chapter 9 shadowing saunders

8/2/2019 Chapter 9 Shadowing Saunders

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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

188 Antennas and Propagation for Wireless Communication Systems


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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

Shadowing 189


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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

Shadowing 193


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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

Shadowing 197


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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

Shadowing 199


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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

Shadowing 201


16/21

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

chapter 9 shadowing saunders

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