## 3 Subchannel Selection Policies
\begin{array}{rcl}C_{M}^{\rho_{ps}}&\approx&\underset{\theta_{\mathbf{Q}}}{ \max}\overset{M}{\underset{j=1}{\sum}}\overset{j-1}{\underset{l=0}{\sum}} \Delta f\overset{N_{j}F_{l}^{j-1}}{N-l}\log\left[(N-l)\,\theta_{\mathbf{Q}_{j} }\right],\\ s.t.&\overset{M}{\underset{j=1}{\sum}}\theta_{\mathbf{Q}_{j}}=GN\theta_{ \mathbf{Q}_{j}},\ \ \ 0\leq\theta_{\mathbf{Q}_{j}}\leq GN\theta_{\mathbf{Q}}.\end{array}  \tag{27}\]  
The Lagrange multiplier algorithm can be used to solve the optimization problem in Equation (28) [19]:  
\[ \begin{array}{rcl}L(\theta_{\mathbf{Q}_{j}},\lambda)&=&\overset{M}{\underset {j=1}{\sum}}\overset{j-1}{\underset{l=0}{\sum}}\overset{N_{j}F_{l}^{j-1}}{ N-l}\log\left[(N-l)\,\theta_{\mathbf{Q}_{j}}\right]\\ &-\lambda\left(\overset{M}{\underset{j=1}{\sum}}\theta_{\mathbf{Q}_{j}}-GN \theta_{\mathbf{Q}}\right)\end{array}  \tag{28}\]  
where, \(\lambda\) is the Lagrangian coefficient. Taking the derivative with respect to the \(\theta_{\mathbf{Q}_{j}}\) on Equation (29) and setting it equal to zero gives:  
\[\theta_{\mathbf{Q}_{j}}^{*}=\frac{1}{\lambda^{*}}v_{j}, \tag{29}\]  
where, \(v_{j}\triangleq\sum_{-0}^{j-1}N_{j}F_{l}^{j-1}/N-l\). Substituting Equation (30) into Equation (28):  
\[\lambda^{*}=\frac{1}{GN\theta_{\mathbf{Q}}}\overset{N}{\underset{j=1}{\sum} }v_{j}. \tag{30}\]  
Substituting Equation (31) into Equation (30):  
\[\theta_{\mathbf{Q}_{j}}^{*}=GN\theta_{\mathbf{Q}}\frac{v_{j}}{\underset{j}{ \sum}}. \tag{31}\]  
Furthermore, according to Equation (3) and \(\theta_{\mathbf{Q}_{j}}\triangleq Q_{j}/KN_{0}\Delta f\), the optimal transmitting power vector \(\mathbf{P}_{s}^{*}\) with selection policy \(\rho_{ps}\) can be obtained as follows:\[\mathbf{P}_{s}^{*}=\frac{GQ}{\sum\limits_{j=1}^{M}v_{j}}\left(\frac{v_{1}}{g_{ps} ^{j}},\frac{v_{2}}{g_{ps}^{j}},\dots,\frac{v_{M}}{g_{ps}^{j}}\right) \tag{32}\]  
Equivalently, substituting Equation (33) into Equation (28) yields the approximated maximum achievable capacity of the SUTCH with selection policy \(\rho_{ps}\), which is shown in Equation (34):  
\[C_{M}^{\rho_{ps}}\approx\sum\limits_{j=1}^{M}\sum\limits_{l=0}^{j-1}\frac{ \Delta fN_{j}F_{l}^{j-1}}{N-l}\log\left[\left(N-l\right)GN\theta_{Q}\frac{v_{ j}}{\sum\limits_{j=1}^{M}v_{j}}\right] \tag{33}\]  
Considering \(\rho_{ss}\), the selection criteria can be assumed as follows:  
\[\psi\left(g_{ps}^{j},g_{ss}^{j}\right)=g_{ss}^{j} \tag{34}\]  
Consequently, \(\mu_{j}=g_{ss}^{j}\) and based on \(\mu_{j}\), \(j=1\), \(2\), \(\dots\), \(M\), the selection sequence is defined as follows:  
\[\gamma_{M}=\left(\mu_{1},\mu_{2},\dots,\mu_{m}\right)\overset{\Delta}{=}\rho_ {ss}(\mu_{1},\mu_{2},\dots\mu_{m}). \tag{35}\]  
where \(\mu_{1}\geq\mu_{2}\geq\dots\geq\mu_{M}\). Using order statistics [21], the probability distribution function of \(\mu_{j}\), \(\forall j\) is shown as follows:  
\[k_{j}(\mu)=N_{j}F_{u}^{N-j}(\mu)\left[1-F_{\mu}(\mu)\right]^{j-1}f_{\mu}(\mu), \tag{36}\]  
Using a binomial expansion to replace \((1-e^{-\mu})^{N-j}\) in Equation (37) one obtains:  
\[k_{J}(\mu)=N_{j}\sum\limits_{l=0}^{N-j}F_{l}^{N-j}e^{-\mu(l+j)}, \tag{37}\]  
where \(F_{l}^{N-j}\triangleq\left( \begin{array}{c}N-j\\ l\end{array} \right)(-1)^{l}\). Thus the optimization problem of maximizing the capacity of the SUTCH while satisfying the tolerable interference constraints of the PUs with selection policy \(\rho_{ps}\) is shown as follows:  
\[ \begin{array}{lll}C_{M}^{\rho_{ss}}&=&\max\limits_{\theta_{Q}}\sum\limits_{ j=1}^{M}\sum\limits_{l=0}^{N-j}\Delta f^{\frac{N_{j}F_{l}^{N-j}}{l+j}}\frac{ \theta_{Q_{j}}}{\frac{\theta_{Q_{j}}}{l+j}}\log\left(\frac{\theta_{Q_{j}}}{l+j }\right)\\ s.t.&\sum\limits_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq\theta_{Q_{j}} \leq GN\theta_{Q}.\end{array}