2-Ray Ground Reflection Model

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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

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

and the ground reflected component may be written as

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

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

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

The power of the signal received is ${\displaystyle r_{los}^{2}+r_{rg}^{2}}$ If the signal is narrow band relative to the delay spread τ then s(t)=s(t-τ) the power equation may be simplified as

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

where Pt 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

{\displaystyle {\begin{aligned}x+x'-l&={\sqrt {(h_{t}+h_{r})^{2}+d^{2}}}-{\sqrt {(h_{t}-h_{r})^{2}+d^{2}}}\\&=d{\sqrt {{\frac {(h_{t}+h_{r})^{2}}{d^{2}}}+1}}-{\sqrt {{\frac {(h_{t}-h_{r})^{2}}{d^{2}}}+1}}\\\end{aligned}}}

Using the Taylor series of Template:Sqrt:

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

and taking the first two terms

${\displaystyle x+x'-l\approx {\frac {d}{2}}\times \left({\frac {(h_{t}+h_{r})^{2}}{d^{2}}}-{\frac {(h_{t}-h_{r})^{2}}{d^{2}}}\right)={\frac {2h_{t}h_{r}}{d}}}$

Phase difference may be approximated as

${\displaystyle \Delta \phi \approx {\frac {4\pi h_{t}h_{r}}{\lambda d}}}$

When d increases asymptotically ${\displaystyle d\approx l\approx x+x',\Gamma (\theta )\approx -1,G_{los}\approx G_{gr}}$

Reflection co-efficient tends to -1 for large d.
${\displaystyle P_{r}=P_{t}\left({\frac {\lambda G}{4\pi d}}\right)^{2}\times (1-e^{-j\Delta \phi })^{2}}$
${\displaystyle e^{x}=1+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots =1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+{\frac {x^{5}}{120}}+\cdots \!=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$

and retaining only the first two terms

${\displaystyle e^{-j\Delta \phi }\approx 1+({-j\Delta \phi })+\cdots }$
{\displaystyle {\begin{aligned}\therefore P_{r}&\approx P_{t}\left({\frac {\lambda G}{4\pi d}}\right)^{2}\times (1-(1-j\Delta \phi ))^{2}\\&=P_{t}\left({\frac {\lambda G}{4\pi d}}\right)^{2}\times (j\Delta \phi )^{2}\\&=P_{t}\left({\frac {\lambda G}{4\pi d}}\right)^{2}\times -\left({\frac {4\pi h_{t}h_{r}}{\lambda d}}\right)^{2}\\&=-P_{t}{\frac {Gh_{t}^{2}h_{r}^{2}}{d^{4}}}\end{aligned}}}

Taking the magnitude

${\displaystyle P_{r}=P_{t}{\frac {Gh_{t}^{2}h_{r}^{2}}{d^{4}}}}$

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

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

${\displaystyle PL\;=P_{T_{dBm}}-P_{R_{dBm}}\;=\;PL_{0}\;+\;10\gamma \;\log _{10}{\frac {d}{d_{0}}}\;+\;X_{g},}$

The path loss of 2-ray ground reflected wave is

${\displaystyle PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40log(d)-10log(Gh_{t}^{2}h_{r}^{2})}$

where

${\displaystyle PL_{0}=40log(d_{0})-10log(Gh_{t}^{2}h_{r}^{2})}$,
${\displaystyle X_{g}=0}$

and

${\displaystyle \gamma =4}$

for ${\displaystyle d,d_{0}>d_{c}}$ 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.