## 3 Second-Order System Performance
### Level Crossing Rate
\begin{split}& N_{z}(z)=\frac{L\sqrt{2\pi}f_{m}}{\text{e}^{ m_{i}x_{i_{1}}+\mu_{i}y_{i_{2}}}}\sum\limits_{i_{1}=0}^{+\infty}\sum\limits_{j_{2}=0}^{+ \infty}\frac{\kappa_{x}\lambda_{y}^{j_{1}}m_{i}^{2j_{1}+m_{i}-\frac{1}{2}}\mu _{i}^{2j_{2}+\mu_{i}-\frac{1}{2}}\big{(}\Omega_{i}\big{(}1+\kappa_{y}\big{)} \big{)}^{j_{2}+\mu_{i}-\frac{1}{2}}}{j_{1}!j_{2}!\Gamma(j_{1}+m_{i})}\\ &\times\frac{\text{g}_{i}^{j_{1}+m_{i}-\frac{1}{2}}z_{i}^{2j_{1}+ 2m_{i}-1}\Gamma(j_{1}+j_{2}+m_{i}+\mu_{i}-1/2)}{\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}+m_{i}+\mu_{i}-1}}\\ &\times\bigg{(}\frac{1}{\text{e}^{m_{i}x_{i_{1}}+\mu_{i}y_{i_{2} }}}\sum\limits_{j_{3}=0}^{+\infty}\frac{(\kappa_{x}m_{i})^{j_{3}}\big{(}\kappa _{y}\mu_{i}\big{)}^{j_{4}}\Gamma(j_{3}+j_{4}+\mu_{i}+m_{i})}{\text{g}_{i}!j_{4 }!\Gamma(j_{3}+m_{i})\Gamma(j_{4}+\mu_{i})}B\frac{m_{i}x_{i}z^{2}}{\mu_{i} \Omega_{i}\big{(}1+\kappa_{y}\big{)}+m_{i}s_{i}z^{2}}\big{(}j_{3}+m_{i},j_{4}+ \mu_{i}\big{)}\bigg{)}^{L-1}\end{split}  \tag{40}\]  
This expression differs from ([31], Equation (22)) because the connection from (3) is used:  
\[\lambda_{i}^{2}=\kappa_{x}\Omega_{i}\]  
and \(\kappa_{x}\) and \(\kappa_{y}\) are different. With these improvements, we increased the generality of the expression. This expression can be compared with the expression ([34], Equation (24)). The expression ([34], Equation (24)) will be obtained by setting the appropriate parameter values, as defined below the expressions for PDFs of BX fading and \(\kappa\)-\(\mu\) CCI, and using \(\lambda\) from (3) and the same values for Rician factor of both, fading and CCI (\(\kappa_{x}=\kappa_{y}=K\)). Now, a few graphical presentations of the normalized LCR at the output of the multi-branch SC receiver given by (40) are shown in Figures 12 and 13 in order to examine the influence of BX fading parameters and \(\kappa\)-\(\mu\) CCI parameters. One can see from Figure 12 that due to the increase in BX fading parameters \(\kappa_{x}\) and \(m\), the LCR decreases for negative values of output SIR \(z\) (higher CCI), while for positive values of SIR, LCR decreases, and the system has better performance for all parameters. In Figure 13, the normalized LCR is presented versus SIR for variable \(\kappa\)-\(\mu\) CCI parameters \(\kappa_{y}\) and \(\mu\), and number of receiver input branches \(L\), while BX fading parameters and powers remain permanent values. It is possible to notice that an increase in \(L\) improves performance because it reduces the LCR for all values of SIR \(z\). When \(\mu\) increases for positive \(z\), the LCR drops, thereby improving the system performance. The influence of CCI parameter \(\kappa_{y}\) is negligible for \(z<0\), while an increase in \(\kappa_{y}\) causes a decrease in LCR for \(z>0\). \(z=10\) dB, the number of elements for convergence is higher and increases from 8 to 15. This is visible from Table 9.  
A similar case is seen for variable parameters \(\kappa_{y}\) and \(\mu\), and number of branches \(L\), presented in Table 10. The number of additions increases from 5 to 16 for all values of used parameters.