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ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 28-Sep-16 1(27)
Indoor Propagation Models
Outdoor models are not accurate for indoor scenarios. Examples
of indoor scenario: home, shopping mall, office building,
factory.
Ceiling structure, walls, furniture and people effect the EM wave
propagation. Large/small number of obstacles, material of the
walls etc.
Modeling approach: classify various environments into few
types and model each type individually. Generic model is very
difficult to build.
Key Model
The average path loss is
( ) 0
0
~A
dL d L const d d
d
ν
ν ν
= = ⋅
(4.1)
or in dB:
( ) 0
0
[ ] [ ] 10 lgA
dL d dB L dB
d
= + ν
where 0L is path loss at reference distance 0d .
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 28-Sep-16 2(27)
ITU Indoor Path Loss Model1
The model is used to predict propagation path loss inside
buildings.
The average path loss in dB is
( )[ ] 20lg 10 lg ( ) 28A fL d dB f d L n= + ν + −
where: f is the frequency in MHz;
d is the distance in m; 1d > m;
ν is the path loss exponent (found from measurements);
( )fL n is the floor penetration loss (measurements);
n is the number of floors (penetrated);
Limits:
900 5200
1 3
1
MHz f MHz
n
d m
≤ ≤
≤ ≤
>
1 Recommendation ITU-R P.1238-8.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 28-Sep-16 3(27)
Variations (fading) around the average are accounted for via log-
normal distribution:
( ) ( )[ ] [ ]AL d dB L d dB Xσ
= + (4.2)
where Xσ is a log-normal random variable (in dB) of standard
deviation [ ]dBσ .
Variations on the order of (2...3) [ ]dBσ should be expected in
practice.
(2..3)σ Rule
(adopted from "Empirical Rule" by Dan Kernler)
Site-specific models will follow this generic model. Additional
factors are included (floors, partitions, indoor-outdoor
penetration etc.).
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 28-Sep-16 4(27)
Note: n = ν is the path loss exponent; 914f = MHz.
T.S
. R
appap
ort
, W
irel
ess
Com
munic
atio
ns,
Pre
ntice
Hal
l, 2
002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 28-Sep-16 5(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 28-Sep-16 6(27)
T.S
. R
appap
ort
, W
irel
ess
Com
munic
atio
ns,
Pre
ntice
Hal
l, 2
002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 25-Sep-13 7(27)
Table 4.1: path loss exponent factor 10ν in various
environments
Table 4.2: Floor penetration loss ( )fL n in various environments
Log-normal fading should be added as well,
( ) ( )[ ] [ ]AL d dB L d dB Xσ
= + (4.2)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 25-Sep-13 8(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 25-Sep-13 9(27)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 10(27)
Ericsson’s Indoor Path Loss Model
(900 MHz)
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 11(27)
Log-Normal Fading (Shadowing)
Reminder: 3 factors in the total path loss,
P A LF SFL L L L= (2.5)
Siw
iak, R
adio
wav
e P
ropag
atio
n a
nd A
nte
nnas
for
Pers
onal C
om
munic
atio
ns,
Art
ech
House
, 1998
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 12(27)
Log-Normal Fading (shadowing): this is a long-term (or large-
scale) fading since characteristic distance is a few hundreds
wavelengths.
Due to various terrain effects, the actual path loss varies about
the average value predicted by the models above,
[ ] dBp pL L L= + ∆ (4.5)
where pL is the average path loss , L∆ - its variation, which can
be described by log-normal distribution.
Overall, pL becomes a log-normal RV,
( )( )
2
22
1
2
p pL L
pL e
−
−
σρ =
πσ
(4.6)
where pL and
pL are in dB, and σ is the standard deviation (in
dB as well).
Physical explanation: multiple diffractions + the central limit
theorem (in dB).
Shadowing: due to the obstruction of LOS path.
The semi-empirical models above can be used together with the
log-normal distribution.
Reasonable physical assumptions result in statistical models for
the PC. This approach is very popular and extensively used in
practice.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 13(27)
Log-Normal Fading: Derivation
Assume signal at Rx is a result of many scattering/diffractions:
0
1
, 1N
t i i
i
E E
=
= Γ Γ ≤Π (4.7)
Total Rx power:
2 2 2 20 0
1 1
0,
1
20 lg
N N
t t i t i
i i
N
dB dB i
i
P E E or P P
P P
= =
=
= Γ = Γ
= + Γ
Π Π
∑
∼
(4.8)
If i
Γ are i.i.d., then 0,( , )dB dB dBP P σ∼ ℕ
Log-normal distribution works well for i.i.d multiple diffractions
( 5N ≥ ) and is used in practice to model large-scale fading
(shadowing).
In practice: 5...10dBσ = dB (can take 8 dB as an average).
System design: allow for 2 dBσ margin (for about 95%
reliability).
Tx Rx
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 14(27)
Small-Scale (multipath) Fading Model
Many multipath components (plane waves) arriving at Rx at
different angles,
1
1 1
( ) cos( )
cos cos sin sin
N
t i i
i
N N
i i i i
i i
E t E t
E t E t
=
= =
= ω +ϕ
= ϕ ω − ϕ ω
∑
∑ ∑
(4.9)
This is in-phase (I) and quadrature (Q) representation
2 2
1 1
( ) cos sin cos( )
: cos , Q: sin
t x y
x y
N N
x i i y i i
i i
E t E t E t E t
where E E E envelope
I E E E E
= =
= ω − ω = ω + ϕ
= + →
= ϕ = ϕ∑ ∑
(4.10)
Assume that i
E are i.i.d., and that [ ]0,2i
ϕ ∈ π are i.i.d.
Rx
E1
E2
E3
E4
E5
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 15(27)
By central limit theorem, 2
, (0, )x yE E σ∼ ℕ
E is Rayleigh distributed with pdf
2
2 2( ) exp , 0
2
x x
x x
ρ = − ≥ σ σ
(4.11)
where 2σ is the variance of Ex (or Ey ),
2 2 2
1
1
2
N
x i
i
E E
=
σ = = ∑ (4.12)
which is the total received power (for isotropic antennas).
For this result to hold, N must be “large” ( 5 10N ≥ ∼ ).
0 1 2 3 4 5
0
0.2
0.4
0.6
0.8Rayleigh PDF
./x σ
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 16(27)
Outage Probability and CDF
Importance of the CDF in wireless system design.
Rx operates well if thE E≥ → the threshold effect.
If thE E< , the link is lost --> this is an outage.
Outage probability = CDF is
0
( ) ( ) Pr( )
x
F x t dt E x= ρ = <∫ (4.13)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 17(27)
Rayleigh Fading
For Rayleigh distribution, the outage probability is
2
2
0
( ) Pr( ) ( ) 1 exp2
x
xF x E x t dt
= < = ρ = − − σ
∫ (4.14)
Introduce the instantaneous signal power 2/ 2,P x=
2P P= = σ = the average power, then
{ }Pr SNR
1 exp 1 exp ( )
outP
PF P
P
= < γ
γ = − − = − − = γ
, (4.15)
so that normalized SNR ( /γ γ ) or power ( /P P) PDF and CDF:
( ) ( ), 1x x
f x e F x e− −
= = − (4.15a)
and asymptotically,
( )out
PP P P F P
P
γ⇒ = ≈ =
γ≪ (4.16)
Note that P
P
γ=γ, where γ is the SNR.
Note in (4.16) the 10db/decade law.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 18(27)
Example:
3 310 10 , 30 w.r.t.
outP P P or dB P
− −
= → = −
i.e. if 310
outP
−
= is desired, the Rx threshold is 30dB below one
without fading (the average).
Complex-valued model:
1
( )N
j j tit i
i
E t E e eϕ ω
=
=∑ (4.17)
results in complex Gaussian variables.
Propagation channel gain – simply normalized received signal,
has the same distribution.
40 30 20 10 0 10 201 .10
4
1 .103
0.01
0.1
1
Rayleigh CDF
Ou
tag
e p
rob
abil
ity
./ , dBγ γ
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 19(27)
Rayleigh Fading Channel
Received power (SNR) norm. to the average [dB] vs distance
(location, time)
0 100 200 300 400 50030
20
10
0
10
SISO 1x1
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 20(27)
T.S
. R
appaport
, W
irele
ss C
om
munic
ati
ons,
Pre
nti
ce H
all
, 2002
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 21(27)
Measured Fading Channels
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 22(27)
LOS and Ricean Model
Motivation: a LOS component. The distribution of in-phase and
quadrature components is still Gaussian, but non-zero mean:
0 0
1
( ) cos( ) cos( )N
t i i
i
E t E t E t
=
= ω +ϕ + ω +ϕ∑ (4.18)
Both 0E and 0ϕ are fixed (non-random constants).
0 0
2 2
0 0
( ) ( )cos ( )sin
0, ( ) ( )
t x x y y
x y x x y y
E t E E t E E t
E E and E E E E E
= + ω − + ω
= = = + + +
(4.19)
Pdf of E has a Rice distribution 0 0( , )x E x E= = :
0
2 2
002 2 2
( ) exp ,2
x x xxxx I
+ ρ = − σ σ σ (4.20)
Note that if 0 0x = , it reduces to the Rayleigh pdf.
Introduce K-factor:
2 2
0/ (2 )K x= σ (4.21)
where 2
0/ 2x is the LOS power, it is called LOS (“steady” or
“specular”) component, 2σ is the scattered (multipath or
diffused) power, it is called diffused component.
K tells us how strong the LOS is.
Total average power = 2 2 2
0/ 2 (1 )x K+ σ = σ +
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 23(27)
PDF of E becomes
2
02 2( ) exp 2
2
x x xx K I K
ρ = − − σ σ σ (4.22)
0 2 4 6 8 100
0.2
0.4
0.6
0.8Rayleigh & Rice Densities
.
K=0
K=1 K=10 K=20
0 2 4 6 8 100
0.2
0.4
0.6
0.8Rayleigh & Rice Densities
.
K=0
K=1 K=10 K=20
Note: normalized to the multipath power only.
Q: do the same graph if normalized to the total average power.
pd
f
/x σ
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 24(27)
Q.: find the CDF of Ricean distribution as a function of K and
total (LOS + multipath) average power or SNR.
40 30 20 10 0 101 .10
8
1 .107
1 .106
1 .105
1 .104
1 .103
0.01
0.1
1
10
Rice CDFO
uta
ge
pro
bab
ilit
y
/ , dBγ γ
0K =
1K =
10K =
20K =
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 25(27)
Applications of Outage Probability
1) Fade margin evaluation for the link budget:
10 0( / ) / 1/ ( )out th th outP F P
−γ γ = ε→ = γ γ = ε
2) Average outage time:
out outT P T=
2) Average # of users in outage:
out outN P N=
To be discussed later on:
• level crossing rates (# of fades per unit time) • average fade duration
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 26(27)
Monte-Carlo Method
It is a powerful simulation technique to solve many statistical
problems numerically in a very efficient way. You should be
familiar with it. Detailed description of the method and many
examples can be found in numerous references, including the
following:
[1] M.C. Jeruchim, P. Balaban, K.S. Shanmugan, Simulation of
Communication Systems, Kluwer, New York, 2000.
[2] W.H. Tranter et al, Principles of Communication System
Simulation with Wireless Applications, Prentice Hall, Upper
Saddle River, 2004.
[3] J.G. Proakis, M. Salehi, Contemporary Communication
Systems Using MATLAB, Brooks/Cole, 2000.
This is used in labs extensively.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Lecture 4 4-Oct-17 27(27)
Summary
• Indoor propagation path loss models.
• Log-normal shadowing.
• Small-scale fading.
• Rayleigh & Rice distributions.
• Physical mechanisms.
Reading:
o Rappaport, Ch. 4.
References:
o S. Salous, Radio Propagation Measurement and Channel
Modelling, Wiley, 2013. (available online)
o J.S. Seybold, Introduction to RF propagation, Wiley, 2005.
o https://en.wikipedia.org/wiki/ITU_model_for_indoor_attenuation
o Other books (see the reference list).
Note: Do not forget to do end-of-chapter problems. Remember
the learning efficiency pyramid!