Seybold J.S. Introduction to RF Propagation

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Since the blockage is below the line of sight, the values of h and v are neg-

ative. The following parameters are known:

Using the expression for the Fresnel–Kirchhoff diffraction parameter, (8.19),

yields

From the plot in Figure 8.12, the diffraction loss can be estimated as 0.75 or

-2.5 dB, or the Lee approximation to the diffraction integral can be used

directly to get

䊐

Unfortunately, most blockages encountered in the real world are not well-

modeled by a single knife-edge diffractor. The multiple knife-edge diffraction

model given in Ref. 11 can be used, but Rappaport indicates that it tends to

oversimplify the geometry and produce somewhat optimistic results. Another

option is the rounded-surface diffraction model, where the diffraction is

treated as a broadside cylinder as shown in proﬁle in Figure 8.14 [12–15].

The diffraction from a rounded hilltop or surface is determined by com-

puting the knife-edge diffraction for the equivalent height, h, and then com-

puting the excess diffraction loss, L

, due to the rounded surface. The ﬁrst step

is to determine the radius, r, of the cylinder that circumscribes the actual dif-

fraction points on the obstacle. Then the extent of the diffraction surface, D

can be found. The expression for the excess diffraction loss is

(8.22)

=-11 7. a

()

=-20 05 062 26log . . .n dB

n=-0 395.

=- =

200 800

182 FADING AND MULTIPATH CHARACTERIZATION

Figure 8.13 Knife-edge diffraction geometry for Example 8.3.

where

and r is estimated by

It is important to note that the excess loss and the knife-edge diffraction loss

are actually expressed as gains. That is to say, a negative loss is indeed a loss,

not a gain in signal strength.

Example 8.4. Consider a point-to-multipoint communications link operating

at 5 GHz over a distance of 1 km shown in Figure 8.15a. There is a pair of

narrow hilltops in between the transmitter and receiver, located 300 m from

from the transmitter. Find the total diffraction loss if the hilltops are 10 m apart

and 3 m above the line of sight.

The individual hilltops are approximated as pointed conductive wedges or

knife edges. From the problem statement and the geometry shown in Figure

8.15b, the following observations can be made:

BB m AB m

ab m CB cb m

===

10 695

295 3

Ddd

()

GROUND-BOUNCE MULTIPATH 183

Figure 8.14 Geometry for rounded-surface diffraction model.

From this information, the angles b and g can be found

and from basic geometry, the angle, a, is found to be

Now it is possible to determine the value of h = ED, by ﬁrst solving the fol-

lowing set of equations for bD:

ED ab bD

ED AB bD

()

[]

()

tan

b 10

abg=+= ∞0 826.

bg=∞ =∞0 243 0 583., .

184 FADING AND MULTIPATH CHARACTERIZATION

(a)

(b)

300 m

700

10 m

Figure 8.15 Diffraction geometry for the double-hill blockage problem. (a) Pictorial

diagram for Example 8.4. (b) Geometric diagram for Example 8.4.

Equating these expressions and solving for bD yields

from which the value of

can be deduced.

The next step is to determine r, the radius of the circle that is tangential to

ae at point c and to AE and point C:

The interior angle of the circle,

cOC

can be determined by looking at the inte-

rior angles of the quadrangle ceCO. The result is that

The cord of this angle is

Thus cOC is an isosolicles triangle with the equal sides being of length r.

Solving for r yields

The one remaining parameter needed to compute the diffraction loss is the

Fresnel–Kirchhoff diffraction parameter. Using

and

h = 303.m

006

297 98

aD m

AD = 702.02 m

r =

()

bd 2

cos 90

a 2

694

cC bB m==10

cOC = = 0.826a∞

r ==Oc OC

ED m==h 303.

bD m= 298.

AB ab

()

tan tan

GROUND-BOUNCE MULTIPATH 185

yields

Thus the basic knife-edge diffraction loss is

The excess diffraction loss is then determined as

where a must, of course, be expressed in radians. Therefore the signal at the

receiver will be about 47.4 dB lower than it would be for the unobstructed

case. This represents a very deep fade, which is to be expected when a 5-GHz

signal is blocked by two hilltops. 䊐

For scenarios where the blockage is not well-modeled as a narrow edge of

a perfect conductor, the analysis should include the effects of nonideal reﬂec-

tion/diffraction and account for any penetration of the blockage. Such analy-

sis is best treated using the uniform theory of diffraction [16], which is beyond

the scope of this text. The UTD permits a variety of shapes, surface textures,

and conductivities to be treated. The use of knife-edge diffraction is often suf-

ﬁcient for simple geometries or cases where the speciﬁc geometry and mate-

rial properties are not known with high conﬁdence. When geometry and

material speciﬁcs are well known, the problem can be treated using the UTD,

often in the context of an advanced modeling program. For many RF propa-

gation applications, however, generalized statistical models (such as log-

normal shadowing covered in the next section) are sufﬁcient.

According to Lee [14], the diffraction loss will be slightly greater for hori-

zontal polarization than for vertical polarization. Thus the wave in the shadow

region for a veritcal polarization signal will be stronger than that of a hori-

zontal polarization signal, all else being equal. In the case of circular polar-

ization, this means that the wave in the shadow region will be elliptically

polarized (i.e., the axial ratio will be increased).

8.3 LARGE-SCALE OR LOG-NORMAL FADING

The ﬁrst step in prediction of the path loss is to determine the median path

loss. Several procedures for estimating the median path loss were presented

in Chapter 7 and in Section 8.2. The second step is to determine the large-

scale fading about that median path loss. Large-scale fading is generally attrib-

=- =-11 7 32 2..a

=---

()

=-20 0 4 0 1184 0 38 0 1 15 2

log . . . . .n dB

n=1 2095.

186 FADING AND MULTIPATH CHARACTERIZATION

utable to blockage (diffraction), which is sometimes called shadowing. In the

case where there is a fair amount of blockage, there may be several diffrac-

tion paths to the receiver.These paths are multiplicative, which means that the

effects are additive if considered in dB. If there are a sufﬁcient number of dif-

fraction points and/or multireﬂection paths on the path to the receiver, the

Central Limit Theorem may be invoked to justify using a Gaussian random

variable to represent the path loss. This is in fact what is frequently done, par-

ticularly in cellular telephony. The result is called log-normal fading, since the

fading follows a normal distribution in dB (the log domain). For this reason,

large-scale fading is modeled as a log-normal random variable. The mean path

loss is the same as the median path loss (since for a normal pdf, the mean and

median are identical). Thus, only a variance is required to characterize the log-

normal fading. This variance is called the location variability and is denoted

by s

The probability of a log-normal fade is used to determine the percentage

of annular area that is covered at a ﬁxed range from the transmitter. Figure

8.16 shows the variation of RSL with position for a ﬁxed distance, d, as the

cell perimeter is traversed. The effects of the normal pdf can be envisioned

from this plot.

LARGE-SCALE OR LOG-NORMAL FADING 187

–5

–10

–15

–20

050

100 150 200 250 300 350 400 450 500

Distance [m]

Signal level relative to median [dB]

Figure 8.16 Received signal levels relative to the median signal level in a shadowing

environment. (Figure 9.2 from Ref. 17, courtesy of Wiley.)

Figure 8.17 shows the histogram and how it compares with the normal pdf.

This provides some insight into the validity of log-normal fading models.

Figure 8.18 shows a representative plot of path loss measurements. This plot

shows the variation of the path loss at any given distance.

Example 8.5. For a given communication system, 90% coverage is desired at

the fringe (edge) of the coverage cell. Assume plane earth propagation plus

a 20-dB allocation for clutter loss, use s

= 6 dB for the shadowing location

variability,

= 1.5 m, h

= 30 m

and use a maximum allowable path loss of 140 dB. Determine the shadowing

fade margin and the distance at which it occurs (i.e., the cell radius). Repeat

the problem if s

= 8dB.

The expression for the path loss is

clutter s

()

++10

log

188 FADING AND MULTIPATH CHARACTERIZATION

0.04

0.035

0.025

0.015

0.005

0.03

0.02

0.01

–30 –20 –10 0 10 20 30 40

Shadowing Level [dB]

Probability Density

Measured

Normal Distribution

Figure 8.17 Histogram of measured shadowing loss compared to a normal pdf. (Figure

9.3 from Ref. 17, courtesy of Wiley.)

where L

is the loss due to shadowing and can be expressed as

with z being a Gaussian random variable. The probability of an outage is

which can be rewritten as

Since the shadowing pdf is normal and zero mean, the standard normal

random variable is

PPLLLL

outage S Max PL clutter

=>--

()

PPLL

outage Max

()

LARGE-SCALE OR LOG-NORMAL FADING 189

–60

–70

–80

–90

–100

–110

–120

–130

–140

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Distance from Base Station [m]

Maximum Cell

Range

Maximum

Acceptable

Path Loss

Median

Path Loss

Fade

Margin, z dB

Total Path Loss [–dB]

Figure 8.18 Total path loss versus distance from the transmitter in a shadowing envi-

ronment. (Figure 9.4 from Ref. 17, courtesy of Wiley.)

which yields

The net fade margin for the link that can be allocated to shadowing is set equal

to the shadowing loss,

and the maximum allowable path loss is

The shadowing fade margin (based on the probability of a shadowing fade) is

The value of s

is known; and from a Q table (see Table A.1) at p = 0.1, z can

be found to be 1.28. Thus the required shadowing margin to ensure 90% cov-

erage when s

= 6 dB is

and the maximum allowable path loss is 112.3 dB, so

which yields a cell radius of

If instead of s

= 6, s

= 8 dB is used, then

and

Ld hh d

PL L m b

()

=- -=-

()

140 1 28 20 20 40

10 24

33 06

. log log

1234

1244344

d = 43.km

20 112 3

log .

= 77.dB

Ld L

PL s

()

=-120

ML L L d L

S Max PL clutter

== -

()

PPz

LLL

PPQ

LLL

outage

Max PL clutter

outage

Max PL clutter

erage outage

Max PL clutter

=- =-

cov

11 1

190 FADING AND MULTIPATH CHARACTERIZATION

䊐

In summary, use 1 - p, the probability of a shadowing fade, and a Q table to

look up z; then z times s

, the shadowing location variability, gives the numer-

ical value of L

, the required shadowing fade margin for the link.

By computing the available link margin as a function of distance, it is pos-

sible to generate a curve of the percent coverage versus distance for a given

location variability. This is shown in Figure 8.19. Note that the curves all cross

at the 50% point as expected since at 50% implies that the probability of being

above or below the median is the same regardless of the location variability

and the path loss is equal to the median.

Instead of computing the percent coverage at the edge of the cell, one can

average over all of the cell area. This can be accomplished by simply comput-

ing the coverage over a series of narrow concentric rings and then summing

them up [18], where it is implicitly assumed that the coverage area is circular.

This is shown conceptually in Figure 8.20. The area of each ring is 2pdDd. Each

ring area is multiplied by its corresponding coverage probability, and then they

are added and divided by the total area to get the overall cell coverage

probability:

d = 37.km

LARGE-SCALE OR LOG-NORMAL FADING 191

0 2000 4000 6000 8000 1

100

Sigma = 6 dB

Sigma = 8 dB

Sigma = 10 dB

Percent Coverage vs. Distance

Distance (m)

Percent Coverage

Figure 8.19 Percent coverage versus distance for log-normal fading.