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```					Fading multipath radio
channels

• Narrowband channel modelling
• Wideband channel modelling
• Wideband WSSUS channel
(functions, variables & distributions)
Low-pass equivalent (LPE) signal

RF carrier frequency


s  t   Re z  t  e j 2 fct   
Real-valued RF signal            Complex-valued LPE signal

z t   x t   j y t   c t  e
j t 

Spectrum characteristics of LPE signal

magnitude
Real-valued time
domain signal
(e.g. RF signal)
0            f

phase
Signal spectrum
is Hermitian
0            f

Complex-valued LPE        Signal spectrum
time domain signal        is not Hermitian

Narrowband modelling        Wideband modelling

Calculation of path loss   Deterministic models
e.g. taking into account     (e.g. ray tracing,
- free space loss        playback modelling)
- reflections
- diffraction             Stochastical models
- scattering                 (e.g. WSSUS)

Basic problem: signal      Basic problem: signal
Signal fading in a narrowband channel

magnitude of complex-valued

distance

propagation
propagation paths cause
Tx                   destructive interference
Rx

phase              resultant (= summation result)
component              of “propagation path vectors”

path delays are not
important
Rx
Tx     in-phase component

Wideband channel modelling: in addition to magnitudes
and phases, also path delays are important.
Propagation mechanisms

A: free space
A: free space
B: reflection
B: reflection
A
C
C: diffraction
C: diffraction
D: scattering
D: scattering
D

B

reflection: object is large
compared to wavelength
scattering: object is small
or its surface irregular

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)
Bit interleaving

Bits are          Fading affects      After de-
interleaved ...      many adjacent    interleaving of
bits        bits, bit errors

... and will be
de-interleaved        Bit errors in
in the receiver                       (better for FEC)
Channel Impulse Response (CIR)

assumed linear!

 h(,t) 
time   t

Channel presented in delay / time
domain (3 other ways possible!)

zero excess delay             delay   
CIR of a wideband fading channel

The CIR consists of L resolvable propagation paths

L 1
h  , t    ai  t  e                   i 
ji  t 

i 0

path attenuation     path phase                path delay

LOS path



Transmitted signal:                s t      b p  t  kT 
k 
k

complex symbol              pulse waveform


r t   h t   s t            h  , t  s t   d


L 1
  ai  t  e               s t  i 
ji  t 

i 0
 f t   t  t  dt  f t 
0            0
The received multipath signal is the sum of L attenuated, phase
shifted and delayed replicas of the transmitted signal s(t)

a0 e j0 s t  0 
a1 e j1 s t 1 
a2 e j2 s  t  2 
T
:

Tm
Normalized delay spread D = Tm / T

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       Important feature
- time dispersive           has many names!
BER vs. S/N performance
In a Gaussian channel (no fading)      BER <=> Q(S/N)
erfc(S/N)

Typical BER vs. S/N curves
BER

Frequency-selective channel
(no equalization)

Gaussian

S/N
BER vs. S/N performance
Flat fading (Proakis 7.3):   BER   BER  S N z  p  z  dz
z   = signal power level

Typical BER vs. S/N curves
BER

Frequency-selective channel
(no equalization)

Gaussian

S/N
BER vs. S/N performance
Frequency selective fading <=> irreducible BER floor

Typical BER vs. S/N curves
BER

Frequency-selective channel
(no equalization)

Gaussian

S/N
BER vs. S/N performance
Diversity (e.g. multipath diversity) <=>    improved
performance

Typical BER vs. S/N curves
BER

Gaussian          Frequency-selective channel
channel          (with equalization) channel

S/N
Time-variant transfer function
L 1
h  , t    ai  t  e                    i 
ji  t 
Time-variant CIR:
i 0

Time-variant transfer function (frequency response):
                             L 1
H  f ,t       h  , t  e j 2 f  d   ai  t  e
ji  t   j 2 f  i

                            i 0
e

L 1
H  f , t    ai  t  e
In a narrowband channel                                                     ji  t 
this reduces to:
i 0
Example: two-ray channel (L = 2)

h    a1 e j1   1   a2 e j2    2 

H  f   a1 e j1 e j 2 f 1  a2 e j2 e j 2 f  2

At certain frequencies the two terms add constructively
(destructively) and we obtain:

H  fconstructive   a1  a2

H  f destructive   a1  a2                              f
Deterministic channel functions

(Inverse)               Time-variant
Fourier             impulse response
transform
h  , t 
Time-                                                 Doppler-
variant
transfer
H  f ,t                        d  ,      variant
impulse
function                                                response
D  f , 
Doppler-variant
transfer function
Stochastical (WSSUS) channel functions

H  t      Td
Channel intensity
profile                 h         Tm

h  ; t 
Frequency
time
correlation
H  f ; t                   Sh  ; 
Scattering
function
function

SH  f ; 
H  f       Bm
Channel Doppler
spectrum                 SH        Bd
Stochastical (WSSUS) channel variables

h  
Tm

defined in several ways.
For this reason, the RMS delay

 h   d     h   d 
2


2

                                  
 h   d   h   d 
              
Stochastical (WSSUS) channel variables

Coherence bandwidth
of channel:
H  f 

Bm  1 Tm
Bm
0            f
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
Stochastical (WSSUS) channel variables

Maximum Doppler spread:        Bd              SH  
The Doppler spectrum is often
U-shaped (like in the figure on
the right). The reason for this
behaviour is the relationship                    0          
(see next slide):                               Bd
V
        cos   f d cos              SH    p  

Task: calculate p() for the case where p() = 1/2 (angle of
arrival is uniformly distributed between 0 and 2).
Physical interpretation of Doppler shift

arriving path                 V

movement
Rx

Doppler frequency shift
V                         = RF wavelength
       cos   f d cos 
                       Angle of arrival of arriving
path with respect to
Maximum Doppler shift        direction of movement
Delay - Doppler spread of channel

delay 
L = 12 components in
delay-Doppler domain

0                            Doppler shift 
Bd
L 1
h  , t    ai  t  e                           i 
j  2 i t i 

i 0

In a flat fading channel, the (time-variant) CIR reduces to a
(time-variant) complex channel coefficient:

c t   a t  e               x t   j y t    ai t  e
j  t                                        ji t 

i

When the quadrature components of the channel coefficient
are independently and Gaussian distributed, we get:

a                                         1
p a              e    a 2 2 2
p   
     2
2

Rayleigh distribution                     Uniform distribution

In case there is a strong (e.g., LOS) multipath component
in addition to the complex Gaussian component, we obtain:

c  t   a0  a  t  e               a0   ai t  e
j  t                         ji t 

i

From the joint (magnitude and phase) pdf we can derive:


a     a 2  a02        2 2       aa0 
p a  2 e                                 I0  2 
                                        
Modified Bessel function of
Rice distribution                first kind and order zero
Representation in complex plane
Complex Gaussian distribution       Complex Gaussian distribution
is centered at the origin of the    is centered around the “strong
complex plane => magnitude          path” => magnitude is Rice
is Rayleigh distributed, the        distributed, probability of deep
larger than in the Rician case

iy            p  x, y        iy

a0
x                                     x
Bell-shaped function
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.

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