LOG-NORMAL
SHADOWING
Shadowing
• Also 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
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 power
measurement
> λ/2
Velocity of antenna
Points where power measurements are made
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-Rx
locations within
a floor of a building
Repeat for Multiple Distances
• Similar collections of average powers are made for other
Tx-Rx distances
Likelihood of Coverage
• At a certain distance, d, what is the probability that the
local average received power is below a certain threshold
γ?
P( Pr (d ) < γ )
Pt Gt Gr
Pr (d ) =
Lt L(d ) X σ Lr
Path Loss Shadowing
Likelihood of Coverage, cont’d
• Since only Xσ is random, the probability can be expressed
as a probability involving it:
P( Pr (d ) > γ ) = P( X σ > β )
Chosen to give a desired quality of service
Typical Macrocell Characteristics
[not real data]
-60 .
. ... ..
... ... . . .
..
-70
.. ... ..
.
. .. . .. .
. . . .. .. . .... ... . . . .. . ...
..
Total Path Loss [-dB]
. .. . . .. . .
. . ..
. . . . . . ... .. .
.
.
-80
. ..... . ... . . . .
. . .
.
. . . . . The set of average
. .. ... ..... . . . . .
.
... ... .
. .. . . .
. . ... . .
... . . . . losses measured
..
. .
. . .... .. . . .
-90
. . . . .
.
.
. . ..
.... ... . ... ... ...... . . . at about 4km
. . . ........ . .
. . . . .. ..
[10log10(4)=6]
.
.
-100
.. ... .. . .
..
.
. . .. .. . . .. ..
.. . . .
.. . .
.
.. . ... .. . . . ..
.
. . .
.... . ... . .
.
-110
.
-120
0 1 2 3 4 5 6 7 9 10
10log10(Distance from Base Station [km])
Path Loss Assumptions
• The mean loss in dB follows the power law :
⎛ d ⎞
L(d ) = L(d o ) + 10n log10 ⎜⎜ ⎟⎟
⎝ d o ⎠
• The measured loss in dB varies about this mean
according to a zero-mean Gaussian RV, Xσ, with
standard deviation σ
⎛ d ⎞
L(d ) = L(d o ) + 10n log10 ⎜⎜ ⎟⎟ + X σ
⎝ d o ⎠
Typical Data Characteristics
[not real data]
-60 .
. ... ..
... ... . . .
..
-70
.. ... ..
.
. .. . .. . Best Fit
. . . .. .. . .... ... . . . .. . ...
..
Total Path Loss [-dB]
. .. . . .. . .
Path Loss
. . ..
. . . . . . ... .. .
.
.
-80
. ..... . ... . . . .
. . .
. Exponent:
. . . . . n=4
. .. ... ..... . . . . .
.
... ... .
. .. . . .
. . ... . .
... . . . .
..
. .
. . .... .. . . .
-90
. . . . .
.
.
. . ..
.... ... . ... ... ...... . . .
. . . ........ . .
. . . . .. ..
.
.
-100
.. ... .. . . σ is usually
..
.
. . .. .. . . .. ..
.. . . . 5-12 dB for
.. . .
.
.. . ... .. . . . ..
. mobile comm
. . .
.... . ... . .
.
-110
.
-120
0 1 2 3 4 5 6 7 9 10
10log10(Distance from Base Station [km])
Probability Calculation
• Since Xσ is Gaussian, we need to know how to calculate
probability involving Gaussian RVs
P( X σ > β )
Q Function
• If X is a Gaussian RV with mean α and standard deviation
σ, then
⎛ b − α ⎞
P( X > b) = Q⎜ ⎟
⎝ σ ⎠
where Q is a function defined as
1
+∞
⎛ x 2 ⎞
Q( z ) = ∫z exp⎜⎜⎝ − 2 ⎟⎟⎠dx
2π
The Problem with Q
• The integrand of Q has no antiderivative
• Q is found tabulated in books
• Q can be calculated using numerical integration
What is Log-normal Shadowing?
• If Y is a Gaussian RV and Z is defined such that Y=logZ,
then Z is a log-normal RV
• Shadowing is log-normal shadowing when the path loss in
dB is Gaussian; this means that the path loss expressed
as a ratio is log-normal
Inverse Q Problems
• Sometimes, the probability is specified and we
must find one of the parameters in the
argument of Q
⎛ b − α ⎞
P( X > b) = Q⎜ ⎟
⎝ σ ⎠
• Suppose the value of P ( X
> b)is given, along
with values of b and α. Solve for σ
• Must look up the argument of Q that gives the
specified value.
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 ⎛ b − (−30) ⎞
P( Pr > b) = Q⎜ ⎟ = 0.95
⎝ 9 ⎠
• The argument of Q that yields 0.05 is about 1.65
Q(z ) = 1 − Q(− z)
b + 30
− = 1.65, and b = −44.85 dBm
9
Boundary Coverage
• Suppose that a cell has radius R and γ is the
minimum acceptable received power level
• Then P( Pr ( R) > γ ) is the “likelihood of coverage”
at the boundary of the cell
• P( Pr ( R) > γ ) is also the “fraction of time” that a
mobile’s signal is acceptable at a distance R from
the transmitter, assuming the car moves around
that circle
Percentage of Useful Service Area
• By integrating these probabilities over all the circles within
a disk, one can compute the fraction of the area within the
cell that will have acceptable power levels
2π R
1
U (γ ) = 2 ∫ ∫ P( P (r ) > γ ) rdrdθ
r
πR 0 0
Integral Evaluated
• Assuming log-normal shadowing and the power
path loss model, the fraction of useful service
area is
1 ⎛ ⎛ 1 − 2ab ⎞⎡ ⎛ 1 − ab ⎞⎤ ⎞
U (γ ) = ⎜⎜1 − erf (a) + exp⎜ 2 ⎟⎢1 − erf ⎜ ⎟⎥ ⎟⎟
where 2 ⎝ ⎝ b ⎠⎣ ⎝ b ⎠⎦ ⎠
γ − Pr ( R) 10n log10 e
a= and b=
σ 2 σ 2
The Error Function
• erf(x) is another form of the Gaussian integral (like
Q(x))
• erf(x) has odd symmetry, with extreme values ±1.
erf1(x)
1
x
2 −t 2
erf ( x) = ∫e dt x
π 0
-1
• Note that some authors may define erf differently
erf and Q
• The erf function and Q are related:
( )
erf ( z ) = 1 − 2Q 2 z
When the Average Boundary Power is
Acceptable
• Suppose
Pr (R) =. γThen
we may use this
graph from
[Rappaport ’96]
to figure the
percent useful
service area
Summary
• The logs of local averages of received power (or
path loss) tend to be Gaussian when the
ensemble is all Tx-Rx locations with the same
distance in the same type of environment
• The mean local average path loss follows the
standard power model (proportional to 10logdn)
• Can use Q or erf to calculate the likelihood of
boundary coverage or the percent of useful
service area
References
• [Rapp, ’96] T.S. Rappaport, Wireless Communications,
Prentice Hall,
• [Saunders,`99] Simon R. Saunders, Antennas and
Propagation for Wireless Communication Systems, John
Wiley and Sons, LTD, 1999.
