## 3 Second-Order System Performance
### Level Crossing Rate  
The LCR of the SIR at the output of a multi-branch SC receiver is actually the mean value of the first derivative of the SIR at the receiver output. Accordingly, it is necessary to average the first derivative by an integration [30]:  
\[N_{z_{i}}(z_{i})=\int\limits_{0}^{\infty}\dot{z}_{i}p_{\dot{z}_{i}z_{i}}(\dot{ z}_{i}z_{i})d\dot{z}_{i} \tag{27}\]  
Since we need \(\dot{z}_{i}\) (the first derivative of SIR z\({}_{\rm I}\)) to obtain LCR, let us repeat the calculation from [31] with \(\lambda\) from (3):  
\[\dot{z}_{i}=\frac{1}{y_{i}}\dot{x}_{i}-\frac{x_{i}}{y_{i}^{2}}\dot{y}_{i}. \tag{28}\]  
The first derivatives of \(\dot{x}_{i}\) and \(\dot{y}_{i}\) are distributed by Gauss, subsequently \(\dot{z}_{i}\) has Gaussian distribution with zero mean value:  
\[\overline{z}_{i}=\frac{1}{y_{i}}\overline{x}_{i}-\frac{x_{i}}{y_{i}^{2}}\overline {y}_{i}=0. \tag{29}\]  
\(\dot{z}_{i}\) has the variance:  
\[\sigma_{z_{i}}^{2}=\frac{1}{y_{i}^{2}}\sigma_{z_{i}}^{2}+\frac{x_{i}^{2}}{y_{ i}^{4}}\sigma_{y_{i}}^{2}. \tag{30}\]  
The variance of \(\dot{x}_{i}\) is ([16], Equation (2.4)):  
\[\sigma_{x_{i}}^{2}=\frac{\pi^{2}f_{m}^{2}\Omega_{i}}{m_{i}}. \tag{31}\]  
CCI's derivative \(\dot{y}_{i}\) has the variance:  
\[\sigma_{y_{i}}^{2}=\pi^{2}f_{m}^{2}\frac{s_{i}}{\mu_{i}(x_{y}+1)}. \tag{32}\]  
In expressions (31) and (32), Doppler frequency is marked with f\({}_{\rm m}\). After substitution of the last two expressions in (30), the variance of \(\dot{z}_{i}\) becomes:  
\[\sigma_{\dot{z}_{i}}^{2}=\frac{\pi^{2}f_{m}^{2}}{y_{i}^{2}}\Bigg{(}\frac{\mu_ {i}\Omega_{i}(x_{y}+1)+m_{i}s_{i}z_{i}^{2}}{m_{i}\mu_{i}(x_{y}+1)}\Bigg{)}. \tag{33}\]  
Let us now determine the conditional PDFs (CPDF) of \(\dot{z}_{i}\) and z\({}_{\rm i}\). They are [32]:  
\[p_{\dot{z}_{i}}\big{(}\dot{z}_{i}|z_{i}y_{i}\big{)}=\frac{1}{\sqrt{2\pi}\sigma _{\dot{z}_{i}}}\mathrm{e}^{-\frac{\dot{z}_{i}^{2}}{2\sigma_{\dot{z}_{i}}^{2}}}, \tag{34}\]  
\[p_{z_{i}}(z_{i}|y_{i})=\left|\frac{\mathrm{d}x_{i}}{\mathrm{d}z_{i}}\right|p_{ x_{i}}(z_{i}y_{i})=y_{i}p_{x_{i}}(z_{i}y_{i}). \tag{35}\]  
Now, we should find the conditional joint PDF of z\({}_{\rm i}\), \(\dot{z}_{i}\) and y\({}_{\rm i}\)[21]:  
\[p_{\dot{z}_{i}\dot{y}_{i}}(\dot{z}_{i}z_{i}y_{i})=p_{\dot{z}_{i}}\big{(}\dot{z} _{i}|z_{i}y_{i}\big{)}\,p_{y_{i}}(y_{i})y_{i}p_{x_{i}}(z_{i}y_{i}). \tag{36}\]Then, the joint PDF of \(z_{i}\) and \(\dot{z}_{i}\) is [21]:  
\[\dot{z}_{i}=\frac{1}{y_{i}}\dot{x}_{i}-\frac{x_{i}}{y_{i}^{2}}\dot{y}_{i}. \tag{28}\]0  
Some replacements are performed in (27), and the LCR of SIR \(z_{i}\) becomes:  
\[ \begin{split}& N_{z_{i}}(z_{i})=\frac{\sqrt{2\pi}f_{m}}{\text{e}^{ m_{i}x_{i_{1}}+\mu_{i}y_{i_{2}}}}\sum\limits_{i_{1}=0}^{+\infty}\sum\limits_{j_{2}=0}^{+ \infty}\frac{\kappa_{x^{j_{1}}}\kappa_{y}^{j_{2}}m_{i}^{2j_{1}+m_{i}-\frac{1}{ 2}}\mu_{i}^{2j_{2}+\mu_{i}-\frac{1}{2}}}{j_{1}!j_{2}!\Gamma(j_{1}+m_{i})}\\ &\times\frac{\big{(}\Omega_{i}\big{(}1+\kappa_{y}\big{)}\big{)}^{ j_{2}+\mu_{i}-\frac{1}{2}}\text{g}_{i}^{j_{1}+m_{i}-\frac{1}{2}}z_{i}^{2j_{1}+2m_{i}-1} \Gamma(j_{1}+j_{2}+m_{i}+\mu_{i}-1/2)}{\Gamma(j_{2}+\mu_{i})\big{(}\mu_{i} \Omega_{i}\big{(}1+\kappa_{y}\big{)}+m_{i}s_{i}z_{i}^{2}\big{)}^{j_{1}+j_{2}+ m_{i}+\mu_{i}-1}}\end{split}  \tag{38}\]  
The LCR of the output SIR \(z\) is calculated using formula ([33], Equation (8)):  
\[N_{z}(z)=L(F_{z_{i}}(z_{i}))^{L-1}N_{z_{i}}(z_{i}). \tag{39}\]  
Using expressions (39) and (13), we obtain the LCR at the output of the multi-branch SC receiver in the form:  
\[