Two-ray ground-reflection model

2-ray Ground Reflected Model is a radio propagation model that predicts path loss when the signal received consists of the line of sight component and multi path component formed predominately by a single ground reflected wave.

2-Ray Ground Reflection diagram including variables for the 2-ray ground reflection propagation algorithm.

Mathematical Derivation

From the figure the received line of sight component may be written as

$r_{los}(t)=Re \left\{ \frac{ \lambda \sqrt{G_{los}} }{4\pi}\times \frac{s(t) e^{-j2\pi l/\lambda}}{l} \right\}$

and the ground reflected component may be written as

$r_{gr}(t)=Re\left\{\frac{\lambda \Gamma(\theta) \sqrt{G_{gr}}}{4\pi}\times \frac{s(t-\tau) e^{-j2\pi (x+x')/\lambda}}{x+x'} \right\}$

where $s(t)$ is the transmitted signal $\Gamma(\theta)$ is ground reflection co-efficient and $\tau$ 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

$x+x'=\sqrt{(h_t+h_r)^2 +d^2}$

and

$l=\sqrt{(h_t - h_r) ^2 +d^2}$,

therefore, the path difference between them

$\Delta d=x+x'-l=\sqrt{(h_t+h_r )^2 +d^2}-\sqrt{(h_t- h_r) ^2 +d^2}$

and the phase difference between the waves is

$\Delta \phi =\frac{2 \pi \Delta d}{\lambda}$

The power of the signal received is

$r_{los}^2 + r_{gr}^2$

If the signal is narrow band relative to the delay spread $\tau$, the power equation $s(t)=s(t-\tau)$ may be simplified to

$|s(t)|^2 \left( {\frac{\lambda}{4\pi}} \right) ^2 \times \left( \frac{\sqrt{G_{los}} \times e^{-j2\pi l/\lambda}}{l} + \Gamma(\theta) \sqrt{G_{gr}} \frac{e^{-j2\pi (x+x')/\lambda}}{x+x'} \right)^2 =P_t \left( {\frac{\lambda}{4\pi}} \right) ^2 \times \left( \frac{\sqrt{G_{los}}} {l} + \Gamma(\theta) \sqrt{G_{gr}} \frac{e^{-j \Delta \phi}}{x+x'} \right)^2$

where $P_t$ is the transmitted power.

When distance between the antennas $d$ is very large relative to the height of the antenna we may expand $x+x'-l$ using Generalized Binomial Theorem

$\begin{align} x+x'-l & = \sqrt{(h_t+h_r)^2 +d^2}-\sqrt{(h_t- h_r )^2 +d^2} \\ & = d \Bigg(\sqrt{\frac{(h_t+h_r) ^2}{d^2}+1}-\sqrt{\frac{(h_t- h_r )^2 }{d^2}+1}\Bigg) \\ \end{align}$

Using the Taylor series of $\sqrt{1 + x}$:

$\sqrt{1 + x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + \textstyle \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + \dots,\!$

and taking the first two terms

$x+x'-l \approx \frac{d}{2} \times \left( \frac{(h_t+ h_r )^2}{d^2} -\frac{(h_t- h_r )^2 }{d^2} \right) = \frac{2 h_t h_r }{d}$

Phase difference may be approximated as

$\Delta \phi \approx \frac{4 \pi h_t h_r }{\lambda d}$

When $d$ increases asymptotically

Reflection co-efficient tends to -1 for large d.

$\begin{align} d & \approx l \approx x+x', \\ \Gamma(\theta) & \approx -1, \\ G_{los} & \approx G_{gr} = G \\ \end{align}$

$\therefore P_r =P_t \left( {\frac{\lambda \sqrt{G}}{4\pi d}} \right) ^2 \times ( 1-e^{-j \Delta \phi})^2$

Expanding $e^{-j\Delta \phi}$using Taylor series

$e^x = 1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}+ \cdots = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots\! = \sum_{n=0}^\infty \frac{x^n}{n!}$

and retaining only the first two terms

$e^{-j\Delta \phi} \approx 1 + ({-j\Delta \phi}) + \cdots$

$\begin{align} \therefore P_r & \approx P_t \left( {\frac{\lambda \sqrt{G}}{4\pi d}} \right) ^2 \times (1 - (1 -j \Delta \phi) )^2 \\ & = P_t \left( {\frac{\lambda \sqrt{G}}{4\pi d}} \right) ^2 \times (j \Delta \phi)^2 \\ & = P_t \left({\frac{\lambda \sqrt{G}}{4\pi d}} \right) ^2 \times -\left(\frac{4 \pi h_t h_r }{\lambda d} \right)^2 \\ & = -P_t \frac{G h_t ^2 h_r ^2}{d^4} \end{align}$

Taking the magnitude

$|P_r| = P_t \frac{G h_t ^2 h_r ^2}{d^4}$

Power varies with inverse fourth power of distance for large $d$.

In logarithmic units

In logarithmic units : $P_{r_{dBm}}=P_{t_{dBm}}+ 10 \log_{10}(G h_t ^2 h_r ^2) - 40 \log_{10}(d)$

Path loss : $PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40 \log_{10}(d)-10 \log_{10}(G h_t ^2 h_r ^2)$

Power vs. Distance Characteristics

Power vs distance plot

When d is small compared to transmitter height two waves add constructively to yield higher power and as d increases these waves add up constructively and destructively giving regions of up-fade and down-fade as d increases beyond the critical distance or first Fresnel zone power drops proportional to inverse fourth power of d. An approximation to critical distance may be obtained by setting Δφ to π as critical distance a local maximum.

As a case of log distance path loss model

The standard expression of Log distance path loss model is

$PL\;=P_{T_{dBm}}-P_{R_{dBm}}\;=\;PL_0\;+\;10\gamma\;\log_{10} \frac{d}{d_0}\;+\;X_g,$

The path loss of 2-ray ground reflected wave is

$PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40 \log_{10}(d)-10 \log_{10}(G h_t ^2 h_r ^2)$

where

$PL_0 = 40 \log_{10}(d_0)-10 \log_{10}(G h_t ^2 h_r ^2)$,

$X_g = 0$

and

$\gamma = 4$

for $d,d_0 > d_c$ the critical distance.

As a case of multi-slope model

The 2-ray ground reflected model may be thought as a case of multi-slope model with break point at critical distance with slope 20 dB/decade before critical distance and slope of 40 dB/decade after the critical distance.

See also

• Radio propagation model
• Free-space path loss
• Friis transmission equation

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. pp. 454–455. ISBN 978-1-4673-4827-0.

Rappaport, Theodore S. (2002). Wireless communications : principles and practice (2. ed.). Upper Saddle River, NJ: Prentice Hall PTR. ISBN 978-0130422323.