## 3 Proposed Sensing Strategy
### Three Classical Sensing Strategies  
Table 1 shows comparisons of three classical sensing strategies, where \(P_{f-0}\) and \(P_{d-0}\), respectively, signify the maximum false-alarm probability and minimum detection probability that a CR could allow. For Sensing Strategy 1, the detection probability is maximized, which indicates the best protection for the PU. However, the corresponding false alarm probability is not the lowest. This manifests that Sensing Strategy 1 is suitable for the case where there is not much requirement for communication capacity and quality of secondary network. The communication capacity of the secondary network is optimal for Sensing Strategy 2, while its protection towards PU is limited. Different from Sensing Strategy 1 and 2, Sensing Strategy 3 concentrates on a tradeoff between PU protection and communication capacity of the secondary network, where \(P_{m}(\varepsilon,\tau)\) denotes probability of miss and \(P_{m}(\varepsilon,\tau)=1-P_{d}(\varepsilon,\tau)\). This is conducive to balancing the performance of a registered user and cognitive user and promotes the development and application of CR. As a result, this paper attempts to make a further optimization on Sensing Strategy 3.  
\[\frac{\partial P_{f}}{\partial\varepsilon} =-\frac{\sqrt{\tau f_{s}}}{\sqrt{2\pi}\sigma_{u}^{2}}\exp\left\{- \frac{1}{2}\left[\left(\frac{\varepsilon}{\sigma_{u}^{2}}-1\right)\sqrt{\tau f _{s}}\right]^{2}\right\}<0. \tag{12}\] \[\frac{\partial G}{\partial\varepsilon} =\frac{\sqrt{\tau f_{s}}}{\sqrt{2\pi}\sigma_{u}^{2}}\left\{\frac{ 1}{\sqrt{2\gamma+1}}\exp\left[-\frac{1}{2}\left(\frac{\varepsilon-\sigma_{u}^ {2}-\sigma_{s}^{2}}{\sigma_{u}^{2}}\sqrt{\frac{\tau f_{s}}{2\gamma+1}}\right)^ {2}\right]\right.\] (13) \[\left.\qquad-\exp\left[-\frac{1}{2}\left(\frac{\varepsilon-\sigma_ {u}^{2}}{\sigma_{u}^{2}}\sqrt{\tau f_{s}}\right)\right\}.\]  
For \(\frac{\sqrt{\tau f_{s}}}{\sqrt{2\pi}\sigma_{u}^{2}}>0\), we have  
\[\frac{\partial G}{\partial\varepsilon}\underset{\geq}{\leq}0 \Leftrightarrow\frac{\exp\left[-\frac{1}{2}\left(\frac{\varepsilon-\sigma_{u} ^{2}-\sigma_{s}^{2}}{\sigma_{u}^{2}}\sqrt{\frac{\tau f_{s}}{2\gamma+1}}\right) ^{2}\right]}{\exp\left[-\frac{1}{2}\left(\frac{\varepsilon-\sigma_{u}^{2}}{ \sigma_{u}^{2}}\sqrt{\tau f_{s}}\right)^{2}\right]}\underset{\geq}{\leq}\sqrt{ 2\gamma+1}, \tag{14}\] \[\Leftrightarrow\left(\varepsilon-\frac{1}{2}\sigma_{u}^{2}\right) \underset{\geq}{\leq}\frac{(2\gamma+1)\sigma_{u}^{4}}{2\gamma \tau f_{s}}\ln(2\gamma+1)+\frac{\gamma}{2}\sigma_{u}^{4}+\frac{1}{4}\sigma_{ u}^{4}. \tag{15}\]  
Then, we get  
\[\frac{\partial G}{\partial\varepsilon}\underset{\geq}{\leq}0 \Leftrightarrow\varepsilon\underset{\geq}{\leq}\left(\frac{1}{2}+\sqrt{ \frac{\gamma}{2}+\frac{1}{4}+\frac{(2\gamma+1)\ln(2\gamma+1)}{2\gamma\tau f_{ s}}}\right)\sigma_{u}^{2}. \tag{16}\]  
So, we can find that \(G\) monotonically increases within the internal \(\varepsilon\in[\varepsilon_{0},+\infty)\) while it monotonically decreases if \(\varepsilon\in(-\infty,\varepsilon_{0})\). That is, \(G\) reaches the minimum, when  
\[\varepsilon=\varepsilon_{0}=\left(\frac{1}{2}+\sqrt{\frac{\gamma}{2}+\frac{1} {4}+\frac{(2\gamma+1)\ln(2\gamma+1)}{2\gamma\tau f_{s}}}\right)\sigma_{u}^{2}.\]