7. Channel Models
Signal Losses due to three Effects:
2. Medium Scale
Fading: due to
shadowing and
obstacles
1. Large Scale
Fading: due to
distance
3. Small Scale
Fading: due to
multipath
Wireless Channel
Frequencies of Interest: in the UHF (.3GHz – 3GHz) and SHF (3GHz – 30 GHz)
bands;
Several Effects:
• Path Loss due to dissipation of energy: it depends on distance only
• Shadowing due to obstacles such as buildings, trees, walls. Is caused by
absorption, reflection, scattering …
• Self-Interference due to Multipath.
10 log 10
Prec
Ptransm
log 10 distance
1.1. Large Scale Fading: Free Space
Path Loss due to Free Space Propagation:
For isotropic antennas:
Transmit
antenna
  
Prec  
 Ptransm
 4 d 
c
wavelength  
F
2
d
Receive
antenna
Path Loss in dB:
 Ptransm 
L  10log10 
  20log10 ( F ( MHz ))  20log10 (d (km))  32.45
 Prec 
2. Medium Scale Fading: Losses due to Buildings, Trees,
Hills, Walls …
The Power Loss in dB is random:
L p  E L p  
expected value
random, zero mean
approximately gaussian with
  6  12 dB
Average Loss
Free space loss at reference
distance
d 
E{L p }  10 log 10    L0
 d0 
dB
Reference distance
• indoor 1-10m
Path loss
exponent
E  L p   L0
• outdoor 10-100m
10
20dB
102
101 100 10
log10 (d / d0 )
Values for Exponent

Free Space
2
Urban
2.7-3.5
Indoors (LOS)
1.6-1.8
Indoors(NLOS)
4-6
:
Empirical Models for Propagation Losses to Environment
• Okumura: urban macrocells 1-100km, frequencies 0.15-1.5GHz,
BS antenna 30-100m high;
• Hata: similar to Okumura, but simplified
• COST 231: Hata model extended by European study to 2GHz
3. Small Scale Fading due to Multipath.
a. Spreading in Time: different paths have different lengths;
Receive
Transmit
x(t )   (t  t0 )
t0
y (t ) 
 hk (t  t0   k )  ...
t0  1  2
time
Example for 100m path difference we have a time delay
100
102
1



3  sec
8
c
3  10
3
Typical values channel time spread:
x(t )   (t  t0 )
t0
channel
Indoor
10  50 n sec
Suburbs
2  101  2  sec
Urban
1  3  sec
Hilly
3-10  sec
t0  1  2  MAX
b. Spreading in Frequency: motion causes frequency shift (Doppler)
x(t )  X T e j 2 Fct
Receive
Transmit
y(t )  YRe
j 2  Fc F t
time
v
for each path
Doppler Shift
f c Fc  F Frequency (Hz)
time
Put everything together
Transmit
x(t )
time
v
time
Receive
y (t )
channel
x(t )
w(t )
y (t )
gT (t )
h(t )
Re{.}
g R (t )
LPF
LPF
e j 2FC t
Each path has …
e  j 2FC t
… attenuation…
…shift in time …


y (t )  Re  a (t ) x(t   )e j 2 ( Fc  F )(t  ) 


paths
…shift in frequency …
(this causes small scale time variations)
2.1 Statistical Models of Fading Channels
Several Reflectors:
x (t )
Transmit
1
t
2
v
y (t )
t

For each path with NO Line Of Sight (NOLOS):
average time delay
v
y (t )

t
v cos( )
• each time delay
  k
• each doppler shift F  FD
t

y (t )  Re   ak e j 2 ( Fc  F )(t 
 k
 k )

x(t     k )  

Some mathematical manipulation …

y (t )  Re   ak e j 2F t e j 2 ( Fc  F )(
 k
 k )
 j 2 Fct 
x(t     k )  e







j 2F t  j 2 ( Fc F )
r (t )    ak e
e
 k
 k 

 x(t   )

Assume: bandwidth of signal << 1 /  k
x(t )  x(t   k )

… leading to this:

y (t )  Re r (t ) e j 2 Fct

r (t )  c (t ) x(t   )
with
c (t )   ak e
k
j 2F t  j 2 ( Fc F )  k 
e
random, time varying
Statistical Model for the time varying coefficients
M
j 2F t  j 2 ( Fc F )  k 
k
k 1
random
By the CLT c (t ) is gaussian, zero mean, with:
c (t )   a e
e
E c (t )c* (t  t )  P J 0 (2 FD t )
with
v
v
FD  FC 
c

the Doppler frequency shift.
Each coefficient c (t ) is complex, gaussian, WSS with autocorrelation
E c (t )c (t  t )  P J 0 (2 FD t )
*
and PSD
1
 2
 F
S ( F )  FT  J 0 (2 FD t )   D 1  ( F / FD ) 2
0

with
FD
if | F | FD
otherwise
maximum Doppler frequency.
S (F )
This is called Jakes
spectrum.
FD
F
Bottom Line. This:
x(t )
y (t )
time
time
v
1

time
N
… can be modeled as:
1
c1 (t )
x(t )

c (t )
time
time
N
delays

y (t )
cN ( t )
For each path
c (t )  P  c (t )
• time invariant
• from power distribution
• unit power
• time varying (from autocorrelation)
Parameters for a Multipath Channel (No Line of Sight):
Power Attenuations:
1
P1
Doppler Shift:
FD
Time delays:
2  L
P2  PL 
Summary of Channel Model:
y(t )   c (t ) x(t    )

c (t )  P  c (t )
c (t ) WSS with Jakes PSD
sec
dB
Hz
Non Line of Sight (NOLOS) and Line of Sight (LOS) Fading Channels
1. Rayleigh (No Line of Sight).
E{c (t )}  0
Specified by:
Time delays
T  [ 1 , 2 ,..., N ]
Power distribution
P  [ P1 , P2 ,..., PN ]
Maximum Doppler
2. Ricean (Line of Sight)
FD
E{c (t )}  0
Same as Rayleigh, plus Ricean Factor
K
K
PTotal
1 K
Power through LOS
PLOS 
Power through NOLOS
PNOLOS 
1
PTotal
1 K
Simulink Example
M-QAM Modulation
Bernoulli
Binary
Rectangular
QAM
Bernoulli Binary
Generator
Rectangular QAM
Modulator
Baseband
Channel
Transmitter
Attenuation
Gain
Multipath Rayleigh
Fading Channel
-KB-FFT
Spectrum
Scope
Bit Rate
Rayleigh Fading Channel
Parameters
-K-
Receiver
Gain
-K-
Rayleigh
Fading
Set Numerical Values:
velocity
carrier freq.
Recall the Doppler Frequency:
Easy to show that:
FD 
v
FC
c
3 108 m / sec
FD Hz  v km / h FC GHz
modulation
power
channel
Channel Parameterization
1. Time Spread and Frequency Coherence Bandwidth
2. Flat Fading vs Frequency Selective Fading
3. Doppler Frequency Spread and Time Coherence
4. Slow Fading vs Fast Fading
1. Time Spread and Frequency Coherence Bandwidth
Try a number of experiments transmitting a narrow pulse p (t ) at different random
times
x(t )  p(t  ti )
We obtain a number of received pulses
yi (t )   c (t ) p(t  ti   )
 c (t

0
transmitted
1
c1 (ti   1 )
0
0
t  t1

c2 (ti   2 )
c (ti   )
2

2

t  ti


1
  ) p(t  ti   )

2

1
i



t  tN
Take the average received power at time
P1
P2
1

P  E | c (t ) |
P

0
  t  ti
2



More realistically:
Received Power
0
10
20
 RMS
 MEAN
time
2

This defines the Coherence Bandwidth.
Take a complex exponential signal
the channel is:
x(t )
with frequency
F . The response of
y(t )   c (t )e j 2F (t  MEAN   )

If
| F |  RMS  1
then

 j 2 F (t  MEAN )
y (t )    c (t )  e


i.e. the attenuation is not frequency dependent
Define the Frequency Coherence Bandwidth as
Bc 
1
5   RMS
This means that the frequency response of the channel is “flat” within
the coherence bandwidth:
Channel “Flat” up to the
Coherence Bandwidth
Bc 
Coherence Bandwidth
Flat Fading
Signal Bandwidth
<
>
1
frequency
5   RMS
Just attenuation, no distortion
Frequency Coherence
Frequency Selective
Fading
Distortion!!!
Example: Flat Fading
Channel :
Delays T=[0 10e-6 15e-6] sec
Power P=[0, -3, -8] dB
Symbol Rate Fs=10kHz
Doppler Fd=0.1Hz
Modulation QPSK
Very low Inter Symbol
Interference (ISI)
Spectrum: fairly uniform
Example: Frequency Selective Fading
Channel :
Delays T=[0 10e-6 15e-6] sec
Power P=[0, -3, -8] dB
Symbol Rate Fs=1MHz
Doppler Fd=0.1Hz
Modulation QPSK
Very high ISI
Spectrum with deep
variations
3. Doppler Frequency Spread and Time Coherence
Back to the experiment of sending pulses. Take autocorrelations:

1
0
transmitted
c1 (ti   1 )
2
0
c2 (ti   2 )
c (ti   )
2

0
2
t  ti


R2 (t )
R (t )
R (t )
1



1
t  t1


1


t  tN
Where:


R (t )  E c (t )c* (t  t )
Take the FT of each one:
S (F )
FD
This shows how the multipath characteristics
F
c (t ) change with time.
It defines the Time Coherence:
TC 
9
16 FD
Within the Time Coherence the channel can be considered Time Invariant.
Summary of Time/Frequency spread of the channel
Frequency Spread
Time
Coherence
TC 
9
16 FD
S (t , F )
F
FD
t
 mean
 RMS
Frequency
Coherence
Bc 
1
5   RMS
Time Spread
Stanford University Interim (SUI) Channel Models
Extension of Work done at AT&T Wireless and Erceg etal.
Three terrain types:
• Category A: Hilly/Moderate to Heavy Tree density;
• Category B: Hilly/ Light Tree density or Flat/Moderate to Heavy Tree density
• Category C: Flat/Light Tree density
Six different Scenarios (SUI-1 – SUI-6).
Found in
IEEE 802.16.3c-01/29r4, “Channel Models for Wireless Applications,”
http://wirelessman.org/tg3/contrib/802163c-01_29r4.pdf
V. Erceg etal, “An Empirical Based Path Loss Model for Wireless
Channels in Suburban Environments,” IEEE Selected Areas in
Communications, Vol 17, no 7, July 1999