Seybold J.S. Introduction to RF Propagation

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specular reﬂection with a 180-degree phase shift. This is a worst-case analysis,

which provides a good upper bound on the predicted path loss. It is note-

worthy that this path-loss expression is independent of the wavelength.

The exact ground-bounce expression is seldom used since the geometry is

rarely precise enough to locate the peaks and nulls accurately. The recom-

mended approach is to compute the free-space loss and the approximate

ground-bounce path loss and use whichever gives greater loss.

(8.14)

The crossover point is deﬁned as the distance at which the 1/d

approximation

and free-space loss are equal. It is sometimes helpful to determine where the

crossover point occurs and then use the approximation for ranges greater than

the crossover range and free-space loss otherwise.The crossover point is found

by equating the approximate ground-bounce path loss (8.13), and the free-

space loss and solving for d.

(8.15)

The computed crossover range for the plot in Figure 8.5 is 12.6 km, which

is consistent with the curves shown. By ﬁrst computing the crossover point,

one can immediately determine which path loss expression applies to a given

situation.

Example 8.1. Consider the following point-to-point communications link:

What is the predicted path loss for this link?

Since no antenna gains are given, normalized gains (i.e., gain = 0dB) for

transmit and receive are used. The ﬁrst step is to compute the free-space loss

using antenna gain values of unity:

FSL

FSL dB

110 5

log

()

2015

GHz ml .

()

tr T R

()

min ,

172 FADING AND MULTIPATH CHARACTERIZATION

Since d >> h

and d >> h

and in fact, dl>>h

, the 1/d

approximation can

be used:

So the approximated path loss is less loss than free-space loss, and the free-

space result should be used: PL =-110.5 dB.

If instead of 4 km, d is 40 km, then

and

So in this case, the path loss from the (h

)

approximation would be used.

䊐

In summary, when operating over a ﬂat reﬂective surface, the path loss may

be more or less than that predicted by free-space loss. The 6-dB improvement

=-144 1.dB

FSL dB=-130 5.

()

=¥ =-

3 91 10 104 1..dB

GROUND-BOUNCE MULTIPATH 173

0 5 10 15 20 25 30

140

120

100

Ground Bounce

Approximation

Free-Space Loss

Effect of Specular Reflection for G=1

Distance (km)

Path Loss (dB)

Crossover

Poin

Figure 8.5 Comparison of free-space loss, ground-bounce loss, and the ground-bounce

loss approximation, for h

= 100m, h

= 3m, and l=0.3 m.

cannot be ensured unless operating under very controlled conditions, such as

a radar instrumentation range. For large distances, the path loss is proportional

to (h

)

and is not a function of the wavelength. One may compute the loss

both ways and use the method that predicts the greatest path loss or deter-

mine the crossover point and use free-space loss for ranges less than the

crossover range and the (h

)

approximation for ranges greater than the

crossover range.

A similar expression for the ground-bounce path loss over a curved reﬂec-

tive surface can also be developed. One such derivation is presented in Ref.

2. The curved reﬂective formulation applies to links that operate near the

earth’s surface and close to the horizon. For most applications, the ﬂat reﬂec-

tive surface formulation is adequate.

8.2.1 Surface Roughness

When determining if ground reﬂection is likely to be signiﬁcant, a means of

quantifying the smoothness (ﬂatness) of the reﬂecting surface is required. The

Rayleigh criterion provides a metric of surface roughness. The Rayleigh rough-

ness is derived based on the terrain variation (Dh) that will provide a 90-degree

phase shift at the receiver between a reﬂection at a terrain peak versus a reﬂec-

tion from a terrain valley at the same distance (see Figure 8.6). The Rayleigh

criterion is given by [3]

(8.16)

where

l is the wavelength

q is the grazing angle of the reﬂection

From the geometry shown in Figure 8.6, it can be determined that

where

sin q

()

+hh

()

q8 sin

174 FADING AND MULTIPATH CHARACTERIZATION

Figure 8.6 Geometry of surface roughness.

is the receiver height

is the transmitter height

d is the communication distance

The Rayleigh criterion can then be expressed as [3]

(8.17)

When the extent of the terrain features, Dh, is less than H

, the surface can be

treated as being smooth. Thus the amount of terrain variation that is tolera-

ble is proportional to the wavelength and the distance traversed and inversely

proportional to the height of the antennas above the surface. When a reﬂec-

tive surface is rough, meaning that the extent of the terrain features, Dh,is

much larger than H

, then the reﬂections from the surface will be diffuse and

are characterized as scattering rather than reﬂection.

8.2.2 Fresnel Zones

Consider the point-to-point link depicted in Figure 8.7. The vector TR is the

line of sight between the transmitter and the receiver and the link distance is

+ d

If there is a diffraction point at P, the signal TPR will combine with TR at

R. TPR traverses a slightly greater distance than TR and therefore will have

a different phase.

The direct and reﬂected/diffracted path lengths can be expressed as

So the path-length difference is

If h << d

, and h << d

, then the binomial expansion can be used:

D= + + + - -dh dhdd

TPR

=+++

dh dh

()

GROUND-BOUNCE MULTIPATH 175

Figure 8.7 Fresnel zone geometry.

Making the appropriate substitutions in the equation for D, yields

The corresponding phase difference is

where

(8.18)

The Fresnel–Kirchhoff diffraction parameter is often used to shorten the nota-

tion in Fresnel zone analyses and is deﬁned as

(8.19)

If the diffraction point is below the line of sight (LOS), then h is negative

and n will also be negative. When the diffraction point is located on the

LOS, h and n are both equal to zero. If the blockage is the horizon, then

the “h and n equal zero” case corresponds to the maximum LOS

distance.

The cases where D=nl/2, where n is an integer, can be found by setting

f=np in (8.18), which yields the following equation:

()

=◊

hdd

2 D

D= +

D= + + + - -d

dh d

+@ +

176 FADING AND MULTIPATH CHARACTERIZATION

Thus

The destructive reﬂection/diffraction points can then be identiﬁed by deﬁning

a term, h

, such that

(8.20)

Reﬂectors/diffraction at h

for odd values of n will cause destructive interfer-

ence. Since the difference in path lengths is on the order of l, the reﬂected/dif-

fracted signal may be as strong as the direct signal and cause cancellation.

The equation for h

deﬁnes a sequence of ellipsoids with the transmit and

receive antennas as the foci. Diffractors or reﬂectors at the odd-numbered

Fresnel zone boundaries will cause destructive interference. Figure 8.8 shows

a diagram of the Fresnel zones deﬁned by a point-to-point link. Note that this

diagram is two-dimensional, whereas the actual Fresnel zones are three-

dimensional ellipsoids. For large h or small d

and d

, the antenna pattern may

attenuate the undesired signal. For omnidirection (vertical) antennas, there

may be attenuation of the undesired signal in elevation, but not in azimuth.

From the preceding analysis, it is clear that any reﬂectors/diffractors within

the ﬁeld of view should not be near an odd Fresenel zone boundary to avoid

signal loss. It is also important that the ﬁrst Fresnel zone be clear of obstruc-

tions because this can seriously degrade the available signal energy. Due to

Huygen’s principle, covered in the next section, the diffracted electromagnetic

energy that ﬁlls the shadow at the receive end of the link reduces the energy

that arrives at the receiver. If the ﬁrst Fresnel zone is not clear, then free-space

loss does not apply and an adjustment term must be included. For most appli-

ndd

GROUND-BOUNCE MULTIPATH 177

Third Fresnel

Zone

Fourth Fresnel

Zone

Second Fresnel Zone

First Fresnel Zone

Figure 8.8 Fresnel zones between a transmitter and receiver.

cations, having 60% of ﬁrst Fresnel zone clear is sufﬁcient. The 60% clear

applies to the radius of the ellipsoid as shown in Figure 8.9. At the 0.6h point,

the Fresnel–Kirchhoff diffraction parameter is n =-0.8* and the resulting dif-

fraction loss will be 0 dB [4]. The quantiﬁcation of diffraction loss is discussed

in Section 8.2.4.

Example 8.2. Consider a point-to-point communication system, with d = 1km

and f = 28 GHz. If there is a building present, 300 m from one end of the link,

how far must it be (in elevation or height) from the LOS to not impede trans-

mission? (That is, ﬁnd 60% of the ﬁrst Fresnel zone radius at 300 m.)

The parameters for determination of the Fresnel zone radius at 300 m are

when eqn (8.20) is applied, the resulting expression for the Fresnel zone radii

is:

So the rooftop must be at least 0.9 m below the LOS to keep the ﬁrst Fresnel

zone 60% clear. It is also important that the rooftop not be near one of the odd

Fresnel zone boundaries to prevent destructive interference from reﬂections.

䊐

It should be emphasized again that reﬂectors or diffractors near the odd

Fresnel zone boundaries produce destructive interference and should be

avoided. In addition, blockage within the ﬁrst Fresnel zone may result in

reduced energy at the receiver.

=◊2 247.

300 700

0 107

.l m

178 FADING AND MULTIPATH CHARACTERIZATION

0.6

0.4

First Fresnel

Zone Ellipsoid

Maximum

Allowable

Blockage

Keep Clear

Figure 8.9 Fresnel zone blockage geometry.

is negative since the blockage is below the LOS, i.e. the LOS is not obstructed.

8.2.3 Diffraction and Huygen’s Principle

Diffraction is the physical phenomenon whereby an electromagnetic wave can

propagate over or around objects that obscure the line of sight. Diffraction

has the effect of ﬁlling in shadows, so that some amount of electromagnetic

energy will be present in the shadowed region. The easiest way to view the

effect of diffraction is in terms of Huygen’s Principle. Huygen’s Principle states

that each point on a wavefront acts as the source of a secondary “wavelet”

and all of these “wavelets” combine to produce a new wavefront in the direc-

tion of propagation [4–6]. This is illustrated in Figure 8.10.

The “wavelets” from a plane wave generate a plane wave (propagate it) if

the extent of the wave front is inﬁnite. If an object blocks part of the wave

front, then the “wavelets” near the blockage are not counterbalanced, and

radiation in directions other than that of the plane wave will occur. This is dif-

fraction, which causes partial ﬁlling of shadows. Regardless of whether the

blockage is conductive or nonconductive, the diffraction still occurs. If the

blockage is nonconductive, it must not pass any of the RF energy for the dif-

fraction analysis to be valid. The same effect can occur when a signal source

is beyond the earth’s horizon.While the line of sight is blocked, a small amount

of RF energy will be diffracted over the surface and will appear at the receiver.

8.2.4 Quantifying Diffraction Loss

Accurate modeling of losses due to diffraction is very challenging for all but

the most basic geometries. One such basic geometry is the knife-edge diffrac-

tor, sometimes called a wedge diffractor. This effect has been mathematically

characterized and can be used to provide an estimate of the diffraction loss

for a single knife edge. The effect of a plane wave incident on a perpendicu-

lar conductive barrier can be divided into three shadow regions as shown in

GROUND-BOUNCE MULTIPATH 179

Reduced wave strength in

direction of propagation

Shadow region is

partially illuminated

Wavefront

Physical blockage

Figure 8.10 Illustration of Huygen’s Principle and diffraction.

Figure 8.11 [7]. Region I contains direct, reﬂected, and diffracted rays, region

II contains direct and diffracted rays only, and region III contains diffracted

rays only. This explains why a knife edge that is below the line of sight may

still affect the received wave.

The analysis in this section is based on a conductive barrier. These results

are often applied to scenarios where the barriers are not conductive; however,

they may not be as precise, depending upon the reﬂection properties of the

barrier. The knife-edge diffracting point may be above, below, or directly on

the line of sight between the transmitter and receiver. The electric ﬁeld due to

the diffracted path is given by the diffraction integral,

where E

is the electric ﬁeld at the receiver based on free-space loss only and

n is the Fresnel–Kirchhoff diffraction parameter deﬁned earlier. Evaluation of

the diffraction integral is usually done numerically or graphically [8]. Lee [9]

provides an approximation to the diffraction integral.

(8.21a)

(8.21b)

(8.21c)

(8.21d)

(8.21e)

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180 FADING AND MULTIPATH CHARACTERIZATION

Reflected field

shadow boundary

Incident field

shadow boundary

III

Conducting

half-plane

P (r,f)

f¢

-•

Figure 8.11 Shadow boundaries for diffraction by an inﬁnite conducting half-plane.

(Figure 12.12 from Ref. 7, courtesy of Wiley.)

Figure 8.12 is a plot of the diffraction loss as a function of the Fresnel–

Kirchhoff diffraction parameter, n. Be aware that some authors deﬁne the

Fresnel–Kirchhoff diffraction parameter with the opposite sign from that used

herein. In that case the sign must be reversed to use the plot in Figure 8.12 or

incorrect results will be obtained. When n is equal to zero, the ﬁrst Fresnel

zone is 50% blocked and the corresponding signal loss is 6 dB.

Example 8.3. Consider a communication link comprised of two 150-MHz

hand-held radios separated by 1 km as shown in Figure 8.13. The barrier

between the two radios runs perpendicular to the line of sight and is 5 m below

the line of sight. Assume that the barrier is a thin, solid fence that is 200m

from one end of the link. How much additional path loss (beyond free-space

loss) can be expected due to the diffraction from the fence?

GROUND-BOUNCE MULTIPATH 181

Figure 8.12 Loss due to diffraction. (Figure 7 from Ref. 10, courtesy of ITU.)