## 2 System Model
For our hypotheses, the joint probability distribution function (PDF) of compressed samples of received signal under hypothesis \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\) can be derived as following:  
\[f_{0}(y|\mathcal{H}_{0}\,)=(2\pi)^{-N/2}\Big{|}\sigma_{n}^{2}\Phi\Phi^{T}\Big{|} ^{-1/2}\exp\left(-\frac{1}{2}y^{T}\Big{(}\sigma_{n}^{2}\[PERSON]\Phi^{T}\Big{)}^{- 1}y\right) \tag{2}\]  
\[f_{1}(y|\mathcal{H}_{1}\,)=(2\pi)^{-N/2}\Big{|}\sigma_{n}^{2}\[PERSON]\Phi^{T}\Big{|} ^{-1/2}\exp\left(-\frac{1}{2}(y-\[PERSON]\textit{ls})^{T}\Big{(}\sigma_{n}^{2}\[PERSON] \Phi^{T}\Big{)}^{-1}(y-\[PERSON]\textit{ls})\right) \tag{3}\]  
According to the likelihood ratio test of Neyman-Pearson (NP) optimal decision rule [38; 39]:  
\[\Lambda(y)=\frac{f_{1}(y|\mathcal{H}_{1})}{f_{0}(y|\mathcal{H}_{0})}\underset{ \mathcal{H}_{0}}{\overset{\mathcal{H}_{1}}{\gtr}}\varphi \tag{4}\]  
After taking logarithms on both sides, we can obtain an equivalent test that simplifies as:  
\[y^{T}\Big{(}\Phi\Phi^{T}\Big{)}^{-1}\Phi\textit{ls}\underset{\mathcal{H}_{0}} {\overset{\mathcal{H}_{1}}{\gtr}}\sigma_{n}^{2}\log(\varphi)+\frac{1}{2}( \textit{ls})^{T}\Phi^{T}\Big{(}\[PERSON]\Phi^{T}\Big{)}^{-1}\Phi\textit{ls}:=\lambda \tag{5}\]  
Figure 1: The system model of the CSS (cooperative spectrum sensing) for LEO (low earth orbit) mobile satellite services. We now define the output of the compressive detector as:  
\[v:=y^{T}\Big{(}\[PERSON]\Phi^{T}\Big{)}^{-1}\Phi hs \tag{6}\]  
Since \(v\) is a complete statistic for our detection problem, it contains complete information required to determine between \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\). It is easy to prove that the PDF of \(v\) can be approximated expressed by Gaussian distribution as follows:  
\[\mathcal{H}_{0}:v\sim N\Big{(}0,\sigma_{n}^{2}(hs)^{T}\[PERSON]^{T}(\[PERSON]\Phi^{T})^{ -1}\Phi hs\Big{)} \tag{7}\]  
\[\mathcal{H}_{1}:v\sim N\Big{(}(hs)^{T}\[PERSON]^{T}(\[PERSON]\Phi^{T})^{-1}\Phi hs, \sigma_{n}^{2}(hs)^{T}\[PERSON]^{T}(\[PERSON]\Phi^{T})^{-1}\Phi hs\Big{)} \tag{8}\]  
Thus, the false alarm probability \(p_{f}\) and detection probability \(p_{d}\) at SU can be computed by:  
\[p_{f}=\Pr(v>\lambda|\mathcal{H}_{0})=Q\Bigg{(}\frac{\lambda}{\sigma_{n}\sqrt{ \left(hs\right)^{T}\[PERSON]^{T}(\[PERSON]\Phi^{T})^{-1}\Phi hs}}\Bigg{)} \tag{9}\]  
\[p_{d}=\Pr(v>\lambda|\mathcal{H}_{1})=Q\Bigg{(}\frac{\lambda-(hs)^{T}\[PERSON]^{T}( \[PERSON]\Phi^{T})^{-1}\Phi hs}{\sigma_{n}\sqrt{\left(hs\right)^{T}\[PERSON]^{T}(\[PERSON] \Phi^{T})^{-1}\Phi hs}}\Bigg{)} \tag{10}\]  
where the function \(Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{+\infty}\exp(-y^{2}/2)dy\). From Equation (9), with target detection probability \(p_{d}^{th}\) to provide sufficient protection to PU, \(p_{f}\) is related \(p_{d}^{th}\) to as follows:  
\[f_{1}(y|\mathcal{H}_{1}\,)=(2\pi)^{-N/2}\Big{|}\sigma_{n}^{2}\[PERSON]\Phi^{T}\Big{|} ^{-1/2}\exp\left(-\frac{1}{2}(y-\[PERSON]\textit{ls})^{T}\Big{(}\sigma_{n}^{2}\[PERSON] \Phi^{T}\Big{)}^{-1}(y-\[PERSON]\textit{ls})\right) \tag{3}\]0  
Firstly, considering a special case in which is an orthoprojector (i.e., \(\Phi\Phi^{T}=I_{M}\)), Theorem 1 gives the bound of the compressive detector performance. **Theorem 1**. : _Let \(\Phi\) be \(M\times N\) random orthoprojector. Then, with probability at least of \(1-\delta\)._  
\[f_{1}(y|\mathcal{H}_{1}\,)=(2\pi)^{-N/2}\Big{|}\sigma_{n}^{2}\[PERSON]\Phi^{T}\Big{|} ^{-1/2}\exp\left(-\frac{1}{2}(y-\[PERSON]\textit{ls})^{T}\Big{(}\sigma_{n}^{2}\[PERSON] \Phi^{T}\Big{)}^{-1}(y-\[PERSON]\textit{ls})\right) \tag{3}\]1  
_where \(\delta\in(0,1)\), \(\varepsilon<\sqrt{12\log(2/\delta)/M}\), the compressed sensing SNR \(\gamma=\frac{\|hs\|_{2}^{2}}{\sigma_{n}^{2}}\), and compression ratio \(\mu=\frac{M}{N}\)._  
**Proof**.