## 4 Optimal Sensing Settings for SU Spectral Efficiency  
### Reporting Scheduling Strategy  
In the proposed CSS framework, if sensing SNR \(\gamma_{i}\) received at each SU is identical, the reporting order of cooperative SU makes no difference to sensing performance. However, the primary signal received at different SU are considered to be unequal due to varying channel conditions and diverse path loss caused by independent distance from PU to SU. To achieve the optimal cooperative sensing performance, we propose reporting scheduling strategy based on the received \(\gamma_{i}\) at the \(i\)-th SU. In general, the optimal reporting scheduling strategy changes with the varied fusion rules. In this section, the optimal scheduling strategy for AFR, OFR and MFR are investigated. The sensing false alarm probability and detection probability dictate the feasibility of SU operation, and there is typically a tradeoff between the two. The PU would prefer a high detection probability to alleviate interference. However, the SU would prefer a low false alarm probability to use the PU's frequency band for secondary data transmission, thereby increasing wireless efficiency. Generally, the Neyman-Pearson (NP) detector is the decision rule that maximizes detection probability \(P_{D}\) subject to the constraint that false alarm probability \(P_{F}\leq\alpha\). In practice, it is of interest to optimize performance by minimizing the false alarm probability given a target detection probability. Our object is to achieve the maximal spectral efficiency with sufficient protection to PU, which is equivalent to minimize the false alarm probability with other parameters constant. In respect of different fusion rules, our main task is to minimize the probability of cooperative false alarm \(Q_{f}\) with the target cooperative detection probability \(Q_{d}^{th}\) during the following formulas deduction. When AFR is applied at FC, according to Equation (17), the individual target detection probability at the \(i\)th SU is \(p_{d,i}^{th}=\sqrt[k]{Q_{d}^{th}}\), then the individual false alarm probability can be derived from Equation (15):  
\[p_{f,i}=Q\Bigg{(}Q^{-1}(p_{d,i}^{th})+\sqrt{\frac{1}{N}M_{i}\gamma_{i}}\Bigg{)} \tag{34}\]Mathematically, the objective optimization function can be modeled as:  
\[ \begin{array}{ll}min:&Q_{f\_AND}\\ s.t.:&Q_{f\_AND}=\prod\limits_{i=1}^{K}p_{f,i}\\ &Q_{d\_AND}\geq Q_{d}^{th}\end{array}  \tag{35}\]  
Primarily, we take \(K=2\) as an entrance to analyze the optimal reporting scheduling strategy. It is equivalent to sort SNR \(\{\gamma,\gamma+\Delta\gamma\}\) for the given compressed samples \(\{M,M+\Delta M\}\) with \(A_{2}^{2}\) kinds of permutation methods: \((\gamma,M;\gamma+\Delta\gamma,M+\Delta M)\) and \((\gamma,M+\Delta M;\gamma+\Delta\gamma,M)\). The false alarm probabilities of two cooperative SUs with AFR are computed by:  
\[Q_{f\_AND}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M)=p_{f,1}(\gamma,M)p_ {f,2}(\gamma+\Delta\gamma,M+\Delta M) \tag{36}\]  
\[Q_{f\_AND}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M)=p_{f,1}(\gamma,M+ \Delta M)p_{f,2}(\gamma+\Delta\gamma,M) \tag{37}\]  
It can be seen from Equation (15) that individual \(p_{f,i}\) decreases as the number of compressed samples \(M_{i}\) or SNR \(\gamma_{i}\) increases with the given \(p_{d,i}^{th}\). Hence, we have \(\frac{\partial p_{f,i}(\gamma_{i},M_{i})}{\partial\gamma_{i}}<0\), \(\frac{\partial p_{f,i}(\gamma_{i},M_{i})}{\partial M_{i}}<0\) and \(p_{f,1}(\gamma,M)=p_{f,2}(\gamma,M)\). According to the binary function differential equation, \(Q_{f\_AND}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M)\) and \(Q_{f\_AND}^{(2)}(\gamma,M+\Delta M;\gamma+\Delta\gamma,M)\) can be approximately expressed as:  
\[ \begin{array}{ll}Q_{f\_AND}^{(2)}(\gamma,M;\gamma+\Delta\gamma,M+\Delta M)=p_ {f,1}(\gamma,M)p_{f,2}(\gamma+\Delta\gamma,M+\Delta 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,1}(\gamma,M)\frac{\partial p_{f,2 }(\gamma,M)}{\partial M}\Delta M\end{array}