## 2 SIR-Based Performance Analysis
### The Outage Probability of the Output SIR  
Then, the CDF is given by dint of [21]:  
\[F_{z_{i}}(z_{i})=\int_{0}^{z_{i}}p_{z_{i}}(t)dt \tag{11}\]  
After replacement, the CDF of SIR \(z_{i}\) is:  
\[ \begin{array}{l}F_{z_{i}}(z_{i})=\frac{2}{\mathrm{e}^{\pi\left[\mu_{i}A_{z_{i} }\gamma_{i}\right]\gamma_{i}\delta_{z_{i}}}}\\ \times\sum\limits_{i=1}^{+\infty}\sum\limits_{j=0}^{+\infty}\sum\limits_{j=0} ^{+\infty}\frac{\kappa_{x_{i}}\mu_{j}^{2}m_{j}^{2j+1}+m_{j}\mu_{j}^{2j+2}+\mu_ {j}\mu_{j}\mu_{j}^{3}+n_{i}}{\int_{1}!\mu_{2}!\Gamma(\mu_{i}+m_{j})\Gamma(\mu_ {2}+\mu_{i})}\\ \times\int_{0}^{z_{i}}\frac{z_{i}z_{i+2}n_{i}^{-1}}{\left(\mu_{i}\Omega_{i} \left(1+\kappa_{y}\right)+m_{j}\mu_{j}\mu_{j}^{2}\right)^{3}+n_{i}^{2}+\mu_{j }\mu_{i}}dt\end{array} . \tag{12}\]  
Figure 3: PDF versus SIR \(z\) at the multi-branch SC receiver output for variable CCI parameters \(\kappa_{y}\) and \(\mu\), and number of branches \(L\). Other parameters are: \(\kappa_{x}=1\), \(m=1\), \(\Omega=1\), and \(s=1\). Figure 2: PDF of SIR \(z\) at the multi-branch SC receiver output for different values of fading parameters \(m\) and \(\kappa_{x}\). Other parameters are: \(\kappa_{y}=1\), \(\mu=1\), \(L=2\), \(\Omega=1\), and \(s=1\). Using the incomplete Beta function \(B_{z}(a,b)\) ([20], Formula (8.38)) to solve the integral in expression (12), we obtain the CDF of SIR \(z_{i}\) as:  
\[F_{z_{i}}(z_{i})=\frac{1}{\mathrm{e}^{\pi_{i}\alpha_{i}z_{i}+\mu_{ i}\mu_{i}}}\sum\limits_{j_{1}=0}^{+\infty}\sum\limits_{j_{2}=0}^{+\infty}\frac{( \kappa_{x}m_{i})^{j_{1}}(\kappa_{y}\mu_{i})^{2}\Gamma(j_{1}+j_{2}+\mu_{i}+m_{i} )}{j_{1}!\gamma_{2}!\Gamma(j_{1}+m_{i})\Gamma(j_{2}+\mu_{i})}\] \[\times B\frac{\pi_{i}\mu_{i}z^{2}}{j_{1}!\gamma_{2}(1+\mu_{i})+\mu_{ i}\mu_{i}z^{2}}\left(j_{1}+m_{i},j_{2}+\mu_{i}\right) \tag{13}\]  
The Pout is defined as the probability that the instantaneous error probability exceeds a defined value or, equivalently, probability that the output SIR falls below a certain specified threshold. Mathematically [1], the Pout presents the CDF of the SIR \(z\) at the multi-branch SC receiver output ([21], Equation (6.82)):  
\[P_{out}(z)=F_{z}(z)=(F_{z_{i}}(z_{i}))^{L}. \tag{14}\]  
For our system model that is also treated in [22], the final exact expression for Pout is:  
\[P_{out}(z)=\left(\frac{1}{\mathrm{e}^{\pi_{i}\alpha_{i}z_{i}+\mu_{ i}\mu_{i}}}\sum\limits_{j_{1}=0}^{+\infty}\sum\limits_{j_{2}=0}^{+\infty}\frac{( \kappa_{x}m_{i})^{j_{1}}(\kappa_{y}\mu_{i})^{j_{2}}}{j_{1}!\gamma_{2}!}\right.\] \[\left.\times\frac{\Gamma(j_{1}+j_{2}+\mu_{i}+m_{i})}{\Gamma(j_{1} +m_{i})\Gamma(j_{2}+\mu_{i})}\,B\frac{\pi_{i}\mu_{i}z^{2}}{j_{1}!\gamma_{2}! \gamma_{2}!}\left(j_{1}+m_{i},j_{2}+\mu_{i}\right)\right)^{L}. \tag{15}\]  
Here, we provided an improved formula for Pout since the BX fading parameter \(\lambda\) is given by (3), and parameters \(\kappa\) for fading and CCI are taken to be different, namely \(\kappa_{x}\
eq\kappa_{y}\). This performance is displayed through the infinite series, which requires a reasonable number of terms to be summed to obtain fast convergence. Again, we point out that BX is a general distribution, and the results obtained here can be compared with the results of earlier published papers [23; 24; 25] by setting the particular parameters values defined above. The obtained result in (15) is reduced to special cases of existing fading channels shown in earlier works. The case of Pout in the presence of BX fading and Nakagami-\(m\) CCI is obtained by replacing the values of the CCI parameters to be \(\kappa_{y}=0\) and \(\mu=m>0.5\). In this case, we obtain expression (9) from [24]. The following published case in [25] can be obtained by replacing the CCI parameters to be \(\kappa_{y}=K\) and \(\mu=1\), when expression (15) will become expression (7) from [25], i.e., we obtain the Pout for the channel disturbed by BX fading and the Rician CCI which was published in [23].