Ground Reflections
3.1
Ground Reflections
Introduction
When dealing with radio waves in a terrestrial environment we almost never observe free-space
propagation of the type we talked about in the last lesson. This is because the waves interact with
the environment: ground, trees, buildings, etc. The result is that we typically have several
reflected, refracted, diffracted waves interfering with one another at any given point. (We’ll talk
about diffraction later.) This complicates things quite a bit. In this lesson we’ll consider the
simple model of a transmitter-receiver pair above an infinite “ground plane.” We don’t live on an
infinite flat plane, but this simple model does have some explanatory power in the real world,
and it will motivate some empirical models that we’ll learn about later.
Geometry
Assume a transmitter is a height ht above a flat, smooth ground and a receiver is at a height hr
while the ground distance between them is r. This is illustrated in Figure 3.1. Let the reflection
coefficient of the ground be Γ. In our model we will assume this is a constant. In fact, as you
know from a course like EE351, this is really a function of the field polarization and the angle of
incidence.
TX
r1
ht
RX
r2
θ
θ
hr
r
Figure 3.1: Geometry for the ground reflection problem (the “tworay model”).
There are two paths for radio waves to take from transmitter to receiver: a direct path and a
reflected path. Call the length of the direct path r1 and that of the reflected path r2. The law of
reflection requires the angle of incidence to equal the angle of reflection (the angles θ in the
figure). The result is that the reflected field appears to come from a “mirror image” of the source
at a distance ht below the ground. Basic geometry gives us the distances
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08
Ground Reflections
3.2
r1 = r 2 + (ht − hr ) 2
(3.1)
r2 = r 2 + (ht + hr ) 2
We’ll be interested in cases where r >> ht , hr . For example r might be hundreds or thousands of
meters while the h’s are only a few or maybe a few tens of meters. Then the following
approximations are good (recall that for small x, 1 + x ≈ 1 + x / 2 ):
(ht − hr ) 2
2r
(h + hr ) 2
r2 ≈ r + t
2r
r1 ≈ r +
(3.2)
The path difference is
∆r = r2 − r1
≈2
(3.3)
ht hr
r
As we’d expect from Figure 3.1, this goes to zero at large distances.
Theory
The receiver will “see” two transmitters – the real transmitter above the ground and a virtual, or
mirror image, transmitter below the ground. The intensity of each of these fields in the absence
of the other would be given by the Friis equation (2.6). When both are present we cannot simply
add the powers because it is the EM fields that add, and the fields have both amplitude and
phase. The amplitude is proportional to the square root of the intensity and the phase is 2π times
the distance traveled in wavelengths. The amplitude of the reflected field is also multiplied by Γ,
the reflection coefficient of the ground. The total power is proportional to the magnitude squared
of this total field, or
PR = PT
−j
2π
r1
λ
GT GR λ e
( 4 π) 2
r1
2
+Γ
e
−j
2π 2
r2
λ
r2
(3.4)
Note that if there is no ground reflection ( Γ = 0 ) we simply get (2.6), as we should expect.
For large r, (3.2) tells us that r1 and r2 both approach r, so the phase terms in (3.4) will approach
one another in both phase and amplitude.
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08
Ground Reflections
3.3
If Γ = −1 , which would correspond to a perfectly conducting ground with a certain polarization,
the two terms will tend to cancel each other and the received power will drop very rapidly. We
have in this special case
e
−j
2π
r1
λ
r1
+Γ
e
−j
2π 2
r2
λ
r2
2π
2π
1 − j r1 − j r2
≈ 2 e λ −e λ
r
2π
− j ∆r
1
= 2 1− e λ
r
π
2
(3.5)
π
− j ∆r
1 j ∆r
= 2 e λ −e λ
r
=
2
2
π∆r
4
sin 2
2
r
λ
so
PR ≈ PT GT GR
λ2
2πht hr
sin 2
2
(2πr )
λr
(3.6)
The sine term oscillates between 0 and 1 while the “envelope” PT GT GR λ2 /(2πr ) 2 has the 1 / r 2
behavior of free-space propagation.
The approximation sin 2 1 / x = 1 / x 2 is good to 1.5 dB for x ≥ 1 . So, for large enough r, we can
write
PR ≈ PT GT GR
2
ht hr
r4
2
(3.7)
Here the received power is falling off as 1 / r 4 , much more rapidly than it would in free space. It
is also interesting to note that there is no wavelength dependence in this expression. The
envelope of (3.6) and (3.7) are equal when r = 2πht hr / λ . If we neglect the sine oscillations in
(3.6) we can create a composite propagation model as follows:
PR =
EE432: RF Engineering for Telecommunications
λ2
( 2 πr ) 2
2 2
ht hr
PT GT GR 4
r
PT GT GR
2 πht hr
λ
2 πht hr
;r ≥
λ
;r <
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(3.8)
08/25/08
Ground Reflections
3.4
This is an example of a breakpoint model, the breakpoint being r = 2πht hr / λ where the model
changes from one type of behavior to another. If we use the breakpoint as the reference distance
r0 then
PR ,dBm
r
r0
=
r
P0,dBm − 40 log
r0
P0,dBm − 20 log
; r < r0
; r ≥ r0
r0 = 2πht hr / λ
(3.9)
P0,dBm = PT ,dBm + GT ,dB + GR ,dB + 20 log
λ2
(2π) 2 ht hr
(Keep in mind that everything we’ve done is valid only for the case Γ = −1 .) The important point
here is that the presence of the ground can cause the field to decay very differently from the 1 / r 2
behavior of free space.
It is instructive to ask what happens to (3.9) if we change the height of one or both of the
antennas. First, notice that the model depends only on the product ht hr . Let’s say we change the
heights so that the new value of this product is (ht hr )' = aht hr . For example, if we double the
height of one of the antennas, then a = 2 . The new breakpoint will be r0 ' = ar0 . Since there is no
change in the transmitted power or antenna gains
P0′,dBm = P0,dBm − 20 log
λ2
λ2
+
20
log
(2π) 2 ht hr
(2π) 2 aht hr
(3.10)
= P0,dBm − 20 log a
If we are at a small distance where r < r0 , r0 ' , then the new model is
PR′,dBm = P0,dBm − 20 log a − 20 log
= P0,dBm − 20 log
= PR ,dBm
r
r0
r
ar0
(3.11)
In other words, for distances less than either breakpoint, there is no change in the received
power. This makes sense because (3.9) models the close-in field as being the same as in freespace. That is, for small distances the ground is assumed to have no effect, hence, the antenna
heights shouldn’t have any effect. On the other hand, at a large distance r > r0 , r0 '
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08
Ground Reflections
3.5
PR′, dBm = P0, dBm − 20 log a − 40 log
r
ar0
= P0, dBm − 20 log a + 40 log a − 40 log
= PR , dBm + 20 log a
r
r0
(3.12)
and received power is changed by the constant amount 20 log a . For example, if you increase
either antenna height by a factor of 2, then received power at large distances increases by 6 dB.
Figure 3.2 shows an example in which ht hr is increased by a factor of 10.
0
20
Pr (dBm)
40
60
80
100
1
0.5
0
0.5
log(r/75m)
1
1.5
2
Figure 3.2: Effect of changing antenna height. Solid curve shows
PR for rt = rr = 2m , GT = 10dB , GR = 3dB and PT = 1W . Dashed
curve is the case where we change to rt = 20m . The effect is to
extend the breakpoint to a larger distance.
Simulation
Figure 3.3 shows a numerical (Mathcad) calculation of (3.4) for the case Γ = −1 and
ht = 10λ, hr = 3λ . Also shown are 1 / r 2 and 1 / r 4 behaviors characteristic of small and large r
values.
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08
Ground Reflections
3.6
0
relative Pwr (dB)
50
100
150
200
2
1
two ray
n=2
n=4
0
1
2
3
log(r/r0)
Figure 3.3: Ground reflection model (3.4) (solid curve), envelope
of (3.6) (dotted line), and (3.7) (dashed line) for ht=10λ, hr=3λ.
The breakpoint is at 188λ.
When received power falls off as 1 / r n , we say that the propagation constant is n. For free space
the propagation constant is n = 2 . At large distances the ground-reflection model with Γ = −1
has a propagation constant of n = 4 . Thus we see that it is quite possible, even in a relatively
simple geometry, to get a propagation constant different than that of free space.
Example 3.1
Assume a propagation model of the form (3.9). Suppose a cellular base station
transmits 30dBm (1W) of power and has an antenna with gain of 7dB and height
10m. A roof-mounted car antenna is 1.5m high and has 3dB gain. The frequency
is 1900MHz. How far away can the car be and still receive –90dBm of signal?
The wavelength is 300/1900 or 0.158m. (3.9) then gives r0 = 597m, P0 =
−47.5dBm. We then solve − 90 = −47.5 − 40 log(r / 597 m) to get r = 6894m. So
the car can go about 7km from the base station.
Experiment
A 915MHz transmitter was placed 2.5ft above a sidewalk. Field strength measurements were
made at various distances and at a height of 3ft. The observed data are show in Figure 3.4 along
with (3.4) and the free-space model. A reference distance of r0 = 5 ft was used. The groundreflection model does capture the general behavior of the data. In Figure 3.5 we plot the observed
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08
Ground Reflections
3.7
data along with 1 / r n models for n = 2,3,4 . This illustrates the breakpoint concept. The
theoretical break point of 2πht hr / λ is about 48 feet in this case. Since log(48 / 5) = 0.98 we
would expect a breakpoint near 1 on the horizontal axis. This is indeed what we observe. The
data are fairly well described by n = 2 for small r and by n = 3 for large r.
30
40
Pr (dBm)
50
60
70
80
0
0.2
0.4
two-ray
observed
free-space
0.6
0.8
1
log(r/r0)
1.2
1.4
1.6
1.8
Figure 3.4: Ground reflection measurements above a sidewalk at
915MHz. Reference distance r0 is 5ft. Circles are observed data.
Solid curve is eqn. (3.4) with a Γ of −0.6. Dashed curve is freespace model.
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08
Ground Reflections
3.8
30
Pr(dBm)
40
50
60
70
80
0
0.2
0.4
n=2
n=3
n=4
observed
0.6
0.8
1
log(r/r0)
1.2
1.4
1.6
1.8
Figure 3.5: Data of Figure 3.2 compared to various 1 / r n models.
n = 2 gives a descent description for small r while n = 3 gives a
descent description of large r.
References
1. Rappaport, T. S., Wireless Communications: Principles and Practice, Prentice Hall,
1996, ISBN 0-13-375536-3.
2. Ishimaru, A., Electromagnetic Wave Propagation, Radiation, and Scattering, Prentice
Hall, 1996, ISBN 0-13-249053-6.
EE432: RF Engineering for Telecommunications
Scott Hudson, Washington State University
08/25/08