## 2 SIR-Based Performance Analysis
### The PDF of the Output SIR  
The input signals in the SC diversity receiver have the Beaulieu-Xie PDF ([2], Equation (4)):  
\[p_{X_{i}}(x_{i})=\frac{2m_{i}x_{i}^{m}}{\Omega_{i}\lambda_{i}^{m-1}}\mathrm{e} ^{-\frac{m_{i}}{\Omega_{i}}(x_{i}^{2}+\lambda_{i}^{2})}I_{m-1}\bigg{(}\frac{2m _{i}\lambda_{i}}{\Omega_{i}}x_{i}\bigg{)} \tag{1}\]  
The parameters \(m_{i}\) and \(\lambda_{i}\) are the fading severity parameter and non-centrality parameter, in a row, and \(\Omega_{i}\) are powers of input signals. For PDF of the BX distribution, \(m\) controls the shape, \(\Omega\) defines the spread, and \(\lambda\) influences the location and height of the mode [16]. The Rician distribution may be obtained from BX distribution if parameter \(m_{i}\) is equal to one, and Nakagami-\(m\) distribution will be obtained if non-central parameter \(\lambda_{i}\) is equal to zero. Further, if non-central parameter \(\lambda_{i}\) is zero, the Rician distribution becomes a Rayleigh, and also if \(m_{i}\) is equal to one in Nakagami-\(m\) distribution. This feature of the BX distribution is shown in Figure 1.3 in [16]. For a more practical presentation of the PDF of input signals, we will use a modified Bessel function of the first kind of real order \(v\), \(I_{v}(z)\), developed in an infinite series ([17], Equation (3)), as it is presented by Formula (2) in our conference paper [18]. Now, the PDF is in the form of a sum:  
\[p_{x_{i}}(x_{i})=2\mathrm{e}^{-\frac{m_{i}}{\Omega_{i}}(x_{i}^{2}+\lambda_{i} ^{2})}\sum_{j_{1}=0}^{+\infty}\frac{\lambda_{i}^{2j_{1}}x_{i}^{2j_{1}+2m_{i}-1 }}{j_{1}\mathrm{i}\Gamma(j_{1}+m_{i})}\bigg{(}\frac{m_{i}}{\Omega_{i}}\bigg{)} ^{2j_{1}+m_{i}} \tag{2}\]  
where the Gamma function is labeled by \(\Gamma(z)\) ([19], p. 255). The parameter \(\lambda\) can be expressed using the following formula [2]:  
\[\lambda_{i}=\sqrt{\kappa_{x}\Omega_{i}}, \tag{3}\]  
where \(\kappa_{x}\) is the \(K\)-factor of the generalized Rician distribution defined as \(s^{2}/n\sigma^{2}\), and representing the power in the LOS component divided by the power in the scatter components, whereby \(x\) means that \(\kappa\) refers to the fading. When \(K=\lambda^{2}/\Omega\), \(\lambda^{2}\) signs the LOS power, and \(\Omega\) marks the NLOS power. In that case, the PDF becomes:  
\[p_{x_{i}}(x_{i})=2\mathrm{e}^{-\frac{m_{i}}{\Omega_{i}}(x_{i}^{2}+\kappa_{x_{ i}}\Omega_{i})}\sum_{j_{1}=0}^{+\infty}\frac{(\kappa_{x}\Omega_{i})^{j_{1}}x_{i} ^{2j_{1}+2m_{i}-1}}{j_{1}\mathrm{i}\Gamma(j_{1}+m_{i})}\bigg{(}\frac{m_{i}}{ \Omega_{i}}\bigg{)}^{2j_{1}+m_{i}} \tag{4}\]  
From this formula, it can be seen that the BX fading distribution becomes Rician when the fading parameter \(m\) is equal to 1 for any value of \(\kappa=K\). Also, it can be further reduced to Rayleigh fading distribution when \(m=1\) and \(\kappa=0\). The CCI appearing here has a \(\kappa\)-\(\mu\) distribution [20]:  
\[p_{y_{i}}(y_{i})=\frac{2\mathrm{e}^{-\frac{\mu_{i}(1+\kappa_{y})}{\delta_{i}}y _{i}^{2}}}{\mathrm{e}^{\mu_{i}(\kappa_{y})}}\sum_{j_{2}=0}^{+\infty}\frac{\mu_ {i}^{2j_{2}+\mu_{i}}x_{y}^{j_{2}}y_{i}^{2j_{2}+2\mu_{i}-1}}{\Gamma(j_{2}+\mu_{ i})j_{2}!}\bigg{(}\frac{1+\kappa_{y}}{s_{i}}\bigg{)}^{j_{2}+\mu_{i}} \tag{5}\]  
It is clear that the \(\kappa\)-\(\mu\) distribution is defined by two parameters, \(\kappa\) and \(\mu\). Here, parameter \(\kappa_{y}\) is the Rician factor equal to the ratio of the dominant and scattered components, describing the CCI, and parameter \(\mu\) is the number of clusters in the wireless environment; the CCI mean square values are denoted by \(s_{i}\), \(i=1,2,\ldots,L\). A modified Bessel function \(I_{v}(z)\) is expanded to series by using ([18], Equation (2)).