# 2-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.

## Mathematical Derivation

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

and the ground reflected component may be written as

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

From the figure and the path difference between then the phase difference between the waves

The power of the signal received is 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

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

Using the Taylor series of Template:Sqrt:

and taking the first two terms

Phase difference may be approximated as

When d increases asymptotically

Expanding using Taylor series

and retaining only the first two terms

Taking the magnitude

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

## In logarithmic units

## Power vs. Distance Characteristics

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

The path loss of 2-ray ground reflected wave is

where

and

## 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

## References

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