## 4 Optimal Sensing Settings for SU Spectral Efficiency
### Optimal Sensing Settings of the Novel ST-CSS Framework with Maximal SU Spectral Efficiency
Thus, the false alarm and detection probabilities of the weighted ST-CSS at \(i\)-th SU is given as:  
\[\hat{\rho}_{f,i}=Q\Bigg{(}\frac{\lambda_{i}-E(v_{i}|\eta=0)}{\sqrt{Var(v_{i}| \eta=0)}}\Bigg{)}=Q\Bigg{(}\frac{\lambda_{i}}{\sqrt{\frac{(\tau+(i-1)t_{i})f_ {s}}{NL}\sigma_{n}^{2}\sigma_{s,i}^{2}\sum\limits_{l=1}^{L}\omega_{i,l}^{2} \big{|}{h_{i,l}}\big{|}^{2}}}\Bigg{)} \tag{49}\]  
\[\hat{\rho}_{d,i}=Q\Bigg{(}\frac{\lambda_{i}-E(v_{i}|\eta=1)}{\sqrt{Var(v_{i}| \eta=1)}}\Bigg{)}=Q\Bigg{(}\frac{\lambda_{i}-\frac{(\tau+(i-1)t_{i})f_{s}}{NL }\sigma_{s,i}^{2}\sum\limits_{l=1}^{L}\omega_{i,l}\big{|}{h_{i,l}}\big{|}^{2} }{\sqrt{\frac{(\tau+(i-1)t_{i})f_{s}}{NL}\sigma_{n}^{2}\sigma_{s,i}^{2}\sum \limits_{l=1}^{L}\omega_{i,l}^{2}\big{|}{h_{i,l}}\big{|}^{2}}}\Bigg{)} \tag{50}\]  
Combining Equation (49) with Equation (50), \(\hat{\rho}_{f,i}\) is related to \(\hat{\rho}_{d,i}\) as follows:  
\[\hat{\rho}_{f,i}=Q\Bigg{(}Q^{-1}(\hat{\rho}_{d,i})+\frac{\sum\limits_{l=1}^{L} \omega_{i,l}\big{|}{h_{i,l}}\big{|}^{2}}{\sqrt{\sum\limits_{l=1}^{L}\omega_{i, l}^{2}\big{|}{h_{i,l}}\big{|}^{2}}}\sqrt{\frac{(\tau+(i-1)t_{r})f_{s}}{NL}\times \frac{\sigma_{s,i}^{2}}{\sigma_{n}^{2}}}\Bigg{)} \tag{51}\]  
In order to achieve the maximal spectrum utilization efficiency for the proposed novel ST-CSS scheme, we seek to minimize \(\hat{\rho}_{f,i}\), the optimization problem is formulated as follows:  
\[\min_{\omega_{i,1},\omega_{i,2},\cdots,\omega_{i,L}}\hat{\rho}_{f,i}=Q\Bigg{(} Q^{-1}(\hat{\rho}_{d,i})+\frac{\sum\limits_{l=1}^{L}\omega_{i,l}\big{|}{h_{i,l}} \big{|}^{2}}{\sqrt{\sum\limits_{l=1}^{L}\omega_{i,l}^{2}\big{|}{h_{i,l}}}\big{|} ^{2}}\sqrt{\frac{(\tau+(i-1)t_{r})f_{s}}{NL}\times\frac{\sigma_{s,i}^{2}}{ \sigma_{n}^{2}}}\Bigg{)} \tag{52}\]  
For a given \(\hat{\rho}_{d,i}\), because \(Q(\cdot)\) is a monotonously decreasing function, Equation (52), is converted to maximize  
\[\max_{\omega_{i,1},\omega_{i,2},\cdots,\omega_{i,L}}\frac{\sum\limits_{l=1}^{L }\omega_{i,l}\big{|}{h_{i,l}}\big{|}^{2}}{\sqrt{\sum\limits_{l=1}^{L}\omega_{i, l}^{2}\big{|}{h_{i,l}}\big{|}^{2}}} \tag{53}\]  
In order to minimize \(\hat{\rho}_{f,i}\) the optimal weighted coefficients should be obtained by letting:  
\[\omega_{i,l}=\frac{\big{|}{h_{i,l}}\big{|}^{2}}{\sqrt{\sum\limits_{l=1}^{L} \big{|}{h_{i,l}}\big{|}^{4}}}\,1\leq l\leq L \tag{54}\]Then, with a given \(\hat{p}_{d,i}\) at \(i\)-th SU, combining Equations (52) and (54), the minimum probability of false alarm \(\hat{p}_{f,i}\) is deduced as following:  
\[\hat{p}_{f,i}=Q\left(Q^{-1}(\hat{p}_{d,i})+\frac{\sum\limits_{l=1}^{L}\left|h_{ ij}\right|^{4}}{\sqrt{\sum\limits_{l=1}^{L}\left|h_{ij}\right|^{6}}}\sqrt{ \frac{(\tau+(i-1)t_{r})f_{s}}{NL}\times\frac{\sigma_{s,j}^{2}}{\sigma_{n}^{2} }}\right) \tag{55}\]  
With counting hard fusion rule deployed at FC, the final probabilities of detection and false alarm of the proposed ST-CSS scheme are given by:  
\[\hat{Q}_{f}=\sum\limits_{j=n}^{K}\left( \begin{array}{c}K\\ j\end{array} \right)\hat{p}_{f,i}^{j}\left(1-\hat{p}_{f,j}\right)^{K-j} \tag{56}\]  
\[\hat{Q}_{d}=\sum\limits_{j=n}^{K}\left( \begin{array}{c}K\\ j\end{array} \right)\hat{p}_{d,i}^{j}(1-\hat{p}_{d,i})^{K-j} \tag{57}\]  
Thus, for the proposed novel ST-CSS scheme with hard decision fusion rule, similar to Equation (20), and the latency \(t_{0}\) due to the very long distances that the signals must travel into space and back, which comprises of transmission from SU to FC and broadcasting the final decision on the state of PU from FC to SU is considered. The average SU transmission spectral efficiency is:  
\[\hat{\mathcal{C}}=\frac{T-\tau-Kt_{r}-t_{0}}{WT}\left(P_{\mathcal{H}_{0}}(1- \hat{Q}_{f})r_{0}+P_{\mathcal{H}_{1}}(1-\hat{Q}_{d})r_{1}\right) \tag{58}\]  
In the scenario of mobile satellite communication confronted with the problem of frequency scarcity, the degraded spectrum utilization will result in a greatly decreasing communication capacity or even interruption for mobile satellite service.