## 4 Optimal Sensing Settings for SU Spectral Efficiency
### Optimal Sensing Settings of the Novel ST-CSS Framework with Maximal SU Spectral Efficiency
In the proposed novel ST-CSS scheme, the interference to PU caused by the secondary network mainly results from missed detection of PU, a high probability of detection is required to ensure tolerable interference caused by SUs. For example, in IEEE 802.22 WRAN, the target detection probability is selected to be 90% for an SNR of \(-\)20 dB for the primary signal at SU [28]. We focus on the joint optimization of the reporting scheduling strategy and fusion scheme to maximize the transmission spectral efficiency with adequate protection to PU. The optimization problem can be modeled as:  
\[ \begin{array}{l}\max:\hat{\mathcal{C}}\\ s.t.,\,\hat{Q}_{d}\geq\hat{Q}_{d}^{th};1\leq n\leq K\end{array}  \tag{59}\]  
**Theorem 4. **\(\hat{\mathcal{C}}\) _is a decreasing function of \(\hat{Q}_{d}\)._  
**Proof**. For the given \(n\), we have:  
\[\frac{d\hat{\mathcal{C}}}{d\hat{Q}_{d}}=-\frac{T-\tau-Kt_{r}-t_{0}}{WT}\left(P _{\mathcal{H}_{0}}r_{0}\frac{d\hat{Q}_{f}}{d\hat{Q}_{d}}+P_{\mathcal{H}_{1}}r _{1}\right) \tag{60}\]  
\[ \begin{array}{ll}\frac{d\hat{\mathcal{O}}_{f}}{d\hat{Q}_{d}}&=\frac{d\hat{ \mathcal{O}}_{f}}{d\hat{p}_{f,i}}\Big{/}\frac{d\hat{Q}_{d}}{d\hat{p}_{d,i}} \times\frac{d\hat{p}_{f,i}}{d\hat{p}_{d,i}}\\ &=\frac{K\left(\begin{array}{c}K-1\\ n-1\end{array}\right)\left(\hat{p}_{f,i}\right)^{n-1}\left(1-\hat{p}_{f,i} \right)^{K-n}}{K\left(\begin{array}{c}K-1\\ n-1\end{array}\right)\left(\hat{p}_{d,i}\right)^{n-1}\left(1-\hat{p}_{d,i} \right)^{K-n}}\times\frac{d\hat{p}_{f,i}}{d\hat{p}_{d,i}}\\ &=\left(\frac{\hat{p}_{f,i}}{d\hat{p}_{d,i}}\right)^{n-1}\left(\frac{1-\hat{p} _{f,i}}{1-\hat{p}_{d,i}}\right)^{K-n}\times\frac{d\hat{p}_{f,i}}{d\hat{p}_{d,i}} \end{array}  \tag{61}\]According to Equations (49) and (50), we have:  
\[\frac{d\hat{p}_{f,i}}{d\hat{p}_{d,i}}=\frac{d\hat{p}_{f,i}}{d\lambda_{i}}\Big{/} \frac{d\hat{p}_{d,i}}{d\lambda_{i}}>0 \tag{62}\]  
Ultimately, we have \(\frac{d\hat{Q}_{f}}{d\hat{Q}_{d}}>0\) and \(\frac{d\hat{\mathcal{C}}}{d\hat{Q}_{d}}<0\). Then, \(\hat{\mathcal{C}}\) is the decreasing function of \(\hat{Q}_{d}\). Theorem 4 is proven. \(\square\)  
The maximal spectral efficiency of the proposed novel ST-CSS scheme can be achieved if and only if when the following relationship establishes:  
\[\Gamma(\hat{p}_{d,i})=\hat{Q}_{d}-\hat{Q}_{d}^{th}=\sum_{j=n}^{K}\left( \begin{array} []{c}K\\ j\end{array} \right)\hat{p}_{d,i}^{j}(1-\hat{p}_{d,i})^{K-j}-\hat{Q}_{d}^{th}=0 \tag{63}\]  
In this section, we propose an algorithm based on Equations (55)-(63) to find the maximal spectral efficiency of the proposed novel ST-CSS scheme with its corresponding optimal reporting scheduling strategy when \(n\) takes \(1\), \(\lfloor K/2\rfloor\), and \(K\), respectively. Replacing \(\hat{p}_{d,i}\) in Equation (55) with the root of Equation (63), we can get the minimal cooperative false alarm from Equation (56), as well the optimal spectral efficiency from Equation (58). In the case of \(n=1\) and \(n=K\), the root of Equation (63) can be calculated via exponential operation. The key of the problem lies in how to calculate the root of Equation (63) when \(n=\lfloor K/2\rfloor\). Several algorithms, such as Newton-Raphson, SECOND and Bi-Section, have been employed to seek the root numerically. The Newton-Raphson has a faster convergence rate than the other algorithms since the explicit expression of derivative of \(\Gamma\) over \(\hat{p}_{d,i}\) exists. The implementation process of optimal sensing settings to achieve maximal SU spectral efficiency for the novel ST-CSS with hard decision fusion is presented in Algorithm 1. The initial values \(\mu_{1}\) and the accuracy tolerance \(\zeta\) f root are the two basic factors that affect the complexity and delay for this algorithm according to the convergence theorem of Newton-Raphson. The value of the former is related to \(\lfloor K/2\rfloor\) in the expression of Equation (63). Obviously, the former lies in the interzone (0.5, 1) because the target cooperative detection probability \(\hat{Q}_{d}^{th}\) is selected to be 90%.