FadingChannels_2 - Introduction to Fading Channels part 2...

Introduction to
Fading Channels, part 2
1
1
Dr. Essam Sourour
Alexandria University, Faculty of
Engineering, Dept. Of Electrical
Engineering

• Local reflections cause multipath
• Each path has a random gain, with random
magnitude and random phase
• Each gain is represented in baseband as
n
j
n
e θ α
Small Scale Fading
2
• Receiver, and/or reflectors, may be moving
Building 2
v

• Assume a group of paths with small relative delay
• Net effect is one path with random gain and phase
n
j j
n
n
e R e
θ φ
α ⇒
∑
• According to Central Limit Theorem, the net gain
Small Scale Rayleigh Fading
3
Building 2
v
Re
j φ
is complex Gaussian with zero mean
• The envelope R is Rayleigh distributed and the
phase φ is uniform [0, 2 π ]

• The net gain is the sum of all closely delayed paths:
( ) ( )
cos sin
n
j
n n n
n n n n n n
e x jy
x and y
θ
α
α θ α θ
= +
= =
( )
e
n
j j
n n n r al imag
R e e x jy g g jg θ φ α = = + = = +
∑ ∑
Rayleigh Fading
4
n n
• Each of g
real
and g
imag
is the sum of many independent
random variables
• Hence g
real
and g
imag
are independent and Gaussian
with zero mean and variance σ 2 each
• Fading gain g = g
real
+ j g
imag
is complex Gaussian with
zero mean and variance 2 σ
2
(sum of two variances)

Rayleigh Distribution
• From probability theory we know:
( )
2 2
1
tan
real imag
real imag
R g g g R is Rayleigh Distributed
g g g isUniformly Distributed φ φ
-
= = + ⇒
= ∠ = ⇒
( ) ( ) ( )
If istransmitted,then willbereceived
j
s t r t Rs t e
φ
=
• Received amplitude follows Rayleigh distribution
• Received power follows Exponential distribution
• Received phase follows Uniform distribution
5
( )
( )
2 2
2
exp 2
, 0
r r
p r r
σ
σ
-
= >
( )
1
, 0 2
2
p φ φ π
π
= < <
( )
( )
2
2
exp 2
, 0
2
y
p y y
σ
σ
-
= >

Amplitude, Rayleigh Distribution
( )
( )
2 2
2
exp 2
, 0
r r
p r r
σ
σ
-
= >
6

Power, Exponential Distribution
( )
( )
2
2
exp 2
, 0
2
y
p y y
σ
σ
-
= >
7

Fading of 16 QAM signal
No fading Faded signals with random amplitude and phase
• Signal has a higher probability of being week
• For example, to receive the 16-QAM signal
we must estimate and compensate for the
amplitude and phase
8

Effect of Mobility
• Fading gain changes with time
g(t)=g
real
(t) + j g
imag
(t)
• Fading change rate depends on the
maximum Doppler frequency
• Coherence time << 1/ f
D c
v v
f f
c λ
= =
D
• Example: f
c
=1GHz, v =100 km/h gives:
f
D
= 92.6 Hz , Coherence time << 10.8 ms
9

Fading Example for R
10

Statistical Properties 1
• Complex fading gain g(t)
• The two parts g
real
(t) and g
imag
(t) are zero
mean
( ) ( ) ( )
e r al imag
g t g t j g t = +
• The two parts g
real
(t) and g
imag
(t) are
statistically independent
11
( ) ( )
( )
0, forany
real imag
E g t g t t t + Δ = Δ
( )
( )
( )
( )
0
real imag
E g t E g t = =

Statistical Properties 2
• Fading gain is correlated over time
• Usually Jakes model is used in mobile comm.