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9 ShadowingBeware lest you lose the substance by grasping at the shadow.
Aesop, Greek slave and fable author
9.1 INTRODUCTION
The models of macrocellular path loss described in Chapter 8 assume that path loss is a
function only of parameters such as antenna heights, environment and distance. The predicted
path loss for a system operated in a particular environment will therefore be constant for a
given base-to-mobile distance. In practice, however, the particular clutter (buildings, trees)
along a path at a given distance will be different for every path, causing variations with respect
to the nominal value given by the path loss models, as shown by the large scatter evident in the
measurements in Figure 8.2. Some paths will suffer increased loss, whereas others will be less
obstructed and have an increased signal strength, as illustrated in Figure 9.1. This phenom-enon is called shadowing or slow fading. It is crucial to account for this in order to predict the
reliability of coverage provided by any mobile cellular system.
9.2 STATISTICAL CHARACTERISATION
If a mobile is driven around a base station (BS) at a constant distance, then the local mean
signal level will typically appear similar to Figure 9.2, after subtracting the median (50%)
level in decibels. If the probability density function of the signal is then plotted, a typical
result is Figure 9.3. The distribution of the underlying signal powers is log-normal; that is, the
signal measured in decibels has a normal distribution. The process by which this distribution
comes about is known as shadowing or slow fading. The variation occurs over distances
comparable to the widths of buildings and hills in the region of the mobile, usually tens or
hundreds of metres.
Antennas and Propagation for Wireless Communication Systems Second Edition Simon R. Saunders andAlejandro Aragon-Zavala
2007 John Wiley & Sons, Ltd
Path 1 Path 2
Path 3
Basestation
1 2
3 Mobilelocation
Figure 9.1: Variation of path profiles encountered at a fixed range from a base station
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0 50 100 150 200 250 300 350 400 450 500
-20
-15
-10
-5
0
5
10
15
20
25
Distance [m]
Signallevelrelativetomedia
n[dB]
Figure 9.2: Typical variation of shadowing with mobile position at fixed BS distance
30 20 10 0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Shadowing level [dB]
P
robabilitydensity
MeasuredNormal distribution
Figure 9.3: Probability density function of shadowing. Measured values are produced by subtract-ing the empirical model shown in Figure 8.2 from the total path loss measurements. Theoretical
values come from the log-normal distribution
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The standard deviation of the shadowing distribution (in decibels) is known as the location
variability, L. The location variability varies with frequency, antenna heights and theenvironment; it is greatest in suburban areas and smallest in open areas. It is usually in the
range 512 dB (Section 9.5); the value in Figures 9.2 and 9.3 is 8 dB.
9.3 PHYSICAL BASIS FOR SHADOWING
The application of a log-normal distribution for shadowing models can be justified as follows.
If contributions to the signal attenuation along the propagation path are considered to act
independently, then the total attenuation A, as a power ratio, due to Nindividual contributions
A1, . . ., AN will be simply the product of the contributions:
A A1 A2 . . . AN 9:1
If this is expressed in decibels, the result is the sum of the individual losses in decibels:
L L1 L2 . . . LN 9:2
If all of theLi contributions are taken as random variables, then the central limit theorem holds
(Appendix A) and L is a Gaussian random variable. Hence A must be log-normal.
In practice, not all of the losses will contribute equally, with those nearest the mobile end
being most likely to have an effect in macrocells. Moreover, as shown in Chapter 8, the
contributions of individual diffracting obstacles cannot simply be added, so the assumption of
independence is not strictly valid. Nevertheless, when the different building heights, spacings
and construction methods are taken into account, along with the attenuation due to trees,
the resultant distribution function is indeed very close to log-normal [Chrysanthou, 90]
[Saunders, 91].
9.4 IMPACT ON COVERAGE
9.4.1 Edge of Cell
When shadowing is included, the total path loss becomes a random variable, given by
L
L50
Ls
9:3
where L50 is the level not exceeded at 50% of locations at a given distance, as predicted by any
standard path loss model (the local median path loss) described in Chapter 8. Ls is the
shadowing component, a zero-mean Gaussian random variable with standard deviation L.The probability density function of Ls is therefore given by the standard Gaussian formula
(Appendix A equation (A.16)):
pLS 1Lffiffiffiffiffiffi2
p exp L2S
22L
!9:4
In order to provide reliable communications at a given distance, therefore, an extra fade
margin has to be added into the link budget according to the reliability required from the
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system. In Figure 9.4, the cell range would be around 9.5 km if shadowing were neglected,
then only 50% of locations at the edge of the cell would be properly covered. By adding the
fade margin, the cell radius is reduced to around 5.5 km but the reliability is greatly increased,
as a much smaller proportion of points exceed the maximum acceptable path loss.
The probability that the shadowing increases the median path loss by at least z [dB] is then
given by
PrLS > z Z1
LSzpLSdLS
Z1LSz
1
L
ffiffiffiffiffiffi2
p exp L2S
22L
!dLS 9:5
It is then convenient to normalise the variable z by the location variability:
PrLS > z Z1
xz=L
1ffiffiffiffiffiffi2
p exp x2
2
!dx Q z
L
9:6
where the Q(.) function is the complementary cumulative normal distribution. Values for Q
are tabulated in Appendix B, or they can be calculated from erfc(.), the standard cumulative
error function, using
Qt 1ffiffiffiffiffiffi2
p Z1
xtexp x
2
2
dx 1
2erfc tffiffiffi
2p 9:7
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-140
-130
-120
-110
-100
-90
-80
-70
-60
Distance from base station [m]
Totalpathloss[-dB]
Maximum
acceptable
path loss
Medianpath loss
Fademargin, z [dB]
Maximum cellrange
Figure 9.4: Effect of shadowing margin on cell range
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Q(t) is plotted in Figure 9.5 and can be used to evaluate the shadowing margin needed for
any location variability in accordance with Eq. (9.7) by putting t z=L, as described inExample 9.1.
Example 9.1
A mobile communications system is to provide 90% successful communications at the
fringe of coverage. The system operates in an environment where propagation can be
described by a plane earth model plus a 20 dB clutter factor, with shadowing of location
variability 6 dB. The maximum acceptable path loss for the system is 140 dB. Antenna
heights for the system are hm
1:5mand hb
30 m. Determine the range of the system.
How is this range modified if the location variability increases to 8 dB?
Solution
The total path loss is given by the sum of the plane earth loss, the clutter factor and the
shadowing loss:
Ltotal LPEL Lclutter LS 40 log r 20 log hm 20 log hb 20 LS
To find LS, we take the value of t z=L for which the path loss is less than themaximum acceptable value for at least 90% of locations, or when Qt 10% 0:1.
0 0.5 1 1.5 2 2.5 3 3.5 410-5
10-4
10-3
Q(t)
10-2
10-1
100
Q
Figure 9.5: The Q function
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From Figure 9.5 this occurs when t% 1.25. Multiplying this by the location variabilitygives
LS z tL 1:25 6 7:5 dB
Hence
log r 140 20log1:5 20 log 30 20 7:540
3:64
So the range of the system is r 103:64 4:4 km.IfL rises to 8 dB, the shadowing margin Ls 10 dB and d 3:8 km. Thus shadowinghas a decisive effect on system range.
In the example above, the system was designed so that 90% of locations at the edge of the
cell have acceptable coverage. Within the cell, although the value of shadowing exceeded for
90% of locations is the same, the value of the total path loss will be less, so a greater
percentage of locations will have acceptable coverage.The calculation in the example may be rearranged to illustrate this as follows. The
probability of outage, i.e. the probability that LT > 140 dB is
Outage probability pout PrLT > 140 PrLPEL Lclutter LS > 140 PrLS > 140 LPEL Lclutter
Q 140 LPEL LclutterL
pout
9:8
Consequently, the fraction of locations covered at a range r is simply
Coverage fraction per 1 pout 9:9
Note that the outage calculated here is purely due to inadequate signal level. Outage may also
be caused by inadequate signal-to-interference ratio, and this is considered in Section 9.6.2. In
general terms, Eq. (9.9) can be expressed as
per 1 Q Lm LrL ! 1 Q
M
L ! 9:10where Lm is the maximum acceptable path loss and L(r) is the median path loss model,
evaluated at a distance r: M Lm Lr is the fade margin chosen for the system.This variation is shown in Figure 9.6, using the same values as Example 9.1. The
shadowing clearly has a significant effect on reducing the cell radius from the value predicted
using the median path loss alone, which would be around 6.7 km. It is also important to have a
good knowledge of the location variability; this is examined in Section 9.5.
9.4.2 Whole Cell
Figure 9.6 shows that, although locations at the edge of the cell may only have a 90% chance
of successful communication, most mobiles will be closer to the base station than this, and
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they will therefore experience considerably better coverage. It is perhaps more appropriate to
design the system in terms of the coverage probability experienced over the whole cell. Thefollowing analysis is similar to that in [Jakes, 94].
Figure 9.7 shows a cell of radius rmax, with a representative ring of radius r, small widthr,
within which the coverage probability is pe(r). The area covered by the ring is 2rr. Thecoverage probability for the whole cell, pcell, is then the sum of the area associated with all
such rings from radius 0 to rmax, multiplied by the corresponding coverage percentages and
0 1000 2000 3000 4000 5000 6000 7000 800030
40
50
60
70
80
90
100
Distance from base station [m]
Percen
tageoflocationsadequatelycovered[%]
sL = 6 dB
8 dB
10 dB
Figure 9.6: Variation of coverage percentage with distance
r
rmax
r
Figure 9.7: Overall cell coverage area by summing contributions at all distances
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0 1 2 3 4 5 6 7 80.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
s /n
Fractionofcellareawithacceptablesignalstrength
pe = 0.5
pe = 0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
Figure 9.8: Probability of availability over whole cell area, with pe as a parameter
0 1 2 3 4 5 6 7 80.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
s/ n
Fractionofcellareaw
ithacceptablesignalstrength
1 dB
10 dB
Figure 9.9: Probability of availability over whole cell area, with fade margin as a parameter,
varying from 110 dB in steps of 1 dB
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divided by the area of the whole cell, r2max. As the radius of the rings is reduced, thesummation becomes an integral in the limit r! 0, and we have
pcell 1r2max
Zrmaxr
0
per 2r dr 2r2max
Zrmaxr0
rperdr 9:11
After substituting using (9.10) and (9.7), this yields
pcell 12
1r2max
ZrmaxD0
r erfLm Lr
Lffiffiffi
2p
dr 9:12
where erfx 1 erfcx.This may be solved numerically for any desired path loss model L(r). In the special case of
a power law path loss model, the result may be obtained analytically. If the path loss model is
expressed as (8.2)
Lr Lrref 10n log rrref
9:13
where n is the path loss exponent, then the eventual result is
pcell permax 12
expA 1 erf B 9:14
where
A Lffiffiffi
2p10n log e
2 2M
10n log eB L
ffiffiffi2p
10n log e M
Lffiffiffi
2p 9:15
Note the direct dependence ofpcell on pe(rmax), the cell edge availability. Results from (9.14)
are illustrated in Figures 9.8 and 9.9.
9.5 LOCATION VARIABILITY
Figure 9.10 shows the variation of the location variability L with frequency, as measured byseveral studies. It is clear there is a tendency for L to increase with frequency and that itdepends upon the environment. Suburban cases tend to provide the largest variability, due to
the large variation in the characteristics of local clutter. Urban situations have rather lower
variability, although the overall path loss would be higher. No consistent variation with range
has been reported; the variations in the [Ibrahim, 83] measurements at 29 km are due to
differences in the local environment. Note also that it may be difficult to compare values from
the literature as shadowing should properly exclude the effects of multipath fading, which
requires careful data averaging over an appropriate distance. See Chapters 10 and 19 for
discussion of this point.
Figure 9.10 also includes plots of an empirical relationship fitted to the [Okumura, 68]
curves and chosen to vary smoothly up to 20 GHz. This is given by
L 0:65logfc2 1:3logfc A 9:16
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where A
5:2 in the urban case and 6.6 in the suburban case. Note that these values apply
only to macrocells; the levels of shadowing in other cell types will be described in Chapters1214.
9.6 CORRELATED SHADOWING
So far in this chapter, the shadowing on each propagation path from base station to mobile has
been considered independently. In this section we consider the way in which the shadowing
experienced on nearby paths is related. Consider the situation illustrated in Figure 9.11. Two
mobiles are separated by a small distance rm and each can receive signals from two base
100 200 300 400 500 700 1000 2000 3000 5000 7000 10000 200002
3
4
5
6
7
8
9
10
11
12
Frequency [MHz]
Egli
Ibrahim 2 km
Ibrahim 9 km
Reudink
Ott
Black
Urban Empirical Model
Suburban Empirical Model
Okumura UrbanOkumura Suburban Rolling Hills
StandarddeviationsL[d
B]
Figure 9.10: Location variability versus frequency. Measured values from [Okumura, 68], [Egli, 57],
[Reudink, 72], [Ott, 74], [Black, 72] and [Ibrahim, 83]. [After Jakes, 94]
Base 1
Base 2
Mobile 1
Mobile 2
S11
S12
S21
S22
rm
Figure 9.11: Definitions of shadowing correlations
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stations. Alternatively, the two mobile locations may represent two positions of a single
mobile, separated by some time interval. Each of the paths between the base and mobile
locations is marked with the value of the shadowing associated with that path. Each of the four
shadowing paths can be assumed log-normal, so the shadowing values S11, S12, S21 and S22 are
zero-mean Gaussian random variables when expressed in decibels. However, they are not
independent of each other, as the four paths may include many of the same obstructions in thepath profiles. There are two types of correlations to distinguish:
Correlations between two mobile locations, receiving signals from a single base station,such as between S11 and S12 or between S21 and S22. These are serial correlations, or
simply the autocorrelation of the shadowing experienced by a single mobile as it moves.
Correlations between two base station locations as received at a single mobile location,such as between S11 and S21 or between S12 and S22. These are site-to-site correlations or
simply cross-correlations.
These two types are now examined individually in terms of their effects on system perfor-
mance and their statistical characterisation.
9.6.1 Serial Correlation
The serial correlation is defined by Eq. (9.17):
rsrm ES11S12
129:17
where 1 and 2 are the location variabilities corresponding to the two paths. It is reasonable
to assume here that the two location variabilities are equal as the two mobile locations willtypically be sufficiently close together that they encounter the same general category of
environment, although the particular details of the environment close to the mobile may be
significantly different. Equation (9.18) may therefore be applied:
rsrm ES11S12
2L9:18
The serial correlation affects the rate at which the total path loss experienced by a mobile
varies in time as it moves around. This has a particularly significant effect on power control
processes, where the base station typically instructs the mobile to adjust its transmit power so
as to keep the power received by the base station within prescribed limits. This process has to
be particularly accurate in CDMA systems, where all mobiles must be received by the base
station at essentially the same power in order to maximise system capacity. If the shadowing
autocorrelation reduces very rapidly in time, the estimate of the received power which the
base station makes will be very inaccurate by the time the mobile acts on the command, so the
result will be unacceptable. If, on the contrary, too many power control commands are issued,
the signalling overhead imposed on the system will be excessive.
Measurements of the shadowing autocorrelation process suggest that a simple, first-order,
exponential model of the process is appropriate [Marsan, 90]; [Gudmundson, 91], charac-terised by the shadowing correlation distance rc, the distance taken for the normalised
autocorrelation to fall to 0.37 (e1), as shown in Figure 9.12. This distance is typically a few
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tens or hundreds of metres, with some evidence that the decorrelation distance is greatest atlong distances (e.g. rc 44 m at 1.6 km range, rc 112 m at 4.8 km range [Marsan, 90]).This corresponds to the widths of the buildings and other obstructions which are found closest
to the mobile. The path profile changes most rapidly close to such obstructions as the mobile
moves around the base station.
Such a model allows a simple structure to be used when simulating the shadowing process;
Figure 9.13 shows an appropriate method. Independent Gaussian samples with zero mean and
unity standard deviation are generated at a rate T, the simulation sampling interval. Individual
samples are then delayed by T, multiplied by the coefficient a and then summed with the new
samples. Finally, the filtered samples are multiplied by Lffiffiffiffiffiffiffiffiffiffiffiffiffi1 a2p
, so that they have a
standard deviation of L as desired.
0
0.2
0.4
0.6
0.8
1
Distance moved by mobile between shadowing samples, rm[m]
S(rm)
1/e
Shadowingautocorrelation
Figure 9.12: Shadowing autocorrelation function
a
T
Independent
gaussian
samples
x
+ 10x/20S [dB]
Linearvoltage
x
L 1 a2
Figure 9.13: Method for generating correlated shadowing process
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The result is a process with the correlation function shown in Figure 9.12. Any desired
correlation distance can obtained by setting a in accordance with
a evT=rc 9:19
where v is the mobile speed in [m s
1
]. A typical output waveform is shown in Figure 9.14.
9.6.2 Site-to-Site Correlation
The site-to-site cross-correlation is defined as follows:
rc ES11S21
129:20
In this case the two paths may be very widely separated and different in length. Although they
may also involve rather different environments, the location variability associated with the
paths may also be different.The two base stations involved in the process may be on the same channel, in which case
the mobile will experience some level of interference from the base station to which it is not
0 10 20 30 40 50 60 70 80 90 10015
10
5
0
5
10
15
20
25
Time [s]
Relativepower
[dB]
Figure 9.14: A simulated correlated shadowing process, generated using the approach shown in
Figure 9.13. Parameters are vehicle speed 5 0 k m h1, correlation distance 100 m, location var-iability 8 dB
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currently connected. The system is usually designed to avoid this by providing sufficient
separation between the base stations so that the interfering base station is considerably further
away than the desired one, resulting in a relatively large signal-to-interference ratio (C/I). If
the shadowing processes on the two links are closely correlated, the C/I will be maintained
and the system quality and capacity is high. If, by contrast, low correlation is produced, the
interferer may frequently increase in signal level while the desired signal falls, significantlydegrading the system performance.
As an example, consider the case where the path loss is modelled by a power law model,
with a path loss exponent n. It can then be shown that the downlink carrier-to-interference
ratio R [dB] experienced by a mobile receiving only two significant base stations is itself a
Gaussian random variable, with mean R and variance 2
R, given by
R ER 10n log r2r1
9:21
2
R ER2
ER2
2
1 2
2 2rc12 9:22where r1 and r2 are the distances between the mobile and base stations 1 and 2, respectively.
Clearly the mean is unaffected by the shadowing correlation, whereas the variance decreases
as the correlation increases, reaching a minimum value of 0 when rc 1. Just as theprobabilities associated with shadowing on a single path were calculated in Section 9.4 using
the Q function, so this can be applied to this case, where the probability of R being less than
some threshold value RT is
PrR < R
T 1
Q
RT RR
9:23In the case where 1 2 L (i.e. the location variability is equal for all paths), Eq. (9.23)becomes
PrR < RT 1 Q RT RL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi21 rc
p
9:24
This is plotted in Figure 9.15 for L
8 dB with various values ofrc. Figure 9.15 effectively
represents the outage probability for a cellular system in which the interference is dominated
by a single interferer. The difference between rc 0:8 and rc 0 is around 7 dB for anoutage probability of 10%. With a path loss exponent n 4, the reuse distance r2 would thenhave to be increased by 50% to obtain the same outage probability. This represents a very
significant decrease in the system capacity compared to the case when the correlation is
properly considered. Further discussion of the effects of the correlation on cellular system
reuse is given in [Safak, 91], including the effects of multiple interferers, where the
distribution of the total power is no longer log-normal, but can be estimated using methods
described in [Safak, 93]. It is therefore clear that the shadowing cross-correlation has a
decisive effect upon the system capacity and that the use of realistic values is essential to
allow accurate system simulations and hence economical and reliable cellular system design.
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Here are some other system design issues which may be affected by the shadowing cross-
correlation:
Optimum choice of antenna beamwidths for sectorisation. Performance of soft handover and site diversity, including simulcast and quasi-synchro-
nous operation, where multiple base sites may be involved in communication with a single
mobile. Such schemes give maximum gain when the correlation is low, in contrast to the
conventional interference situation described earlier.
Design and performance of handover algorithms. In these algorithms, a decision to handover to a new base station is usually made on the basis of the relative power levels of the
current and the candidate base stations. In order to avoid chatter, where a large number
of handovers occur within a short time, appropriate averaging of the power levels must be
used. Proper optimisation of this averaging window and of the handover process in general
requires knowledge of the dynamics of both serial and site-to-site correlations, particu-
larly for fast-moving mobiles.
Optimum frequency planning for minimised interference and hence maximised capacity. Adaptive antenna performance calculation (Chapter 18).
Unfortunately, there is currently no well-agreed model for predicting the correlation. Here anapproximate model is proposed which has some physical basis, but which requires further
testing against measurements. It includes two key variables:
The angle between the two paths between the base stations and the mobile. If this angle issmall, the two path profiles share many common elements and are expected to have high
correlation. Hence the correlation should decrease with increasing angle-of-arrival dif-
ference.
The relative values of the two path lengths. If the angle-of-arrival difference is zero, thecorrelation is expected to be one when the path lengths are equal. As one of the path
lengths is increased, it incorporates elements which are not common to the shorter path, sothe correlation decreases.
c
c
-20 -15 -10 -5 010-5
10-4
10-3
10-2
10-1
100
Difference between threshold and mean C/I [dB]
Probabilityofina
dequateC/I
12
=0
12
=0.2
0.4
0.6
0.8
Figure 9.15: Effect of shadowing correlation on interference outage statistics
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An illustration of these points is given in Figure 9.16. The angle-of-arrival difference is
denoted by and each of the paths is made up of a number of individual elements whichcontribute to the shadowing process with sizes r along the path and t transverse to it. If
all the elements are assumed independent and equal in their contribution to the overall
scattering process, then the following simple model for the cross-correlation may be deduced:
rc ffiffiffi
r1r2
qfor 0 < T
T
ffiffiffir1r2
qfor T
8