This document discusses radio channel modeling and the effects of multipath fading. It describes narrowband and wideband channel modeling approaches. Narrowband channels cause signal fading due to destructive interference from multiple propagation paths. Wideband channels cause signal dispersion in addition to fading. Channel characteristics like delay spread, Doppler spread, and coherence bandwidth are defined. Common fading distributions like Rayleigh and Rice are also summarized. Techniques to mitigate fading effects in narrowband and wideband systems are outlined.

1. Fading multipath radio
channels
• Narrowband channel modelling
• Wideband channel modelling
• Wideband WSSUS channel
(functions, variables & distributions)

2. Low-pass equivalent (LPE) signal
Real-valued RF signalReal-valued RF signal Complex-valued LPE signalComplex-valued LPE signal
RF carrier frequencyRF carrier frequency
In-phase signal componentIn-phase signal component Quadrature componentQuadrature component
( ) ( ){ }2
Re cj f t
s t z t e π
=
( ) ( ) ( ) ( ) ( )j t
z t x t j y t c t e
φ
= + =

3. Spectrum characteristics of LPE signal
f
f
magnitude
phase
Real-valued time
domain signal
(e.g. RF signal)
Signal spectrum
is Hermitian
0
0
Complex-valued LPE
time domain signal
Signal spectrum
is not Hermitian

4. Radio channel modelling
Narrowband modellingNarrowband modelling Wideband modellingWideband modelling
Deterministic models
(e.g. ray tracing,
playback modelling)
Deterministic models
(e.g. ray tracing,
playback modelling)
Stochastical models
(e.g. WSSUS)
Stochastical models
(e.g. WSSUS)
Calculation of path loss
e.g. taking into account
- free space loss
- reflections
- diffraction
- scattering
Calculation of path loss
e.g. taking into account
- free space loss
- reflections
- diffraction
- scattering
Basic problem: signal
fading
Basic problem: signal
dispersion

5. Signal fading in a narrowband channel
magnitude of complex-valued
LPE radio signal
distance
propagation
paths fade <=> signal replicas
received via different
propagation paths cause
destructive interferenceTx
Rx

6. Fading: illustration in complex plane
in-phase component
quadrature
phase
component
Tx
Rx
Received signal in vector form:
resultant (= summation result)
of “propagation path vectors”
Wideband channel modelling: in addition to magnitudes
and phases, also path delays are important.
path delays are not
important

7. Propagation mechanisms
C
A
D
B
ReceiverTransmitter
A: free space
B: reflection
C: diffraction
D: scattering
A: free space
B: reflection
C: diffraction
D: scattering
reflection: object is large
compared to wavelength
scattering: object is small
or its surface irregular

8. Countermeasures: narrowband fading
Diversity (transmitting the same signal at different
frequencies, at different times, or to/from different
antennas)
- will be investigated in later lectures
- wideband channels => multipath diversity
Interleaving (efficient when a fade affects many bits
or symbols at a time), frequency hopping
Forward Error Correction (FEC, uses large overhead)
Automatic Repeat reQuest schemes (ARQ, cannot be
used for transmission of real-time information)

9. Bit interleaving
TransmitterTransmitter ChannelChannel ReceiverReceiver
Bits are
interleaved ...
Fading affects
many adjacent
bits
After de-
interleaving of
bits, bit errors
are spread!
Bit errors in
the receiver
... and will be
de-interleaved
in the receiver (better for FEC)

10. Channel Impulse Response (CIR)
time t
|h(τ,t)|
zero excess delay
delay spread Tm
delay τ
Channel is
assumed linear!
Channel presented in delay / time
domain (3 other ways possible!)

11. CIR of a wideband fading channel
τ
path delaypath delaypath attenuationpath attenuation path phasepath phase
LOS pathLOS path
( ) ( ) ( )
( )
1
0
, i
L
j t
i i
i
h t a t e
φ
τ δ τ τ
−
=
= −∑
The CIR consists of L resolvable propagation paths

12. Received multipath signal
( ) ( )k
k
s t b p t kT
∞
=−∞
= −∑
( ) ( ) ( ) ( ) ( ),r t h t s t h t s t dτ τ τ
∞
−∞
= ∗ = −∫
( ) ( )
( )
1
0
i
L
j t
i i
i
a t e s t
φ
τ
−
=
= −∑ ( ) ( ) ( )0 0f t t t dt f tδ − =∫
pulse waveformpulse waveformcomplex symbolcomplex symbol
Transmitted signal:
Received signal:

13. Received multipath signal
The received multipath signal is the sum of L attenuated, phase
shifted and delayed replicas of the transmitted signal s(t)
T
Tm
( )0
0 0
j
a e s tφ
τ−
( )1
1 1
j
a e s tφ
τ−
( )2
2 2
j
a e s tφ
τ−
Normalized delay spread D = Tm / T
:

14. Received multipath signal
The normalized delay spread is an important quantity.
When D << 1, the channel is
- narrowband
- frequency-nonselective
- flat
and there is no intersymbol interference (ISI).
When D approaches or exceeds unity, the channel is
- wideband
- frequency selective
- time dispersive
Important feature
has many names!

15. BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel
(no equalization)
Flat fading channel
Gaussian
channel
(no fading)
In a Gaussian channel (no fading) BER <=> Q(S/N)
erfc(S/N)

16. BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel
(no equalization)
Flat fading channel
Gaussian
channel
(no fading)
Flat fading (Proakis 7.3): ( ) ( )BER BER S N z p z dz= ∫
z = signal power level

17. BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel
(no equalization)
Flat fading channel
Gaussian
channel
(no fading)
Frequency selective fading <=> irreducible BER floor

18. BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Flat fading channel
Gaussian
channel
(no fading)
Diversity (e.g. multipath diversity) <=>
Frequency-selective channel
(with equalization)
improved
performance

19. Time-variant transfer function
( ) ( ) ( ) ( )
1
22
0
, , i i
L
j t j fj f
i
i
H f t h t e d a t e e
φ π τπ τ
τ τ
∞ −
−−
=−∞
= = ∑∫
( ) ( ) ( )
( )
1
0
, i
L
j t
i i
i
h t a t e
φ
τ δ τ τ
−
=
= −∑Time-variant CIR:
Time-variant transfer function (frequency response):
In a narrowband channel
this reduces to:
( ) ( ) ( )
1
0
, i
L
j t
i
i
H f t a t e
φ
−
=
= ∑

20. Example: two-ray channel (L = 2)
( ) ( ) ( )1 2
1 1 2 2
j j
h a e a eφ φ
τ δ τ τ δ τ τ= − + −
( ) 1 1 2 22 2
1 2
j j f j j f
H f a e e a e eφ π τ φ π τ− −
= +
( ) 1 2constructiveH f a a= +
( ) 1 2destructiveH f a a= −
At certain frequencies the two terms add constructively
(destructively) and we obtain:
f

21. Deterministic channel functions
Time-variant
impulse response
Time-variant
impulse response
Time-
variant
transfer
function
Time-
variant
transfer
function
Doppler-variant
transfer function
Doppler-variant
transfer function
Doppler-
variant
impulse
response
Doppler-
variant
impulse
response
( ),h tτ
( ),H f t ( ),d τ ν
( ),D f ν
(Inverse)
Fourier
transform

22. Stochastical (WSSUS) channel functions
Channel intensity
profile
Channel intensity
profile
Frequency
time
correlation
function
Frequency
time
correlation
function
Channel Doppler
spectrum
Channel Doppler
spectrum
Scattering
function
Scattering
function
( );h tφ τ ∆
( );H f tφ ∆ ∆ ( );hS τ ν
( );HS f ν∆
( )hφ τ
( )HS ν( )H fφ ∆
( )H tφ ∆ Td
Bm
Tm
Bd
τσ

23. Stochastical (WSSUS) channel variables
Maximum delay spread: Tm
Maximum delay spread may be
defined in several ways.
For this reason, the RMS delay
spread is often used instead:
( )
( )
( )
( )
2
2
h h
h h
d d
d d
τ
τ φ τ τ τ φ τ τ
σ
φ τ τ φ τ τ
 
 = −
  
∫ ∫
∫ ∫
( )hφ τ
τTm

24. Stochastical (WSSUS) channel variables
Coherence bandwidth
of channel:
1m mB T≈
( )H fφ ∆
Bm
f0
Implication of
coherence bandwidth:
If two sinusoids (frequencies) are spaced much less apart
than Bm , their fading performance is similar.
If the frequency separation is much larger than Bm , their
fading performance is different.

25. Stochastical (WSSUS) channel variables
Maximum Doppler spread:
The Doppler spectrum is often
U-shaped (like in the figure on
the right). The reason for this
behaviour is the relationship
(see next slide):
ν
Bd
0
Bd
( )HS ν
cos cosd
V
fν α α
λ
= =
Task: calculate p(ν) for the case where p(α) = 1/2π (angle of
arrival is uniformly distributed between 0 and 2π).
( ) ( )HS pν ν≈

26. Physical interpretation of Doppler shift
Vαarriving patharriving path
direction of receiver
movement
direction of receiver
movement
Rx
cos cosd
V
fν α α
λ
= =
Maximum Doppler shiftMaximum Doppler shift
Angle of arrival of arriving
path with respect to
direction of movement
Angle of arrival of arriving
path with respect to
direction of movement
V = speed of receiver
λ = RF wavelength
V = speed of receiver
λ = RF wavelength
Doppler frequency shiftDoppler frequency shift

27. Delay - Doppler spread of channel
Doppler shift ν
delay τ
0
L = 12 components in
delay-Doppler domain
L = 12 components in
delay-Doppler domain
Bd
( ) ( ) ( )
( )
1
2
0
, i i
L
j t
i i
i
h t a t e
πν φ
τ δ τ τ
−
+
=
= −∑

28. Fading distributions (Rayleigh)
( )
2 2
2
2
a
a
p a e σ
σ
−
= ( )
1
2
p φ
π
=
( ) ( ) ( )
( ) ( ) ( ) ( )ij t j t
i
i
c t a t e x t j y t a t e
φ φ
= = + = ∑
In a flat fading channel, the (time-variant) CIR reduces to a
(time-variant) complex channel coefficient:
When the quadrature components of the channel coefficient
are independently and Gaussian distributed, we get:
Rayleigh distributionRayleigh distribution Uniform distributionUniform distribution

29. Fading distributions (Rice)
In case there is a strong (e.g., LOS) multipath component
in addition to the complex Gaussian component, we obtain:
From the joint (magnitude and phase) pdf we can derive:
Rice distributionRice distribution
Modified Bessel function of
first kind and order zero
Modified Bessel function of
first kind and order zero
( ) ( ) ( )
( ) ( )
0 0
ij t j t
i
i
c t a a t e a a t e
φ φ
= + = + ∑
( ) ( )22 2
0 2 0
02 2
a aa aa
p a e I
σ
σ σ
− +  
=  ÷
 

30. Representation in complex plane
iy
Complex Gaussian distribution
is centered at the origin of the
complex plane => magnitude
is Rayleigh distributed, the
probability of a deep fade is
larger than in the Rician case
Complex Gaussian distribution
is centered around the “strong
path” => magnitude is Rice
distributed, probability of deep
fade is extremely small
( ),p x yiy
0a
Bell-shaped functionBell-shaped function
x x

31. Countermeasures: wideband systems
Equalization (in TDMA systems)
- linear equalization
- Decision Feedback Equalization (DFE)
- Maximum Likelihood Sequence Estimation
(MLSE) using Viterbi algorithm
Rake receiver schemes (in DS-CDMA systems)
Sufficient number of subcarriers and sufficiently long
guard interval (in OFDM or multicarrier systems)
Interleaving, FEC, ARQ etc. may also be helpful in
wideband systems.