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Duke UniversityDepartment of Electrical and Computer Engineering

ECE 186Lecture 8

Two Ray Ground Reflection Model

Even when there is a direct line of sight (LOS) path between the mobile phone and the base station there istypically a strong multipath signal resulting from ground reflection. The angle of incidence is always equal tothe angle of reflection. The ground is typically treated as a perfect reflector as a first order approximation. It iscritical to understand what happens when a transverse electromagnetic wave is incident on a good conductor.The underlying boundary condition is that the tangential component of the electric field must be zero at theinterface. This means that a perpendicularly polarized EM wave must undergo a 180 phase shift at the surface.A parallel polarized EM wave simply reflects with no change in phase due to the reflection.

In summary,

for parallel polarization,

for perpendicular polarization, and

.

To analyze the ground reflection process, we must first write the expression for the transmitted electric field,which is both a function of time and space:

, Where E is the electric field magnitude, is the electric field measured at a distant , d is any distance

beyond , is the carrier angular frequency, and c = 310 8 m/s is the free space velocity of light.

The total electric field E TOT is the summation of E LOS and the ground reflected wave E g.

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To obtain the magnitude of E TOT , we must obtain the path length difference between E g and E LOS .This is easily accomplished using image theory. Consider the following diagram

By applying Pythagoreans theorem twice, we can obtain the distance the LOS wave travels and the groundreflected wave. The line of sight distance will be denoted as d and the distance traveled by the ground reflectedwave will be denoted by d.

, cos The ground reflected wave can be written as

, cos where is the reflection coefficient. The boundary conditions at the air-ground interface have already beendiscussed and are represented by: || 1 and 1. Assuming parallel polarization

, cos cos

The path length difference When the TX-RX separation distance d is very large compared to (d >> ), the path lengthdifference can be approximated by

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2 by using a Taylor series expansion on the form 1 for small . The equation form for simplification is

1 as well as 1 The Taylor series expansion of 1 about 0 (also known as the Maclaurin expansion) is 1 . becomes

1 12 2 1 12 2 2 The phase difference

and the time delay between the two wavefronts is

.

As d becomes large compared to , the and can be approximated as d for the magnitude, but not for the phase terms.

The received E field is a sinusoid with angular frequency . To find the magnitude of the E field, we can

choose any convenient time and use phasors. Consider and the use of phasors , 0 1 1 0

| | |1 1 0| 1 2 2 Using the trigonometric identity 2 2 4

| | 2 2 This is an approximation of the magnitude of the received E field for the two ray ground reflection model for

parallel polarization.

2 2