Lecture 5 Log Normal Shadowing

8/9/2019 Lecture 5 Log Normal Shadowing

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LOG-NORMAL

SHADOWING


2/23

ShadowingAlso called slow-fading

Accounts for random variations in received power

observed over distances comparable to the widths of

buildings Extra transmit power (a fading margin) must be provided

to compensate for these fades


3/23

Local Average Power Measurements Take power measurements in Watts as the antenna is moved in a on

the order of a few wavelengths

Average these measurements to give a local average powermeasurement

Velocity of antenna

Points where power measurements are made

> !/2


4/23

Same-Distance Measurements

Local averages are made for many different locations,keeping the same transmitter-receiver distance

These local averages will vary randomly with location

Example Tx-Rxlocations within

a floor of a building


5/23

Repeat for Multiple Distances Similar collections of average powers are made for other

Tx-Rx distances


6/23

Likelihood of CoverageAt a certain distance, d, what is the probability that the

local average received power is below a certain threshold

!?

))(( !


7/23

Likelihood of Coverage, contd

Since onlyX"is random, the probability can be expressed

as a probability involving it:

)())(( !" # >=> XPdPP r

Chosen to give a desired quality of service


8/23

Typical Macrocell Characteristics

TotalPath

Loss[-dB

]

10log10(Distance from Base Station [km])

0 2 3 4 5 6 7 9 101

-60

-70

-80

-90

-100

-110

-120

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[not real data]

The set of averagelosses measured

at about 4km[10log10(4)=6]


9/23

Path Loss AssumptionsThe mean loss in dB follows thepower law :

The measured loss in dB varies about this mean

according to a zero-mean Gaussian RV,X", with

standard deviation "

!

!

"

#

$

$

%

&+=

o

o

d

dndLdL 10log10)()(

!X

d

dndLdL

o

o +""

#

$%%&

'+= 10log10)()(


10/23

Typical Data Characteristics

TotalPath

Loss[-dB

]

10log10(Distance from Base Station [km])

0 2 3 4 5 6 7 9 101

-60

-70

-80

-90

-100

-110

-120

..

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Best FitPath Loss

Exponent:n=4

[not real data]

"is usually

5-12 dB formobile comm


11/23

Probability Calculation SinceX

"is Gaussian, we need to know how to calculate

probability involving Gaussian RVs

)( !" >XP


12/23

QFunction If X is a Gaussian RV with mean #and standard deviation", then

where Qis a function defined as

!

"

#$

%

& '=>

(

)bQbXP )(

dx

x

zQz!

+"

##$

%

&&'

(

)=

2exp2

1

)(

2

*


13/23

The Problem with Q The integrand of Qhas no antiderivative

Qis found tabulated in books

Qcan be calculated using numerical integration


14/23

Inverse Q Problems

Sometimes, the probability is specified and wemust find one of the parameters in the

argument of Q

Suppose the value of is given, along

with values of band #. Solve for " Must look up the argument of Qthat gives thespecified value.

!"

#$%

& '=>

(

)bQbXP )(

)( bXP >


15/23

Example Inverse Q Problem Suppose the mean of the local average received powers at a certain

distance is -30dBm, that the standard deviation of shadow fading is 9

dB, and that the observed received power is above the threshold 95%

of the time. What is the threshold power?

Q is usually tabulated for arguments of 0.5 and less, so we must use

the fact that

The argument of Qthat yields 0.05 is about 1.65

95.09

)30()( =!

"

#$%

& ''=>

bQbPP r

dBm85.44and,65.19

30!==

+

! bb

( ) )(1 zQzQ !!=


16/23

Boundary Coverage

Suppose that a cell has radiusRand !is the

minimum acceptable received power level

Then is the likelihood of coverage

at the boundary of the cell

is also the fraction of timethat a

mobiles signal is acceptable at a distanceRfrom

the transmitter, assuming the car moves around

that circle

))(( !>RPPr

))(( !>RPPr


17/23

Percentage of Useful Service Area

By integrating these probabilities over all the circles withina disk, one can compute the fraction of the area within the

cell that will have acceptable power levels

!"#

"

#

drdrrPPR

U

R

r$ $ >=2

0 0

2 ))((1)(


18/23

Integral Evaluated

Assuming log-normal shadowing and the power

path loss model, the fraction of useful service

area is

where!!"

#$$%

&'(

)*+

,!"

#$%

& --!

"

#$%

& -+-=

b

ab

b

abaU

1erf1

21exp)(erf1

2

1)(

2.

2

)(

!

" RPa

r#

=

2

log10 10

!

enb =and


19/23

The Error Function

erf(x)is another form of the Gaussian integral (likeQ(x))

erf(x)hasodd symmetry, with extreme values 1.

Note that some authors may define erfdifferently

! "

=

x

tdtex

0

22)(erf

#

1

erf1(x)

x

-1


20/23

erfand Q

The erffunction and Qare related:

zQz 221)(erf !=


21/23

When the Average Boundary Power is

Acceptable

Suppose

. Then

we may use thisgraph from

[Rappaport 96]

to figure the

percent useful

service area

!=)(RPr


22/23

Summary

The logs of local averages of received power (orpath loss) tend to be Gaussian when theensemble is all Tx-Rx locations with the same

distance in the same type of environmentThe mean local average path loss follows thestandard power model (proportional to 10logdn)

Can use Qor erfto calculate the likelihood of

boundary coverage or the percent of usefulservice area


23/23

References

[Rapp, 96] T.S. Rappaport, Wireless Communications,Prentice Hall,

[Saunders,`99] Simon R. Saunders,Antennas and

Propagation for Wireless Communication Systems, JohnWiley and Sons, LTD, 1999.

Documents

Lecture 5 Log Normal Shadowing