## 4 Distributions of Path Delay, Path Attenuation, Normalized Doppler Shift of the Scattering Path
### The Distribution of \(\tau_{\text{DIF}}\left(t\right)\)
 \tag{41}\]  
where  
\[\mu_{1}\left(t\right)=2\,{y_{\text{R}}}^{2}\left(t\right)+4\,{y_{\text{R}}} \left(t\right)\,{y_{\text{T}}}\left(t\right)-2\,{y_{\text{T}}}^{2}\left(t\right) \tag{42}\]  
\[\mu_{2}\left(t\right)={y_{\text{R}}}^{2}\left(t\right)-2\,{y_{\text{R}}}\left(t \right)\,{y_{\text{T}}}\left(t\right)+{y_{\text{T}}}^{2}\left(t\right)+{z_{ \text{R}}}^{2}\left(t\right)-2\,{z_{\text{R}}}\left(t\right)\,{z_{\text{T}}} \left(t\right)+{z_{\text{T}}}^{2}\left(t\right) \tag{43}\]  
\[\mu_{3}\left(t\right)={y_{\text{R}}}^{2}\left(t\right)-2\,{y_{\text{R}}}\left(t \right)\,{y_{\text{T}}}\left(t\right)+{y_{\text{T}}}^{2}\left(t\right)+{z_{ \text{R}}}^{2}\left(t\right)+2\,{z_{\text{R}}}\left(t\right)\,{z_{\text{T}}} \left(t\right)+{z_{\text{T}}}^{2}\left(t\right) \tag{44}\]  
\[\mu_{4}\left(t\right)={z_{\text{T}}}\left(t\right)-{z_{\text{R}}}\left(t\right) \tag{45}\]  
\[\mu_{5}\left(t\right)={z_{\text{T}}}\left(t\right)+{z_{\text{R}}}\left(t\right) \tag{46}\]  
The details of calculating \(l_{l,T}\left(\tau_{\text{DIF}},t\right)\) and \(l_{s,T}\left(\tau_{\text{DIF}},t\right)\) can be found in Appendix B. Thus, the PDF \(f\left(\tau_{\text{DIFF}},t\right)\) can be obtained with Equation (47), where \(\frac{\partial l_{l,T}\left(\tau_{\text{DIFF}},t\right)}{\partial\tau_{\text{DIFF}}}\) is calculated as in (48). \[ \begin{split} f_{\text{TDEF}}\left(\tau_{\text{DIFF}},t\right)& =\frac{\partial F_{\text{TDEF}}\left(\tau_{\text{DIFF}},t\right)}{ \partial\tau_{\text{DIFF}}}\\ &=\frac{\partial S_{T}\left(\tau_{\text{DIFF}},t\right)}{S\left(t \right)\partial\tau_{\text{DIFF}}}\\ &=\frac{\pi}{S\left(t\right)}\left[l_{s,T}\left(\tau_{\text{DIFF }},t\right)\frac{\partial l_{l,T}\left(\tau_{\text{DIFF}},t\right)}{\partial \tau_{\text{DIFF}}}+l_{l,T}\left(\tau_{\text{DIFF}},t\right)\frac{\partial l_{ s,T}\left(\tau_{\text{DIFF}},t\right)}{\partial\tau_{\text{DIFF}}}\right] \end{split}  \tag{47}\]  
\[ \begin{split}\frac{\partial l_{\text{L,T}}\left(\tau_{\text{DIFF }},t\right)}{\partial\tau_{\text{DIFF}}}&=\frac{1}{\xi_{1}\left(t \right)}\left(\varsigma\sqrt{-\xi_{5}\left(t\right)\,\xi_{6}\left(t\right)}+2 c^{2}\,\tau_{\text{DIFF}}\,y_{\text{R}}\left(t\right)+2c^{2}\,\tau_{\text{DIFF}}\,y_{\text{T}} \left(t\right)-\frac{d_{\text{LOS}}\left(t\right)\,\xi_{2}\left(t\right)}{ \xi_{3}\left(t\right)}\\ &+2\,c\,d_{\text{LOS}}\left(t\right)\,y_{\text{R}}\left(t\right) +2\,c\,d_{\text{LOS}}\left(t\right)-\frac{c\,\tau_{\text{DIFF}}\,\xi_{2} \left(t\right)}{\xi_{3}\left(t\right)}\right)-\frac{1}{\xi_{1}^{2}}\left( \left(4\,\tau_{\text{DIFF}}\,c^{2}+4\,d_{\text{LOS}}\left(t\right)\,c\right) \right.\\ &\cdot\left(d_{\text{LOS}}\left(t\right)\,\sqrt{-\xi_{5}\left(t \right)\,\xi_{6}\left(t\right)}+c\,\tau_{\text{DIFF}}\,\sqrt{-\xi_{5}\left(t \right)\,\xi_{6}\left(t\right)}+c^{2}\,\tau_{\text{DIFF}}^{2}\,y_{\text{R}} \left(t\right)+c^{2}\,\tau_{\text{DIFF}}^{2}\,y_{\text{T}}\left(t\right)\right. \\ &\left.+2\,c\,d_{\text{LOS}}\left(t\right)\,\tau_{\text{DIFF}}\,y_ {\text{R}}\left(t\right)+2\,c\,d_{\text{LOS}}\left(t\right)\,\tau_{\text{DIFF }}\,y_{\text{T}}\left(t\right)\right)\end{split}