## 3 Proposed Sensing Strategy
### Proposed Sensing Strategy  
Detection probability is beneficial to secure communication of PU and available throughput indicates the communication capacity of SU, while the false-alarm probability is to the disadvantage of CR. Consequently, we proposed an improved sensing strategy as follows:  
\[ \begin{array}{ll}F&=\left(P_{d}+R(\varepsilon,\tau)-P_{f}\right)_{\max}\\ &=\left.\left[P_{d}+\text{C}(1-P_{f})-P_{f}\right]_{\max}\right.\\ &=\left.\left[P_{d}-(C+1)P_{f}+\text{C}\right]_{\max},\right.\\ \\ &\text{C}=\frac{T-\tau}{T}\text{C}_{0}P(H_{0}).\end{array}  \tag{18}\]  
**Theorem 2**. : _Given \(\gamma,\sigma_{n}^{2},\tau\) and \(f_{s},\)\(F\) reaches the maximum, when_  
\[\varepsilon=\varepsilon_{1}=\left(\frac{1}{2}+\sqrt{\frac{\gamma}{2}+\frac{1} {4}+\frac{(2\gamma+1)\ln(\text{C}+1)\sqrt{2\gamma+1}}{\gamma\tau f_{s}}}\right) \sigma_{u}^{2}.\]  
**Proof. ** According to (3), (4) and (18),  
\[\frac{\partial F}{\partial\varepsilon}=\frac{\partial P_{d}}{\partial\varepsilon }-(\mathrm{C}+1)\frac{\partial P_{f}}{\partial\varepsilon}. \tag{19}\]  
Based on (3), (4) and (5), we can get  
\[\frac{\partial F}{\partial\varepsilon}\ \leq\ 0\Leftrightarrow\varepsilon\ \geq\ \sqrt{\frac{1}{4}+\frac{\gamma}{2}+\frac{(2\gamma+1)\ln(\mathrm{C}+1)\sqrt{2 \gamma+1}}{\gamma\tau f_{\mathrm{s}}}}\sigma_{u}^{2}. \tag{20}\]  
Thus, \(F\) monotonically decreases with the internal \(\varepsilon\in[\varepsilon_{1},+\infty)\), while it monotonically increases as \(\varepsilon\in(-\infty,\varepsilon_{1})\). Finally, \(F\) obtains the maximum value when  
\[\varepsilon=\varepsilon_{1}= \left(\frac{1}{2}+\sqrt{\frac{\gamma}{2}+\frac{1}{4}+\frac{(2 \gamma+1)\ln(\mathrm{C}+1)\sqrt{2\gamma+1}}{\gamma\tau f_{\mathrm{s}}}} \right)\sigma_{u}^{2}.\]  
Compared with the classical sensing strategy in (9), the main difference between them is shown as follows:  
\[\varepsilon=\varepsilon_{0} = \left(\frac{1}{2}+\sqrt{\frac{\gamma}{2}+\frac{1}{4}+\frac{(2 \gamma+1)\ln(2\gamma+1)}{2\gamma\tau f_{\mathrm{s}}}}\right)\sigma_{u}^{2} \tag{21}\] \[\Leftrightarrow\left(P_{m}+P_{f}\right)_{\min};\]  
\[\varepsilon=\varepsilon_{1} = \left(\frac{1}{2}+\sqrt{\frac{\gamma}{2}+\frac{1}{4}+\frac{(2 \gamma+1)\ln(\mathrm{C}+1)\sqrt{2\gamma+1}}{\gamma\tau f_{\mathrm{s}}}}\right) \sigma_{u}^{2} \tag{22}\] \[\Leftrightarrow\left(P_{d}-(\mathrm{C}+1)P_{f}+\mathrm{C}\right) _{\max}.\]  
From (21) and (22), \(\varepsilon_{1}>\varepsilon_{0}\) (\(\mathrm{C}>0\)). According to (3), (4) and (8), \(P_{d}|\varepsilon_{1}<P_{d}|\varepsilon_{0}\); \(P_{f}\big{|}\varepsilon_{1}<P_{f}\big{|}\varepsilon_{0}\); \(R(\varepsilon,\tau)\big{|}\varepsilon_{1}>R(\varepsilon,\tau)\big{|}\varepsilon _{0}\). When \(\varepsilon\) varies from \(\varepsilon_{0}\) to \(\varepsilon_{1}\), the variation percentage for detection probability, the false-alarm probability and achievable throughput in equal weight is approximately defined as  
\[\Delta P_{d}=P_{d}|\varepsilon=\varepsilon_{1}-P_{d}|\varepsilon=\varepsilon _{0}<0; \tag{23}\]  
\[\Delta P_{f}=P_{f}\big{|}\varepsilon=\varepsilon_{1}-P_{f}\big{|}\varepsilon= \varepsilon_{0}<0; \tag{24}\]  
\[\frac{\partial F}{\partial\varepsilon}=\frac{\partial P_{d}}{\partial\varepsilon }-(\mathrm{C}+1)\frac{\partial P_{f}}{\partial\varepsilon}. \tag{19}\]0  
Note that the fall of detection probability is to the disadvantage of protecting PU for a CR while the decline of false alarm probability is beneficial to available throughput of SU. In consequence, we define (26) to compare \(\Delta P_{d}\Delta P_{f}\) and \(\Delta R(\varepsilon,\tau)\).