## 2 SIR-Based Performance Analysis
### The PDF of the Output SIR
The \(\kappa\)-\(\mu\) is also general distribution: the \(\kappa\)-\(\mu\) distribution becomes one-sided Gaussian distribution if \(\mu=0.5\) and \(\kappa=0\); if \(\mu=1\) and \(\kappa=0\), the \(\kappa\)-\(\mu\) distribution becomes Rayleighdistribution; when \(\mu=1\) and \(\kappa=K\), the \(\kappa\)-\(\mu\) distribution is Rician distribution, with \(K\) representing Rician \(K\) parameter; and if \(\mu=m\) and \(\kappa=0\), the \(\kappa\)-\(\mu\) distribution becomes Nakagami-\(m\) distribution, where \(m\) represents Nakagami-\(m\) fading severity parameter and \(m\geq 1/2\)[3]. The SIR \(z_{i}\) has the PDF defined in [21]:  
\[p_{z_{i}}(z_{i})=\int_{0}^{\infty}p_{x_{i}}(z_{i}y_{i})p_{y_{i}}(y_{i})y_{i}dy_ {i}. \tag{6}\]  
For our case of disturbances, it is:  
\[ \begin{split}& p_{z_{i}}(z_{i})=\frac{2}{\mathrm{e}^{m_{i}x_{i} +\mu_{i}x_{i_{j}}}}\\ &\times\sum\limits_{i=0}^{+\infty}\sum\limits_{j_{2}=0}^{+\infty }\frac{\kappa_{x}y_{i}^{1}\kappa_{y}^{2}m_{i}^{2}m_{i}^{2j_{1}+m_{i}}\mu_{i}^{ 2j_{2}+\mu_{i}}z_{i}^{2j_{1}+m_{i}}(\Omega_{i}\big{(}1+\kappa_{y_{i}}\big{)})^ {j_{2}+\mu_{i}}z_{i}^{2j_{1}+2m_{i}-1}\Gamma(j_{1}+j_{2}+\mu_{i}+m_{i})}{j_{1 }!j_{2}!\Gamma(j_{1}+m_{i})\Gamma(j_{2}+\mu_{i})\big{(}\mu_{i}\Omega_{i}\big{(} 1+\kappa_{y}\big{)}+m_{i}s_{i}z_{i}^{2}\big{)}^{j_{1}+j_{2}+\mu_{i}+m_{i}}} \end{split}  \tag{7}\]  
The PDF of the SIR \(z\) from SC receiver output is calculated by dint of formula [22]:  
\[p_{z_{i}}(z)=Lp_{z_{i}}(z_{i})(F_{z_{i}}(z_{i}))^{L-1}. \tag{8}\]  
By substitutions of (4) and (7) in above expression, the PDF of the output SIR \(z\) becomes:  
\[ \begin{split}& p_{z_{i}}(z)=\frac{2L}{\mathrm{e}^{(m_{i}x_{i_{ 1}}+\mu_{i}x_{i_{2}})L}}\\ &\times\sum\limits_{j_{1}=0}^{+\infty}\sum\limits_{j_{2}=0}^{+ \infty}\frac{\kappa_{x}y_{i}^{1}\kappa_{y}^{2}m_{i}^{2j_{1}+m_{i}}\mu_{i}^{2j_{ 2}+\mu_{i}}\big{(}1+\kappa_{y_{i}}\big{)}^{j_{2}+\mu_{i}}z_{i}^{2j_{1}+2m_{i}- 1}\Gamma(j_{1}+j_{2}+\mu_{i}+m_{i})}{j_{1}!j_{2}!\Gamma(j_{1}+m_{i})\Gamma(j_{ 2}+\mu_{i})\big{(}\mu_{i}\big{(}1+\kappa_{y}\big{)}\big{(}\Omega_{i}/s_{i} \big{)}+m_{i}z_{i}^{2}\big{)}^{j_{1}+j_{2}+\mu_{i}+m_{i}}}\Big{(}\frac{\alpha_ {\chi}}{s_{i}}\Big{)}^{2+\mu_{i}}\\ &\times\left(\sum\limits_{j_{3}=0}^{+\infty}\sum\limits_{j_{4}=0}^{+ \infty}\frac{(\kappa_{x}m_{i})^{j_{3}}(\kappa_{y}\mu_{i})^{j_{4}}\Gamma(j_{3}+ j_{4}+\mu_{i}+m_{i})}{j_{3}!j_{4}!\Gamma(j_{3}+m_{i})\Gamma(j_{4}+\mu_{i})}\ B\frac{m_{i}z^{2}}{\mu_{i}(1+\kappa_{y})(\frac{\Omega_{i}}{s_{i}})+ m_{i}z^{2}}\frac{(j_{3}+m_{i},j_{4}+\mu_{i})}{\mu_{i}(1+\kappa_{y})(\frac{\Omega_{i}}{s_{i}})+ m_{i}z^{2}}\right)^{L-1}\end{split}  \tag{9}\]  
The incomplete Beta function from the previous expression is represented by ([20], Equation (8.391)), as in [23] through Equations (6) and (7). Based on these formulas, the PDF of SIR \(z\) is:  
\[ \begin{split}& p_{z_{i}}(z)=\frac{2L}{\mathrm{e}^{(m_{i}x_{i_{ 1}}+\mu_{i}x_{i_{2}})L}}\sum\limits_{j_{1}=0}^{+\infty}\sum\limits_{j_{2}=0}^{+ \infty}\frac{\kappa_{x}y_{i}^{1}\kappa_{y}^{j_{2}}m_{i}^{2j_{1}+m_{i}}\mu_{i} ^{2j_{2}+\mu_{i}}\big{(}1+\kappa_{y_{j}}\big{)}^{j_{2}+\mu_{i}}z_{i}^{2j_{1}+ 2m_{i}-1}\Gamma(j_{1}+j_{2}+\mu_{i}+m_{i})}{j_{1}!j_{2}!\Gamma(j_{1}+m_{i}) \Gamma(j_{2}+\mu_{i})\big{(}\mu_{i}\big{(}1+\kappa_{y}\big{)}(\Omega_{i}/s_{i} \big{)}+m_{i}z^{2}\big{)}^{1/\Omega_{1}+\mu_{i}+m_{i}}}\Big{(}\frac{\Omega_{i} }{s_{i}}\Big{)}^{2+\mu_{i}}\\ &\times\left(\sum\limits_{j_{3}=0}^{+\infty}\sum\limits_{j_{4}=0}^{+ \infty}\sum\limits_{j_{5}=0}^{+\infty}\frac{(\kappa_{x}m_{i})^{j_{3}}(\kappa_{y} \mu_{i})^{j_{4}}\Gamma(j_{3}+j_{4}+\mu_{i}+m_{i})(j_{3}+m_{i})_{j_{5}}(1-j_{4} -\mu_{i})_{j_{5}}}{j_{3}!j_{4}!j_{5}!\Gamma(j_{3}+m_{i})!(j_{3}+m_{i})(j_{3}+m_{ i}+1)_{j_{5}}}\ \left(\frac{m_{i}z^{2}}{\mu_{i}\big{(}1+\kappa_{y}\big{)}\big{(}\frac{\Omega_{i}}{s_{i}} \big{)}+m_{i}z^{2}}\right)^{j_{3}+j_{5}+m_{i}}\right)^{L-1}\end{split}  \tag{10}\]  
In the next two figures (Figures 2 and 3), the PDF is presented versus SIR \(z\) for different values of fading and CCI parameters, powers, and number of input branches at the SC receiver.