Wireless Communication
Channels: Small-Scale Fading
Clarke’s Model for Flat Fading
Assumptions:




z
Mobile traveling in x
direction
Vertically polarized wave
Multiple waves in the x-y
plane arrive at the mobile
antenna at the same time
Waves arrive at different
angles α
y
in x-y
plane
α
x
For N waves incident at the mobile antenna
Each wave arriving at an angle αn will experience a different Doppler shift fn
v
f n  cos αn
λ
 Ez  E0  Cn cos  2πfc t  θn  θn  2πfn t  φn
N
n 1
E0 amplitude of the local average E-field
Cn random variable representing the amplitude of individual waves
fc carrier frequency
φn random phase shift due to distance traveled by the nth wave
© Tallal Elshabrawy
2
Clarke’s Model for Flat Fading
 N

 N

Ez  t   E0   Cn cos θn  cos  2πfc t   E0   Cn sinθn  sin  2πf c t 
 n 1

 n 1

Ez  t   Tc  t  cos  2πfc t   Ts  t  sin  2πfc t 
 N

Tc  t   E0   Cn cos  2πf n  φn  
 n 1

 N

Ts  t   E0   Cn sin  2πf n  φn  
 n 1

Given that:
 Φn uniformly distributed over 2π
 N is sufficiently large (i.e., the central limit theorem is
applicable)
Therefore:
Both Tc(t) and Ts(t) may be modeled as:
Gaussian Random Processes
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3
Clarke’s Model for Flat Fading
Ez  t   Tc  t  cos  2πfc t   Ts  t  sin  2πfc t 
Tc  t  cos  2πfc t 
E z  t   Tc2  t   Ts2  t   r  t 
Ts  t  sin  2πfc t 
Ez  t   r  t  cos  2πfc t  ψ  t 
N
If
2
2
2
2
2
C

1
T

T

σ

E
 n
c
s
0 2
n 1
Power received at mobile antenna  E z  t  2  r 2
 r
 r2 
 2 exp   2  0  r  
pr  σ
 2σ 

0
r 0

© Tallal Elshabrawy
Rayleigh
Distribution
4
Rayleigh Fading Distribution
Main Assumption:
- No LOS
- All waves at the mobile
receiver experience
approximately the same
attenuation
z
y
dα
α
N
E z  E0  Cn cos  2πf c t  θn 
in x-y
plane
x
n 1
constant
N
C
n 1
2
n
1
p(r)
0.6065/σ
 r
 r2 
 exp   2  0  r  
pr  σ2
 2σ 

0
r 0

σ2: Time average received power
σ : rms value of received voltage
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σ
r
5
Rayleigh Fading Statistics
Probability the received signal
does not exceed a value R
 R2 
Pr  r  R    p  r  dr  1  exp   2 
 2σ 
0
R

Mean value of the Rayleigh
distribution
rmean
π
 E  r    rp  r  dr  σ
 1.2533σ
2
0

Variance of the Rayleigh
distribution
σ r  E  r 2   E 2  r    r 2 p  r  dr  σ 2
0
π
2
π

σ r  σ  2    0.4292σ 2
2

2
Median of the Rayleigh
distribution
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1

2
rmedian
 p  r  dr  r
median
 1.177σ
0
6
Ricean Fading Distribution
Main Assumption:
- LOS
- There is a dominant
wave component at the
mobile receiver in addition
to experience multiple
waves that experience
approximately the same
attenuation
 r
 r 2  A2
 2 exp  
2
p r  σ
2σ


0

z
y
dα
α
in x-y
plane
x
  Ar 
 I0  2  A  0,0  r  
 σ 
r 0
A : Peak amplitude of the dominant signal
I(.): Modified Bessel function of the first kind and zero-order
σ2: Time average received power of the non-dominant components
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7
Riciean & Rayleigh Fading
Define K called the Ricean
Factor:
The ratio between the deterministic
signal power and the power of the
non-dominant waves
p(r)
A2
A2
K  2  K  dB  10 log 2
2σ
2σ
K=-∞ dB
Rayleigh
Distribution
K=6 dB
r
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8
Level Crossing Rate and Mean fade Duration for
Rayleigh Fading Signals
Level Crossing Rate Statistic:
The expected rate at which Rayleigh
fading envelope normalized to local
rms level crosses a specified level in a
positive–going direction
Mean Fade Duration Statistic:
The average period of time for which
the received signal is below a
specified level R
Mean Fade duration is a very
important statistic that helps define
the time correlation behavior of BER
performance at the receiver
© Tallal Elshabrawy
NR  2π f m ρe
 ρ2
ρ:= R/Rrms
fm: Maximum Doppler shift
τ
1
Pr  r  R 
NR
R
Pr  r  R    p  r  dr  1  exp   ρ 2 
0
τ
exp  ρ 2   1
2π f m ρ
9
How Wireless Channels Components Fit Together
100
90
Distance Pathloss
Mobile Speed 3 Km/hr
PL=137.744+
35.225log10(DKM)
80
70
60
50
40
30
0
10
20
30
40
50
60
0
10
20
30
40
50
60
10
20
30
40
50
60
d
15
Lognormal
Shadowing
Mobile Speed 3 Km/hr
ARMA Correlated
Shadow Model
10
5
0
-5
-10
-15
d
20
10
0
Small-Scale Fading
Mobile Speed 3 Km/hr
Jakes’s Rayleigh Fading
Model
-10
-20
-30
-40
-50
-60
© Tallal Elshabrawy
0
d
10
How Wireless Channels Components Fit Together
PTGT
GR
Wireless Channel
PR=PTGTGR x Distance Pathloss x Shadowing
Parameters x Small-Scale Fading Power
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11
System Modeling of Wireless Networks: Example
P  b' 
Pk b  , Pb
OFF
b'
ON
OFF
ON
OFF
b
ON
b
k
K S Active Sessions
Pk b 
P
b
 kb   TH : Packets Lost (Outage)
 kb  TH : Packets Received Correctly
Target
Signal
H kbb , kbb
b b
H kb'
, kb'  b' 
P
b
 Pk
H kbb , kbb
Intra-cell
Interference
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k
Inter-cell
Interference
GAPk  H kb  kb 
b
b
k 
b
k
P
b
b
 Pk
H
b
NB
b b
 kb   Pb'  H kb'
 kb'  N
b b
kb
b
b' 1
b' b
12
Diversity Techniques
What is Diversity?
 Diversity techniques offer two or more inputs at the receiver
such that the fading phenomena among these inputs are
uncorrelated
 If one radio path undergoes deep fade at a particular point
in time, another independent (or at least highly
uncorrelated) path may have a strong signal at that input
 By having more than one path to select from, both the
instantaneous and average SNR at the receiver may be
improved
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Diversity Techniques: Space Diversity
 Receiver Space Diversity
 M different antennas appropriately separated
deployed at the receiver to combine uncorrelated
fading signals
0
1
Transmitter
2
Receiver
M
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Diversity Techniques: Space Diversity
 Transmitter Space Diversity
 M different antennas appropriately separated
deployed at the transmitter to obtain uncorrelated
fading signals at the receiver
 The total transmitted power is split among the antennas
0
1
Transmitter
2
Receiver
M
© Tallal Elshabrawy
16
Diversity Techniques: Frequency Diversity
 Modulate the signal through M different carriers
 The separation between the carriers should be at least
the coherent bandwidth Bc
 Different copies undergo independent fading
 Only one antenna is needed
 The total transmitted power is split among the
carriers
f
Δf>Bc
t
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17
Diversity Techniques: Time Diversity
 Transmit the desired signal in M different periods
of time i.e., each symbol is transmitted M times
 The interval between transmission of same symbol
should be at least the coherence time Tc
 Different copies undergo independent fading
f
Δt>Tc
t
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Diversity Combining Techniques
Selection Combining
 Select the strongest signal
SNR
Monitor
Select MAX
SNR=γmax
Channel 1
Channel 2
Transmitter
Receiver
Channel M
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20
Selection Combining
 Consider M independent Rayleigh fading channels
available at the receiver
Average SNR at all Diversity Branches
SNR = Γ
Instantaneous SNR at Diversity Branch i
SNR = γi
Rayleigh Fading Voltage means
Exponentially Distributed Power
Outage Probability of
a Single Branch
Outage Probability of
of Selection Diversity Combining
© Tallal Elshabrawy
 f  γi 
1  γΓi
 e
Γ
γ
γ
γ
 i
 

1
Pr  γ i  γ    e Γ dγ i   1  e Γ 
Γ0


M
Pr  γ max
γ
 

 γ    1  e Γ 


21
Maximal Ratio Combining
 Selection Combining does not benefit from power received
across all diversity branches
 Maximal Ratio Combining conducts a weighted sum across
all branches with the objective of maximizing SNR
Channel 1
Channel 2
r1
r2
G1
G2
∑
Transmitter
Receiver
GM
Channel M
© Tallal Elshabrawy
rM
22
Maximal Ratio Combining
 Consider M independent Rayleigh fading channels
available at the receiver
Envelope applied to
receiver detector
Total Noise Power
applied to detector
SNR at the
receiver detector
Cauchy’s Inequality
γ
i1
i
i
M
NMRC  N G2i
i1
γMRC
r 
 MRC
NMRC
  ab
i i 
2
 M ri

NG

i
N
i1

 

M
N G2i
i1
 rG
rMRC 
2
 γMRC
M
2
2
 M

   rG
i i
i

1


N G2i
i1
 a b
2
i
2
i
2 M
 ri 



i1  N 
M
M

i1
M
N G
i1
2
i
NGi

2
1 M
2
   ri  
N i1
M
γ
i1
i
is maximized when Gi=ri (MRC requires channel measurements)
23
© Tallal MRC
Elshabrawy
Maximal Ratio Combining
γMRC is maximized when Gi=ri
(MRC requires channel measurements)
γMRC 
M
γ
i1
i
Rayleigh Fading Voltage means
Exponentially Distributed Power
 f  γi 
1  γΓi
 e
Γ
γMRC
Γ
SNR γMRC is Gamma distributed (sum
γMRC 
e

of M exponential random variables)  f  γMRC  
ΓM  M  1 !
M1
Outage Probability of
of Maximal Ratio Diversity Combining Pr  γMRC  γ  
© Tallal Elshabrawy
γ

0
γ
MRC


M1
e

γMRC
Γ
ΓM  M  1 !
dγMRC
24
Equal Ratio Combining
 Maximal Ratio Combining requires estimation of the
channel across all diversity branches
 Equal Gain Combining conducts a sum across all branches
(i.e. Gi=1 for all i)
Channel 1
Channel 2
r1
r2
∑
Transmitter
Receiver
Channel M
© Tallal Elshabrawy
rM
25
Equal Gain Combining
 Consider M independent Rayleigh fading channels
available at the receiver
M
Envelope applied to
receiver detector
rEGC 
Total Noise Power
applied to detector
NEGC  MN
SNR at the
receiver detector
r
i1
i
rEGC 


2
γEGC
NEGC
2


   ri 
 i1 
M
MN
EGC is a special case of MRC with Gi=1
SNR and outage probability performance in
EGC is inferior to that of MRC
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26