## 3 Subchannel Selection Policies
Actually, an appropriate selection policy should consider the interference of the PUs receivers caused by SUs transmission. Such policy should select the lower subchannel gain of \(g_{ps}^{j}\), because it will create a lower interference in the receivers of the PUs. Therefore, a lower \(g_{ps}^{j}\) will give the SUs the flexibility of allocating a higher power, which will result in a higher capacity. Such a selection policy is named as SU-PU-based selection policy, which is simplified as \(\rho_{ps}\). In order to implement \(\rho_{ps}\), the SUs requires \(g_{ps}^{j}\) during each time-slot. Therefore, a signaling channel between the receivers of the PUs and the transmitters of the SUs is required. Similar to \(\rho_{ps}\), another selection policy can be derived. It will select those subchannels which achieve the highest capacity corresponding to allocating the transmitting power of SUs. Such policy selects the subchannel with the higher \(g_{ss}^{j}\), because it will create a higher power in the receivers of the SUs. Such selection is name as SU-SU-based selection policy, which is simplified as \(\rho_{ss}\). In order to implement \(\rho_{ss}\), the SUs requires \(g_{ss}^{j}\) during each time-slot. Therefore, a signaling channel between the receivers of the SUs and the transmitters of the SUs is also required. In the following, the maximum capacity is derived with different selection policy \(\rho_{ps}\) and \(\rho_{ss}\). Considering \(\rho_{ps}\), the selection criteria can be assumed as follows:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]2  
Consequently, \(\mu_{j}=g_{ps}^{j}\) and based on \(\mu_{j}\), \(j=1\), \(2\), \(\ldots\), \(M\), the selection sequence is defined as follows:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]3  
where \(\mu_{1}\leq\mu_{1}\leq\ldots\leq\mu_{M}\). Using order statistics [21], the probability distribution function of \(\mu_{j}\), \(\forall j\) is shown as follows:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]4  
where:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]5  
and \(f_{\mu}(\mu)\), \(F_{\mu}(\mu)\) are the probability density function and probability distribution function of \(\mu\). Assuming the same assumption as discussed above in Equation (9) we obtain:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]6Equivalently:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]7  
Using a binomial expansion to replace \((1-e^{-\mu})^{j-1}\) in Equation (25) gives:  
\[s.t.\quad\sum_{j=1}^{M}\theta_{Q_{j}}=GN\theta_{Q_{j}},\qquad 0\leq \theta_{Q_{j}}\leq GN\theta_{Q}\]8  
where, \(F_{l}^{j-1}\triangleq\left( \begin{array}{c}j-1\\ l\end{array} \right)(-1)^{j-1-l}\). Thus, the optimization problem of maximizing the capacity of SUTCH, while satisfying the tolerable interference constraint of the PUs with selection policy \(\rho_{ps}\) is shown as follows:  
\[ \begin{array}{rcl}C_{M}^{\rho_{ps}}&=&\underset{\theta_{\mathbf{Q}}}{\max} \overset{M}{\underset{j=1}{\sum}}\overset{j-1}{\underset{l=0}{\sum}}\Delta fN _{j}F_{l}^{j-1}\overset{\theta_{\mathbf{Q}_{j}}\log\left[(N-l)\theta_{\mathbf{ Q}_{j}}\right]}{(N-l)\theta_{\mathbf{Q}_{j}}-1},\\ s.t.&\overset{M}{\underset{j=1}{\sum}}\theta_{\mathbf{Q}_{j}}=GN\theta_{ \mathbf{Q}_{j}},\ \ 0\leq\theta_{\mathbf{Q}_{j}}\leq GN\theta_{\mathbf{Q}}.\end{array}  \tag{26}\]  
However, in practice, \(M<N\), thus, \(N\theta_{\mathbf{Q}_{j}}\gg 1\). Therefore, Equation (27) can be approximated by Equation (28):  
\[