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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.
[[File:Two ray ground reflected model.svg|400px|thumbnail|Ray tracing of 2-ray ground reflected model]]


==Mathematical Derivation==
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From the figure the received line of sight component may be written as
 
<math>r_{los}(t)=Re\{\frac{\lambda \sqrt{G_{los}}}{4\pi}\times \frac{s(t) e^{-j2\pi l/\lambda}}{l} \}                      </math>
 
and the ground reflected component may be written as
 
<math>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'} \}                      </math>
 
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 <math>\Gamma(\theta)= \frac{sin \theta - X}{sin \theta + X }</math>
 
where  <math>X_{v}= </math><math>{\sqrt{\varepsilon_g-{cos}^{2}\theta}}\over {\varepsilon_{g}} </math>
 
<math>X_{h}=\sqrt{\varepsilon_{g}-{cos}^2 \theta}</math>
 
From the figure <math>x+x'=\sqrt{(h_t+h_r)^2 +d^2}</math> and  <math>l=\sqrt{(h_t - h_r) ^2 +d^2}</math>
 
the path difference between then <math>\Delta d=x+x'-l=\sqrt{(h_t+h_r )^2 +d^2}-\sqrt{(h_t- h_r) ^2 +d^2}</math>
 
the phase difference between the waves <math>\Delta \phi =\frac{2 \pi \Delta d}{\lambda}</math>
 
The power of the signal received is  <math>r_{los}^2 + r_{rg}^2</math>
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
 
<math>|s(t)|^2  {( \frac{\lambda}{4\pi}} ) ^2 \times ( \frac{G_{los} e^{-j2\pi l/\lambda}}{l} + \Gamma(\theta) G_{gr} \frac{e^{-j2\pi (x+x')/\lambda}}{x+x'})^2 </math>
 
<math>=P_t  ( {\frac{\lambda}{4\pi}} ) ^2 \times ( \frac{G_{los}} {l} + \Gamma(\theta) G_{gr} \frac{e^{-j \Delta \phi}}{x+x'})^2 </math>
 
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|Binomial Theorem]]
 
<math>x+x'-l = \sqrt{(h_t+h_r)^2 +d^2}-\sqrt{(h_t- h_r )^2 +d^2} </math>
 
<math>= d \sqrt{\frac{(h_t+h_r) ^2}{d^2}+1}-\sqrt{\frac{(h_t- h_r )^2 }{d^2}+1}</math>
 
<math>\approx \frac{d}{2} *( \frac{(h_t+ h_r )^2}{d^2} -\frac{(h_t- h_r )^2 }{d^2}) </math>
 
<math>= \frac{2 h_t h_r }{d}</math>
 
Phase difference may be approximated as
 
<math>\Delta \phi \approx \frac{4 \pi h_t h_r }{\lambda d}</math>
 
When d increases asymptotically <math> d \approx l \approx x+x', \Gamma(\theta) \approx -1,G_{los} approx G_{gr}</math>
[[File:Reflection co-efficient ground reflection of radio waves.jpg|thumbnail|Reflection co-efficient tends to -1 for large d.|377x377px]]
<math>P_r =P_t ( {\frac{\lambda G}{4\pi d}} ) ^2 \times ( 1-e^{-j \Delta \phi})^2 </math>
 
Expanding <math>e^{-j\Delta \phi} </math>using [[Taylor series]] an retaining only the first term
 
<math>P_r =P_t  ( {\frac{\lambda G}{4\pi d}} ) ^2 \times (-j \Delta \phi)^2</math>
 
<math>P_r \approx P_t ({\frac{\lambda G}{4\pi d}} ) ^2 \times (\frac{4 \pi h_t h_r }{\lambda d})^2</math>
 
<math>P_r \approx P_t \frac{G h_t ^2 h_r ^2}{d^4}</math>
 
Power varies with inverse fourth power of distance for large d.
 
==In logarithmic units==
In logarithmic units : <math>P_{r_{dBm}}=P_{t_{dBm}}+ 10 log(G h_t ^2 h_r ^2) - 40 log(d)</math>
 
Path loss : <math>PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40 log(d)-10 log(G h_t ^2 h_r ^2) </math>
 
==Power vs. Distance Characteristics==
[[File:2-ray ground reflected Power vs distance.jpg|400px|thumbnail|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 a local maximum.
 
==As a case of log distance path loss model==
The standard expression of [[Log-distance path loss model|Log distance path loss model]] is
: <math>PL\;=P_{T_{dBm}}-P_{R_{dBm}}\;=\;PL_0\;+\;10\gamma\;\log_{10} \frac{d}{d_0}\;+\;X_g,</math>
The path loss of 2-ray ground reflected wave is : <math>PL\;=P_{t_{dBm}}-P_{r_{dBm}}\;=40 log(d)-10 log(G h_t ^2 h_r ^2) </math>
 
where <math> PL_0 = 40 log(d_0)-10 log(G h_t ^2 h_r ^2) </math>
<math>X_g = 0</math> and <math> \gamma = 4  for d,d_0 > d_c </math> 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==
{{cite book|last=Goldsmith|first=Andrea|title=Wireless communications|year=2004|publisher=Cambridge University Press|location=Cambridge, U.K.|isbn=978-0521837163|edition=1. publ.}}
 
{{cite book|last=Institute|first=organized by The Korean Institute of Communications and Information Sciences ; Electronics and Telecommunications Research|title=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|publisher=IEEE|location=Piscataway, NJ|isbn=978-1-4673-4827-0|pages=454–455}}
 
{{cite book|last=Rappaport|first=Theodore S.|title=Wireless communications : principles and practice|year=2002|publisher=Prentice Hall PTR|location=Upper Saddle River, NJ|isbn=978-0130422323|edition=2. ed.}}
 
[[Category:Radio frequency propagation]]

Latest revision as of 16:21, 27 December 2014


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