## 4 Optimal Sensing Settings for SU Spectral Efficiency
### Reporting Scheduling Strategy
 \tag{38}\]  
\[ \begin{array}{ll}Q_{f\_AND}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M)=p _{f,1}(\gamma,M+\Delta M)p_{f,2}(\gamma+\Delta\gamma,M)\\ =p_{f,1}(\gamma,M)p_{f,2}(\gamma,M)+p_{f,1}(\gamma,M)\frac{\partial p_{f,2}( \gamma,M)}{\partial\gamma}\Delta\gamma+p_{f,2}(\gamma,M)\frac{\partial p_{f,1 }(\gamma,M)}{\partial M}\Delta M\\ +\frac{\partial p_{f,1}(\gamma,M)}{\partial M}\frac{\partial p_{f,2}(\gamma,M )}{\partial\gamma}\Delta\gamma\Delta M\end{array}  \tag{39}\]  
Then, we have  
\[Q_{f\_AND}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M)\leq Q_{f\_AND}^{(2) }(\gamma,M+\Delta M;\gamma+\Delta\gamma,M) \tag{40}\]  
In the case of two SUs, when the sensing results of SU with higher received SNR is reported later, which gets more compressed samples under AFR, it will achieve better cooperative false alarm probability. The reporting turn of \(K\) SUs can be attained by comparing that of any two SUs. Thus, in order to achieve better performance of spectrum reuse with the sufficient protection to PU, reporting sensing results to FC in the ascending sort of received SNR \(\gamma_{i}\) would be the obligatory scheduling strategy for the proposed novel CSS framework. When OFR is used at FC, according to Equation (18), the individual target detection probability at the \(i\)-th SU is \(p_{d,i}^{th}=1-\sqrt[k]{1-Q_{d}^{th}}\) then the individual false alarm probability \(p_{f,i}\) can be derived from Equation (15). Mathematically, the optimization goal can be formulized as:  
\[ \begin{array}{ll}min:&Q_{f\_OR}\\ s.t.:&Q_{f\_OR}=1-\prod\limits_{i=1}^{K}(1-p_{f,i})\\ &Q_{d\_OR}\geq Q_{d}^{th}\end{array}  \tag{41}\]Similarly, we take two SUs to investigate the reporting scheduling strategy for the novel CSS framework. The cooperative false alarm probabilities of two SUs under OFR are, respectively, given as:  
\[Q_{f\_\_\_\_\_\_\_\_\_}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M)=1-\Big{(}1 -p_{f,1}(\gamma,M)\Big{)}\Big{(}1-p_{f,2}(\gamma+\Delta\gamma,M+\Delta M)\Big{)} \tag{42}\]  
\[Q_{f\_\_\_\_}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M)=1-\Big{(}1-p_{f, 1}(\gamma,M+\Delta M)\Big{)}\Big{(}1-p_{f,2}(\gamma+\Delta\gamma,M)\Big{)} \tag{43}\]  
Theoretically, \(Q_{f\_\_\_\_\_}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M)\) and \(Q_{f\_\_\_\_\_\_}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M)\) can be approximately represented as:  
\[ \begin{array}{l}Q_{f\_\_\_\_\_}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M )=1-\Big{(}1-p_{f,1}(\gamma,M)\Big{)}\Big{(}1-p_{f,2}(\gamma+\Delta\gamma,M+ \Delta M)\Big{)}\\ =p_{f,1}(\gamma,M)+p_{f,2}(\gamma,M)-p_{f,1}(\gamma,M)p_{f,2}(\gamma,M)\\ +\Big{(}1-p_{f,1}(\gamma,M)\Big{)}\Big{(}\frac{\partial p_{f,2}(\gamma,M)}{ \partial M}\Delta M+\frac{\partial p_{f,2}(\gamma,M)}{\partial\gamma}\Delta \gamma\Big{)}\end{array}  \tag{44}\]  
\[ \begin{array}{l}Q_{f\_\_\_\_\_\_}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M )=1-\Big{(}1-p_{f,1}(\gamma,M+\Delta M)\Big{)}\Big{(}1-p_{f,2}(\gamma+\Delta \gamma,M)\Big{)}\\ =p_{f,1}(\gamma,M)+p_{f,2}(\gamma,M)-p_{f,1}(\gamma,M)p_{f,2}(\gamma,M)+\Big{(} 1-p_{f,1}(\gamma,M)\Big{)}\\ \times\Big{(}\frac{\partial p_{f,2}(\gamma,M)}{\partial M}\Delta N+\frac{ \partial p_{f,2}(\gamma,M)}{\partial\gamma}\Delta\gamma\Big{)}-\frac{ \partial p_{f,2}(\gamma,M)}{\partial M}\frac{\partial p_{f,2}(\gamma,M)}{ \partial\gamma}\Delta\gamma\Delta M\end{array}  \tag{45}\]  
Apparently, we have  
\[Q_{f\_\_\_\_\_\_}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M)\leq Q_{f\_\_ \_\_\_\_\_}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M) \tag{46}\]  
In the case of two SUs under OFR, it performs better in cooperative false alarm probability when higher SNR gets less compressed samples. Then we can attain \(K\) SUs reporting turn by comparing any two SUs. In order to utilize spectrum fully with the sufficient protection to PU, it is required to deliver sensing results to FC in the descending sort of received SNR \(\gamma_{i}\) for the novel CSS framework.