# Distributed-Satellite-Clusters-Based Spectrum Sensing with Two-Stage Phase Alignment
## 4 Sensing Performance Analysis  
The final objective of this paper is to improve the sensing performance of weak signals. In this section, by using the method of probability theory, we derive the closed-form \(P_{d}\) expression for the considered SS_DSC system. There are two parameters associated with spectrum sensing: \(P_{d}\) and the false alarm probability \(P_{f}\), and energy detection is the most popular spectrum sensing scheme. Therefore, \(P_{d}\) and \(P_{f}\) for energy detection on the non-fading channel gain can be expressed as  
\[ \begin{split} P_{d}&=Pr(Y>\lambda|H_{1})=Q_{u}( \sqrt{2\gamma},\sqrt{\lambda})\\ P_{f}&=Pr(Y>\lambda|H_{0})=\frac{\Gamma(u,\frac{ \lambda}{2})}{\Gamma(u)}\end{split}  \tag{11}\]  
where \(\lambda\) is detection threshold, \(u\) is the time-bandwidth product, \(Q_{u}(a,b)\) represents the Marcum-Q function, and \(\Gamma(u,\frac{\lambda}{2})\) refers to incomplete gamma function. After the RPC_TF algorithm, the sensing signal of each satellite achieve phase alignment at the master satellite. At this time, the SNR \(\gamma\) is approximately equal to \(\gamma_{opt}^{EGC}\). Thus, the PDF of \(\gamma\) can be shown as  
\[f_{\gamma}(\gamma)=\frac{1}{2}\sqrt{\frac{L}{\overline{\gamma}\gamma}}f_{Z} \bigg{(}\sqrt{\frac{L\gamma}{\overline{\gamma}}}\bigg{)} \tag{12}\]  
where \(Z=\sum_{i=1}^{L}h_{i}\), denotes the summation of \(L\) SR random variables. By substituting (12) in (11), \(P_{d}\) can be expressed as  
\[P_{d}=\int_{\gamma}Q_{u}\big{(}\sqrt{2\gamma},\sqrt{\lambda}\big{)}f(\gamma)d \gamma=\frac{1}{2}\int_{\gamma}Q_{u}(\sqrt{2\gamma},\sqrt{\lambda})\sqrt{ \frac{L}{\overline{\gamma}\gamma}}f_{Z}\bigg{(}\sqrt{\frac{L\gamma}{\overline {\gamma}}}\bigg{)}d\gamma \tag{13}\]  
Figure 2: The two-stage phase alignment frame diagram for SS_DSC system. From (13), we can see that the PDF of the combined signal amplitude \(Z\) is the key to solving the problem. Especially, the PDF of \(Z\) has been given as (2) when \(L=1\), with the help of (2) and ([25], Equation (22)), the closed-form formula for \(P_{d}\) under SR fading channel when \(L=1\) can be obtained, yielding  
\[ \begin{split} P_{d}^{1}=a\sum_{k=0}^{\infty}\bigg{(}\frac{(m_{s} )_{k}\delta^{k}}{(1)_{k}k!}&\bigg{(}\frac{\Gamma(k+1)}{\beta^{k +1}}-\bigg{(}\frac{1}{\eta\overline{\gamma}}\bigg{)}^{k+1}\exp\bigg{(}-\frac{ \lambda}{2}\bigg{)}\\ &\times\sum_{j=u}^{\infty}\bigg{[}\bigg{(}\frac{\lambda}{2}\bigg{)} ^{j}\frac{\Gamma(k+1)}{\Gamma(j+1)}\ _{1}F_{1}\big{(}k+1;j+1;\frac{\lambda u}{2\eta}\big{)}\bigg{]}\bigg{)} \bigg{)}\end{split}  \tag{14}\]  
where \(\eta=u+\frac{\beta}{\gamma}\). Next, we consider the summation of \(L(L\geqslant 2)\) random variables. An accurate PDF expression of the amplitude of the combined signal is given by Theorem 1. **Theorem 1**. : _An accurate PDF expression of the sum of i.i.d. SR fading random variables \(Z=\sum_{i=1}^{L},L\geqslant 2\) is given by_  
\[f_{L}(Z)=\Xi e^{-\beta Z^{2}}Z^{2\big{(}\sum_{i=1}^{L}k_{i}+\sum_{i=1}^{L-1}m_ {i}\big{)}+2L-1} \tag{15}\]  
_where_  
\[ \begin{split}\Xi=&(2a)^{L}\sum_{k_{1}=0}^{\infty} \bigg{(}\frac{(m_{s})_{k_{1}}\delta^{k_{1}}}{(1)_{k_{1}}k_{1}!}\bigg{)}\prod_{ \rho=2}^{L}\bigg{(}\sum_{k_{p}=0}^{\infty}\frac{(m_{s})_{k_{\rho}}\delta^{k_{ \rho}}}{(1)_{k_{p}}k_{p}!}\\ &\sum_{l_{2p-3}=0}^{2\big{(}\sum_{i=1}^{p-1}k_{i}+\sum_{i=1}^{p- 2}m_{i}\big{)}+2\rho-3\big{)}}\bigg{(}2\big{(}\sum_{l=1}^{p-1}k_{i}+\sum_{i=1} ^{p-2}m_{i}\big{)}+2\rho-3\bigg{)}\sum_{l_{2p-2}=0}^{2k_{p}+1}\binom{2k_{\rho} +1}{t_{2\rho-2}}\\ &\frac{(-1)^{t_{2\rho-3}}+(-1)^{t_{2\rho-2}}}{2}\bigg{(}\frac{1} {2}\bigg{)}^{2\big{(}\sum_{i=1}^{p}k_{i}+\sum_{i=1}^{p-2}m_{i}\big{)}+2\rho-1 }\\ &\sum_{m_{\rho-1}=0}^{\infty}\frac{\Gamma\big{(}\frac{t_{2\rho-3} +t_{2\rho-2}+1}{2}\big{)}}{\Gamma\big{(}\frac{t_{2\rho-3}+t_{2\rho-2}+3}{2}+m_ {\rho-1}\big{)}}\bigg{(}\frac{\beta}{2}\bigg{)}^{m_{\rho-1}}\bigg{)}\end{split}  \tag{16}\]  
The proof is relegated to Appendix A. In Figure 3, the analytical and simulated PDFs of the sum of i.i.d.