## 3 Subchannel Selection Policies
 \tag{5}\]where, \(Q\) is the interference threshold of the PUs, which is the maximum allowable temporal interference in the receiver of the PUs caused by concurrent activity of the SUs in the same subchannel. \(P_{s}\)\(N_{0}\) is the power spectral density, \(\Delta f\) is the subchannel coherence bandwidth. \(K\) is a system parameter related to the characteristics of PUs network [15] within the range of 2-8. Equation (5) is derived from [PERSON]'s Capacity formula with the SUs power vector \(P_{s}\) and \(P_{sc}\). Equation (6) is the constraint function of interference threshold of the PUs and maximum transmitting power of the SUs. Actually, in contrast to the constraint of maximum transmitting power of the SUs, the constraint function of interference threshold of the PUs is much tighter [18]. Therefore, in this paper, the constraint of maximum transmitting power of SUs is not considered. At the same time, as mentioned above, the SUCCH spreads over an ultra-wide bandwidth to exploit the underused spectrum with a very low PSD, therefore, the interference caused by SUCCH is very low. In this paper, in order to simplify the analysis, the effect of SUCCH will not be considered, and Equation (5) will be further simplified as follows:  
\[ \begin{array}{rcl}C_{M}^{\psi}&=&\max\limits_{P_{s}}\sum\limits_{j=1}^{M} \Delta f\int\limits_{g_{ps}^{j}S_{ss}^{j}}\log\left(1+\frac{g_{ss}^{j}P_{sj}}{KN _{0}\Delta f}\right)\\ &&\times f_{ss}^{j}\left(g_{ss}^{j}\right)f_{ps}^{j}\left(g_{ps}^{j}\right)dg_ {ss}^{j}dg_{ps}^{j}\\ s.t.&\frac{1}{G}\left(\sum\limits_{j=1}^{M}g_{ps}^{j}P_{sj}\right)\leq Q\\ \end{array}  \tag{6}\]  
Suppose \(\psi\left(g_{ps}^{j},g_{ss}^{j}\right)=1\), thus the SUs will randomly choose \(M\) out of \(N\) subchannels without any priori information by \(\rho_{1}\), which is a uniform subchannel selection policy. Consequently, substituting \(P_{sj}=Q_{j}/g_{ps}^{j},\ j=1,\ldots,M\) and defining \(\theta_{Q_{j}}\triangleq Q_{j}/KN_{0}\Delta f\) Equation (6) can be simplified as follows:  
\[ \begin{array}{rcl}C_{M}^{\rho_{1}}&=&\max\limits_{Q}\sum\limits_{j=1}^{M} \Delta f\int\limits_{v_{j}}\log\left(1+v_{j}\theta_{Q_{j}}\right)h_{j}\left(v _{j}\right)dv_{j}\\ s.t.&\sum\limits_{j=1}^{M}Q_{j}=GQ,\ \ \ \ \ 0\leq Q_{j}\leq GQ\\ \end{array}  \tag{7}\]  
where \(v_{j}\triangleq g_{ss}^{j}/g_{ps}^{j}\), \(0\leq v_{j}\leq\infty\), \(v_{j}\) is the reward factor of the subchannel \(j\). \(\theta_{Q_{j}}\) is defined as the spectrum sharing load factor of the subchannel \(j\). Suppose the statistics characteristics of \(\sqrt{g_{ps}^{j}}\sqrt{g_{ss}^{j}}\) is i.i.d. Rayleigh random variables, \(g_{ps}^{j}\) and \(g_{ss}^{j}\) are exponentially distributed random variables with unit-mean, therefore, the PDF of \(v_{j}\) can be converted into [17]:  
\[ \begin{array}{rcl}h_{j}(v_{j})&=&\frac{d}{dv_{j}}\int_{0}^{\infty}\int_{0}^{ g_{ps}^{j}v_{j}}e^{-g_{ps}^{j}e^{-g_{ss}^{j}}}dg_{ps}^{j}dg_{ss}^{j}\\ &=&\int_{0}^{\infty}g_{ps}^{j}e^{-g_{ps}^{j}e^{-g_{ps}^{j}\frac{g_{ss}^{j}}{g_ {ps}^{j}}}}dg_{ps}^{j}\\ &=&\int_{0}^{\infty}g_{ps}^{j}e^{-g_{ps}^{j}(1+v_{j})}dg_{ps}^{j}\\ &=&-\frac{1}{1+v_{j}}\left\{\left.\left[g_{ps}^{j}e^{-g_{ps}^{j}(1+v_{j})} \right]\right|_{0}^{\infty}-\int_{0}^{\infty}e^{-g_{ps}^{j}(1+v_{j})}dg_{ps}^ {j}\right\}\\ &=&-\frac{1}{\left(1+v_{j}\right)^{2}}\left.\left[e^{-g_{ps}^{j}(1+v_{j})} \right]\right|_{0}^{\infty}\\ &=&\frac{1}{\left(1+v_{j}\right)^{2}}\ 0<v_{j}<\infty\\ \end{array}