## 4 Mathematical Representation of the Proposed System
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Parameter or Function Definition** \\ \hline \(T_{b}\), \(E_{b}\) & Bit duration, bit energy (rectangular pulse). \\ \(m_{j}(t)\) & Message signal that is constant within each \(T_{b}\) time interval. \\ \(P_{u}\) & User-transmitted power. \\ \(P_{p}\), \(P_{s}\) & The transmitted power from the PU and SU towers. \\ \(U_{p}\), \(U_{s}\) & Number of PU system subscribers and SU system subscribers. \\ K & The increase in PU-RX SNR due to donations from the SU system. \\ \(\gamma\) & Donation power from SU to the PU (power incentive). \\ \(\lambda\) & Min. SNR level at the SU-RX to guarantee accepted QoS. \\ \(T_{c}\) & Symbol (chip) duration, \(T_{c}=\frac{T_{b}}{N_{c}}\), where \(N_{o}\) is the spreading factor. \\ \(n(t)\) & Noise function, assumed AWGN, zero mean, with \(PSD=\frac{T_{b}}{2}\). \\ \(\tau_{u}\) & Random phase offset due to propagation (delay). \\ \(\theta_{u}\) & Random phase, where \(\theta_{u}=-\omega_{c}\tau_{u}+\phi_{u}\). \\ \(E[.]\) & Expectation operator. \\ \(h_{pj}\) & The channel gain from the PU-TX to the SU-RX. \\ \(h_{si}\) & The channel gain from the SU-TX to the PU-RX. \\ \(\alpha_{u}\in[0.7,0.95]\) & The channel gain of the desired signal for the PU and SU subscribers. \\ \(\alpha_{\bar{u}}\in[0.2,0.3]\) & The channel gain of the interferers within the same system. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and functions used in the proposed DSCA-MC-CR technique mathematical representation. where \(n(t)\) is the AWGN term at the reciever. Figure 6 shows the correlator receiver of the \(i^{th}\) SU. It receives \(m_{i}\) by converting the signal from the RF band to the base-band level, then decoding it, then correlating it. \(s(t)\) is the desired \(SU_{i}\) RF received signal. Thus, Equation (3) can be expressed as  
\[r(t) = \sqrt{\frac{2E_{p}}{T_{p}}}m_{i}(t-\tau_{i})c_{i}(t-\tau_{i})cos( \omega_{c}(t-\tau_{i})+\phi_{i})\] \[+ \sum_{ \begin{subarray}{c}u=0\\ u\
eq i\end{subarray} }^{U}\sqrt{\frac{2E_{p}}{T_{p}}}m_{u}(t-\tau_{u})c_{u}(t -\tau_{u})cos\left(\omega_{c}(t-\tau_{u})+\phi_{u}\right)\ +n(t).\]  
So, by processing the signal down from the RF band to the base-band level for user \(SU_{1}\), the correlated signal can be classified into three terms:  
\[z(t)=\int_{0}^{T_{b}}r(t)c_{i}(t-\tau_{i})cos(\omega_{c}(t-\tau_{i})+\phi_{i}) dt\ =v_{i}(t)+v_{MAI}(t)+\eta(t). \tag{5}\]  
where \(v_{i}(t)\) is the desired received signal term by subscriber \(SU_{i}\), \(v_{MAI}(t)\) is the multiple-access interference (MAI) due to \(U-1\) users within the system coverage area, and \(\eta(t)\) is the channel noise term. Furthermore, the derivations of \(v_{i}(t)\), \(v_{MAI}(t)\), and \(\eta(t)\) are implemented using [41] in Appendix A as functions in the signal and the proposed system parameters. The derivation of the \(\gamma\) limits of the proposed DSCA-MC-CR system is obtained by using the practical asynchronous model. So, to derive the lower boundary of \(\gamma\), the \(PU_{i}\) receiver must have an increase of \(K\) in the SNR value due to the power donation from the SU system, which creates the following inequality:  
\[\left(\alpha_{u}\sqrt{P_{p}}+h_{si}\sqrt{\gamma P_{s}}\right)^{2}\geq\alpha_{ u}^{2}P_{p}(1+K), \tag{6}\]  
After several manipulations of Equation (6), the lower boundary of \(\gamma\) can be expressed as  
\[\gamma\geq\frac{\alpha_{u}^{2}P_{p}\big{(}K-2\sqrt{K+1}+2\big{)}}{h_{si}^{2}P _{s}}. \tag{7}\]  
On the other hand, to derive the upper boundary of \(\gamma\), the secondary user receiver \(SU_{j}\) should be guaranteed an acceptable minimum level of QoS, which creates the following inequality:  
\[\frac{\alpha_{u}^{2}\frac{E_{p}T_{p}}{2}(1-\gamma)}{\sum_{ \begin{subarray}{c }d=1,d\
eq u\\ \end{subarray} }^{U}\frac{T_{c}E_{p}}{6}\alpha_{\bar{d}}^{2}+\frac{\eta_{s}T_{ p}}{4}}\geq\lambda, \tag{8}\]  
After several manipulations of Equation (8), the upper boundary of \(\gamma\) can be expressed as  
\[\gamma\leq 1-\frac{2\lambda\left(\sum_{