## 3 Subchannel Selection Policies
At a given time-slot, the power allocation policy of SUTCH is defined as \(\rho_{\psi}\), which is based on a selection criterion \(\psi(...)\), and set:  
\[\mu_{j}\stackrel{{\Delta}}{{=}}\psi\left(g_{ps}^{j},g_{ss}^{j}\right) \tag{1}\]  
For the observing random variables \(\mu_{j}\), \(j=1\), \(\ldots\), \(M\), the selection sequence \(\gamma_{M}\) is defined as follows:  
\[\gamma_{M}=\left(\mu_{\tau_{1}},\mu_{\tau_{2}},\ldots,\mu_{\tau_{M}}\right) \stackrel{{\Delta}}{{=}}\rho_{\psi}\left(\mu_{1},\mu_{2},\ldots, \mu_{M}\right) \tag{2}\]  
The \(M\)-tuple selection sequence is arranged, so that its first element is the most suitable subchannel for SUTCH based on the selection criteria in Equation (1). The probability distribution function of random variable \(\gamma_{j}\) is defined as \(k_{j}(\gamma)\), \(j=1\), \(\ldots\), \(M\). It is important to note that if \(j_{1},j_{2}\) are entities in \(\gamma_{M}\) and \(j_{1}<j_{2}\), then it can be considered that compared to the choice \(j_{2}\), the SUs can get a better performance by choosing subchannel with index \(j_{1}\). Suppose \(\psi(g_{ps}^{j},g_{ss}^{j})\) is constant, which means subchannels are considered equally. The SUs will randomly choose \(M\) out of \(N\) subchannels without any a priori information. This selection strategy is defined as the uniform subchannel selection, whereas, if the prior information of the subchannel obtained by cooperation or other techniques is \(-1\), the SUs will choose the corresponding value of \(\psi(g_{ps}^{j},g_{ss}^{j})\). This selection strategy is defined as the non-uniform selection strategy. The transmission power of the SUTCH in the subchannel \(j\) is referred to \(P_{sj}\). \(P_{s}(P_{s1},\ldots,P_{sM})\) is defined as the transmission power vector of SUTCH over \(M\) subchannels. Suppose that SUTCH accesses to the chosen subchannel \(j\) with the transmission power of \(P_{sj}\), and the corresponding interference at the receiver of the PUs is \(Q_{j}\), where:  
\[Q_{j}=g_{ps}^{j}P_{sj} \tag{3}\]  
Since the PUs utilize DS-CDMA with spreading gain \(G\), therefore, the narrow-band interference \(Q_{j}\) spreads over the whole bandwidth and manifests itself as an equivalent wide-band interference equal to \(G^{-1}Q_{j}\) at the receiver of the PUs. Suppose the SUTCH transmits with the transmission power vector \(P_{s}(P_{s1},P_{s2},\ldots,P_{sM})\) in \(M\) accessible subchannels. Correspondingly, an equivalent narrow-band interference vector \(\mathbf{Q}=(Q_{1},\,Q_{2},\ldots,\,Q_{M})\) will be imposed on the receivers of the PUs. Meanwhile, the SUCCH transmits with the transmission power vector \(\mathbf{P}_{sc}(P_{sc1},P_{s\leq 2},\ldots,\,P_{scN})\). Therefore, in order to comply with the interference threshold \(Q\) of the PUs, the constraint function is as follows:  
\[\frac{1}{G}\left(\sum_{j=1}^{M}g_{ss}^{j}P_{sj}+\sum_{i=1}^{N}g_{ps}^{j}P_{ sci}\right)\leq Q \tag{4}\]  
In this paper, the objective is to achieve the maximum capacity of SUTCH. As discussed above, the transmitting power of SUTCH in each accessible subchannel should be optimally allocated. Meantime, the interference threshold constraint should also be considered. Consequently, according to selection policy \(\rho_{\psi}\), for a given \(Q\) and for \(M\) accessible subchannels, the maximum capacity of SUTCH is defined as \(C_{M}^{\psi}\), which can be obtained by the following constrained optimization problem:  
\[ \begin{array}{ll}C_{M}^{\psi}&=\ \max\limits_{P_{s}}\sum\limits_{j=1}^{M} \Delta f\int\limits_{g_{ps}^{j}g_{ss}^{j}}\log\left(1+\frac{g_{ps}^{j}P_{sj}}{ KN_{0}\Delta f+g_{ps}^{j}P_{scj}}\right)\\ &\times f_{ss}^{j}\left(g_{ss}^{j}\right)f_{ps}^{j}\left(g_{ps}^{j}\right)dg_ {ss}^{j}dg_{ps}^{j}\\ s.t.&\frac{1}{G}\left(\sum\limits_{j=1}^{M}g_{ps}^{j}P_{sj}+\sum\limits_{i=1}^{N }g_{ps}^{i}P_{sci}\right)\leq Q\\ &\sum\limits_{j=1}^{M}P_{sj}+\sum\limits_{i=1}^{N}P_{sci}\leq P_{s}\end{array}