}{2}}e^{2}d\varrho\end{split} \tag{13}\]  
with the help of ([28], equation (3.383.1)), the integral can be evaluated as  
\[\begin{split} f_{2}(Z)=&(2a)^{2}\sum_{k_{1}=0}^{ \infty}\frac{(m_{s})_{k_{1}}\delta^{k_{1}}}{(1)_{k_{1}}k_{1}!}\sum_{k_{2}=0}^ {\infty}\frac{(m_{s})_{k_{2}}\delta^{k_{2}}}{(1)_{k_{2}}k_{2}!}\\ &\sum_{t_{1}}^{2k_{1}+1}\binom{t_{1}}{2k_{1}+1}\sum_{t_{2}}^{2k_ {2}+1}\binom{t_{2}}{2k_{2}+1}\frac{\big{(}(-1)^{t_{1}}+(-1)^{t_{2}}\big{)}}{2 }\bigg{(}\frac{1}{2}\bigg{)}^{2k_{1}+2k_{2}+3}\\ &\sum_{p=0}^{\infty}\frac{\Gamma\bigg{(}\frac{t_{1}+t_{2}+1}{2} \bigg{)}}{\Gamma\bigg{(}\frac{t_{1}+t_{2}+3}{2}+p\bigg{)}}\bigg{(}\frac{\beta} {2}\bigg{)}^{p}Z^{2k_{1}+2k_{2}+2p+3}e^{-\beta Z^{2}}\end{split} \tag{14}\]  
Next, let \(Z=h_{1}+h_{2}+h_{3}\). Using (2) and (14) into (11), the PDF of \(Z\) is given by\[\begin{split}& f_{3}(Z)=(2a)^{3}\sum_{k_{1}=0}^{\infty}\frac{(m_{s})_{k_{ 1}}\dot{g}^{k_{1}}}{(1)_{k_{1}}k_{1}!}\sum_{k_{2}=0}^{\infty}\frac{(m_{s})_{k_{ 2}}\dot{g}^{k_{2}}}{(1)_{k_{2}}k_{2}!}\sum_{k_{3}=0}^{\infty}\frac{(m_{s})_{k_{ 3}}\dot{g}^{k_{3}}}{(1)_{k_{3}}k_{3}!}\\ &\sum_{t_{1}}^{2k_{1}+1}\binom{t_{1}}{2k_{1}+1}\sum_{t_{2}}^{2k_ {2}+1}\binom{t_{2}}{2k_{2}+1}\frac{\left((-1)^{t_{1}}+(-1)^{t_{2}}\right)}{2} \left(\frac{1}{2}\right)^{2k_{1}+2k_{2}+3}\\ &\sum_{p=0}^{\infty}\frac{\Gamma\left(\frac{t_{1}+t_{2}+1}{2} \right)}{\Gamma\left(\frac{t_{1}+t_{2}+3}{2}+p\right)}\left(\frac{\beta}{2} \right)^{p}\\ &\sum_{t_{3}}^{2k_{1}+2k_{2}+2p+3}\binom{t_{3}}{2k_{1}+2k_{2}+2p+ 3}\sum_{t_{4}}^{2k_{3}+1}\binom{t_{4}}{2k_{3}+1}\frac{\left((-1)^{t_{3}}+(-1)^ {t_{4}}\right)}{2}\left(\frac{1}{2}\right)^{2k_{1}+2k_{2}+2k_{3}+2p+5}\\ &\sum_{r=0}^{\infty}\frac{\Gamma\left(\frac{t_{1}+t_{4}+1}{2} \right)}{\Gamma\left(\frac{t_{3}+t_{4}+3}{2}+r\right)}\left(\frac{\beta}{2} \right)^{r}\!Z^{2k_{1}+2k_{2}+2k_{3}+2p+2r+5}e^{-\beta Z^{2}}\end{split} \tag{10}\]  
Following a similar procedure for L steps, the generalized formulation of \(Z\) is obtained, as presented in (15).