Two-ray ground-reflection model

2-ray Ground Reflected Model is a radio propogation model that predicts path loss when the signal recieved consists of the line of sight component and multipath component formed predominatly by a single ground ground reflected wave.

Ray tracing of 2-ray ground reflected model

Analysis

From the figure the recieved line of sight component may be written as $r_{los}(t)=Re\{{\frac {\lambda {\sqrt {G_{los}}}}{4\pi }}\times {\frac {s(t)e^{-j2\pi l/\lambda }}{l}}\}$

and the ground reflected component may be written as $r_{gr}(t)=Re\{{\frac {\lambda \Gamma (\theta ){\sqrt {G_{gr}}}}{4\pi }}\times {\frac {s(t-\tau )e^{-j2\pi (x+x')/\lambda }}{x+x'}}\}$

where s(t) is the transmitted signal Γ(θ) is ground reflection co-efficient and τ is the delay spread of the model and equals (x+x'-l)/c

Ground Reflection $\Gamma (\theta )={\frac {sin\theta -X}{sin\theta +X}}$

where $X_{v}=$${\sqrt {\varepsilon _{g}-{cos}^{2}\theta }} \over {\varepsilon _{g}}$

$X_{h}={\sqrt {\varepsilon _{g}-{cos}^{2}\theta }}$

From the figure $x+x'={\sqrt {(h_{t}^{2}+h_{r}^{2}+d^{2}}}$ and $l={\sqrt {(h_{t}^{2}-h_{r}^{2}+d^{2}}}$

the path difference between then $\Delta d=x+x'-l={\sqrt {(h_{t}^{2}+h_{r}^{2}+d^{2}}}-{\sqrt {(h_{t}^{2}-h_{r}^{2}+d^{2}}}$

the phase difference betwwen the waves $\Delta \phi ={\frac {2\pi \Delta d}{\lambda }}$

References

Goldsmith, Andrea (2004). Wireless communications (1. publ. ed.). Cambridge, U.K.: Cambridge University Press. ISBN 978-0521837163.

Institute, organized by The Korean Institute of Communications and Information Sciences ; Electronics and Telecommunications Research. ICTC 2012 2012 International Conference on ICT Convergence : "Global Open Innovation Summit for Smart ICT Convergence," October 15-17, 2012, Ramada Plaza Jeju Hotel, Jeju Island, Korea. Piscataway, NJ: IEEE. p. 454 - 455. ISBN 978-1-4673-4827-0.{{cite book}}: CS1 maint: multiple names: authors list (link)

Rappaport, Theodore S. (2002). Wireless communications : principles and practice (2. ed. ed.). Upper Saddle River, NJ: Prentice Hall PTR. ISBN 978-0130422323. {{cite book}}: |edition= has extra text (help)