## 2 System Model
Since \(\Phi\) is an orthoprojector, Equation (11) can be rewritten:  
\[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-\[PERSON]\textit{ls})\right) \tag{3}\]2  
Based on the Johnson-Lindenstrauss (JL) lemma [PERSON] (1963), then with probability at least \(1-\delta\),  
\[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-\[PERSON]\textit{ls})\right) \tag{3}\]3  
Combining Equations (13) and (14), the result follows. \(\square\)It can be concluded that Equation (12) tells us how much information we lose via compressed samples in terms of our ability to solve our detection problem, precisely. Moreover, it is absolutely feasible to use Equation (15) to predict the probability of false alarm because \(\varepsilon\) is so small that the impact on \(p_{f}\) can be neglected with high probability. Although, for a given instance of \(\Phi\), the curve of relationship between \(p_{f}\) and compression ratio \(M/N\) may be better or worse than the predicted curve by Equation (15). \[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-\[PERSON]\textit{ls})\right) \tag{3}\]4  
Obviously, it is tightly concentrated around the expected performance curve as shown in \[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-\[PERSON]\textit{ls})\right) \tag{3}\]5  
Additionally, large numbers of trials of methods based on random orthoprojector, matrices with independent Gaussian entries and matrices with independent Bernoulli entries show that there is almost no difference among these three methods, and all perform highly consistent with expected. Actually, it is not necessary to limit \(\Phi\) to be orthoprojector to satisfy the foregoing Theorem 1 and Equation (15). During local compressed sensing, the \(i\)-th SU makes a binary decision (i.e., \"1\" for presence and \"0\" for absence) with probabilities of detection \(p_{d,i}\), \(i=1,2,...,K\) and false alarm \(p_{f,i}\), \(i=1,2,...,K\). Then, the FC combines all \"one bit\" decisions to make a global decision on the state of PU according to the hard counting fusion rule: if \(\Lambda<n\), decide \(\mathcal{H}_{0}\); else if \(\Lambda\geq n\), decide \(\mathcal{H}_{1}\), where \(\Lambda\) is the number of SU which reports presence of PU, and \(i=1,2,...,K\) is the decision threshold at FC. The expressions for cooperative false alarm probability \(Q_{f}\) and cooperative detection probability \(Q_{d}\) will be given as:  
\[Q_{f}=\sum_{j=n}^{K}\left( \begin{array}{c}K\\ j\end{array} \right)\times\left(\prod_{i=1}^{K}p_{f,i}\right)\times\left(\prod_{ i=j+1}^{K}(1-p_{f,i})\right),\,Q_{d}=\sum_{j=n}^{K}\left( \begin{array}{c}K\\ j\end{array} \right)\times\left(\prod_{i=1}^{j}p_{d,i}\right)\times\left(\prod_ {i=j+1}^{K}(1-p_{d,i})\right) \tag{16}\]  
In general, the counting fusion rule is broadly classified as OR fusion rule (OFR), AND fusion rule (AFR) and MAJORITY fusion rule (MFR). For AFR, \(n\) takes \(K\), we have:  
\[Q_{f\_AND}=\prod_{i=1}^{K}p_{f,i},\,Q_{d\_AND}=\prod_{i=1}^{K}p_{d,i} \tag{17}\]  
Figure 2: Effect of compression ratio on \(p_{f}\) at several different SNR (signal to noise ratio) levels (\(p_{d}^{th}=0.9\)). For OFR, \(n\) takes 1, we have:  
\[Q_{f\_OR}=1-\prod\limits_{i=1}^{K}\left(1-p_{f,i}\right)\text{, }Q_{d\_OR}=1-\prod\limits_{i=1}^{K}\left(1-p_{d,i}\right) \tag{18}\]  
For MFR, \(n\) takes the minimum integer greater than or equal to \(\frac{K}{2}\), denoted by \(\left\lfloor\frac{K}{2}\right\rfloor\).