## 3 System Model
### Joint Channel and Power Assignment
\begin{array}{ll}\frac{B}{\alpha_{k}}-\frac{L_{0}B+\frac{M}{m+ k}\sum\limits_{n=1}^{M}P_{n}h_{mk}}{h_{lk}}\\ b\leq n\leq c\\ \frac{B}{\gamma_{k}}-\frac{L_{0}B+\sum\limits_{n=1}^{M}P_{n}h_{mk}}{h_{lk}}\\ a\leq n\leq b,\ c\leq n\leq N\end{array} \right. \tag{15}\]  
In Equation (15), \(\sum\limits_{m\
eq k,\ m=1}^{M}P_{m}h_{mk}\) means that other nodes interfere with user \(k\). Its value can be detected when it is received by user \(k\). So, it can be taken as a constant. It can be seen from the formula that when the power (\(P_{m}\)) of the interference node increases, the value of the user's transmitting power (\(P_{lk}\)) will decrease, thus improving the revenue of the whole network in the game. Then, the maximum value of power \(P_{lk}\) can be calculated. When \(t=0\) and \(b\leq n\leq c\), the user's initial power \(P_{lk}(0)=0\), and let \(t=t+1\), then:  
\[P_{lk}(t)=BR(P_{lk}(t-1)); \tag{16}\]  
After Formula (16) is calculated iteratively, once the result \(P_{lk}(t_{1})=P_{lk}(t_{1}-1)\) is obtained in a calculation, then t\({}_{1}\) is the moment when the Nash equilibrium is achieved. Similarly, in the non-central frequency band \(a\leq n\leq b\), \(c\leq n\leq N\), the Nash equilibrium time (\(t_{2}\)) of this frequency band can be obtained. In Formula (15), In order to achieve bio-friendliness, the user's transmitting power (\(P_{lk}\)) cannot be greater than the limited power. When the cognitive user is located on the channel of the central frequency band, the optimal power of \(P_{lk}\) increases with the decrease in the interference power \(P_{m}\). When \(t\geq t_{1}\)the value of \(P_{lk}\) will increase to greater than the limited power (\(P_{lim1}\)), in which case, \(P_{lk}=P_{lim1}\) can only be taken. When the cognitive user is located on a channel in a non-central frequency band, it can be seen that when \(t\geq t_{2}\), the \(P_{k}\) value will increase to greater than the limited power (\(P_{lim2}\)), and only \(P_{lk}=P_{lim2}\) can be taken at this time. In summary, the power \(P_{k}\) of cognitive users can finally be further optimized to:  
\[P_{lk}=\left\{ \begin{array}{ll}\frac{B}{\alpha_{k}}-\frac{L_{0}B+\sum\limits \limits_{m\
eq k,\ m-1}^{K}P_{m}h_{mk}}{h_{lk}}&,&t<t_{1}\\ P_{lim1}&,&t\geq t_{1}\\ \frac{B}{\gamma_{k}}-\frac{L_{0}B+\sum\limits_{m\
eq k,\ m-1}^{K}P_{m}h_{mk}}{h _{lk}}&,&t<t_{2}\\ P_{lim2}&,&t\geq t_{2}\\ \end{array} \right. \tag{17}\]  
The parameters \(\alpha_{k}\) and \(\gamma_{k}\) in Formula (17) can be calculated according to Formulas (12) and (13), and the calculation results are as follows:  
\[\alpha_{k}=\frac{BN_{k}}{P_{lim1}+\sum\limits_{n=1}^{N}\frac{L_{0}B\alpha_{lk }+\sum\limits_{m\
eq k,\ m-1}^{K}P_{m}h_{mk}}{h_{lk}}} \tag{18}\]  
\[\gamma_{k}=\frac{BN_{k}}{P_{lim2}+\sum\limits_{n=1}^{N}\frac{L_{0}B\alpha_{lk }+\sum\limits_{m\
eq k,\ m-1}^{K}P_{m}h_{mk}}{h_{lk}}} \tag{19}\]  
where \(N_{k}\) is the number of channels that user \(k\) accesses. Put Equations (18) and (19) into \(P_{lk}\) to get the final value of \(P_{lk}\), which is as follows:  
\[P_{lk}=\left\{ \begin{array}{ll}\frac{P_{lim1}+\sum\limits_{n=1}^{N}\frac{L_ {0}B\alpha_{lk}+\sum\limits_{m\
eq k,\ m-1}^{M}P_{m}h_{mk}}{h_{lk}}}{-\frac{L_ {0}B+\sum\limits_{m\
eq k,\ m-1}^{M}P_{m}h_{mk}}{h_{lk}}}{-\frac{L_{0}B+\sum \limits_{m\
eq k,\ m-1}^{M}P_{m}h_{mk}}{h_{lk}}},&t<t_{1}\\ P_{lim1},&t\geq t_{1}\\ \frac{P_{lim2}+\sum\limits_{n=1}^{N}\frac{L_{0}B\alpha_{lk}+\sum\limits_{m \
eq k,\ m-1}^{M}P_{m}h_{mk}}{h_{lk}}}{-\frac{L_{0}B+\sum\limits_{m\
eq k,\ m-1}^{M}P_{m}h_{mk}}{h_{ lk}}}{-\frac{L_{0}B+\sum\limits_{m\
eq k,\ m-1}^{M}P_{m}h_{mk}}{h_{lk}}},&t<t_{2}\\ P_{lim2},&t\geq t_{2}\\ \end{array} \right.