## 3 Subchannel Selection Policies
 \tag{8}\]  
Substituting Equation (9) into Equation (7), and integrating by part, Equation (10) can be gotten as follows, which is the simplified optimization problem of \(C_{M}^{\rho_{1}}\):\[C_{M}^{\rho_{1}}=\underset{\theta_{Q_{j}}}{\text{max}}\sum_{j=1}^{M} \Delta f\frac{\theta_{Q_{j}}}{\theta_{Q_{j}}-1}\text{log}\left(\theta_{Q_{j}}\right) \tag{9}\] \[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]  
where, \(\theta_{Q}\) is defined as the spectrum sharing load factor, and \(\theta_{Q}=(\theta_{Q_{1}},\theta_{Q2},\ldots,\theta_{QM})\) is defined as the spectrum sharing load vector:  
\[\theta_{Q}\overset{\Delta}{=}\frac{Q}{KN_{0}N\Delta f}=\frac{Q}{KN_{0}B} \tag{10}\]  
Furthermore, the following pseudo linear approximation is used to get an approximate solution for Equation (10) [16]:  
\[\frac{x}{x-1}\text{log}(x)\approx-1.2015-0.0052x+1.0772\times\text{log}\left(3.0262x+308829\right) \tag{11}\]  
Substituting Equation (12) into Equation (10), the Lagrangian function of the optimization problem Equation (10) is shown as follows [19; 20]:  
\[L(\theta_{Q},\lambda)= \sum_{j=1}^{M}-1.2015+-0.0052\times\theta_{Q_{j}}+1.0772\times \text{log}\left(3.0262\times\theta_{Q_{j}}+3.8829\right) \tag{12}\] \[-\lambda\left(\sum_{j=1}^{M}\theta_{Q_{j}}-GN\theta_{Q}\right)\]  
where \(\lambda\) is the Lagrangian coefficient. The derivative with respect to the \(\theta_{Q_{j}}\) on Equation (13) is taken, and then it is equal to zero, the following formula can be obtained:  
\[\theta_{Q_{j}}^{*}=\frac{1.0772}{\lambda^{*}+0.0052}-\frac{3.8829}{3.0262}. \tag{13}\]  
Substituting Equation (14) into Equation (10), the following formula can obtained:  
\[\sum_{j=1}^{M}\left[\frac{1.0772}{\lambda^{*}+0.0052}-\frac{3.8829}{3.0262} \right]=GN\theta_{Q} \tag{14}\]  
Equivalently, Equation (16) can be derived from Equation (15):  
\[\lambda^{*}=-0.0052+\frac{1.0772}{\frac{GN\theta_{Q}}{M}+\frac{3.8829}{3.0262}}. \tag{15}\]  
Eventually, substituting Equation (16) into Equation (14) gives:  
\[\theta_{\theta_{j}}^{*}=\frac{GN\theta_{Q}}{M},\quad j=1,2,\ldots,M \tag{16}\]  
Note that Equation (17) suggests that for given \(G,N,M\) and \(\theta_{Q}\), the maximum capacity is achieved by dividing the total acceptable interference \(GN\theta_{Q}\) into equal portions for \(M\) accessible subchannels. Actually, it is a direct consequence of selecting \(M\) out of \(N\) subchannels without any prior knowledge. Furthermore, according to Equation (3) and \(\theta_{Q_{j}}\triangleq Q_{j}/KN_{0}\Delta f\), the optimal transmitting power vector \(\mathbf{P}_{s}^{*}\) can be obtained as follows:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]0Equation (18) suggests that the interference share for each accessible subchannel \(j\), \(\theta_{Q_{j}}\) is mapped to the corresponding transmission power \(P_{sj}\), proportional to \(1/g_{ps}^{j}\). So, if \(g_{ps}^{j}\) is large, then the SUs will creates a large interference in the receivers of Pus. In this case, Equation (18) suggests a lower SUs transmission power in accessible subchannel \(j\). Equivalently, substituting Equation (18) into Equation (10), Equation (19) can be derived:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]1  
In a practical case, \(Q=G^{-1}N_{0}B\) and \(M<N\), the spectrum sharing load factor can be obtained from Equation (17) as \(\theta_{g_{j}}=N/KM\), which is much higher than unity. As mentioned above, \(\rho_{1}\) randomly choose subchannels, which ignores the fact that it is more reasonable for the SUs to allocate higher transmission power to certain subchannels because of their corresponding CSIs, so it is essential to discuss the non-uniform selection policy for SUTCH with a prior knowledge of CSIs pair (\(g_{ss}^{j}\), \(g_{ps}^{j}\)), since it will lead to a larger capacity or a smaller interference on the PUs.