### 7-Channel Models

```7. Channel Models
Signal Losses due to three Effects:
2. Medium Scale
obstacles
1. Large Scale
distance
3. Small Scale
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
Prec
10
Ptransm
log
10
distance
1.1. Large Scale Fading: Free Space
Path Loss due to Free Space Propagation:
For isotropic antennas:
Transmit
antenna
2
d
antenna
  
Prec  
 Ptransm
 4 d 
wavelength  
c
F
Path Loss in dB:
 Ptransm 
L  10 log 10 
  20 log 10 ( F ( M H z ))  20 log 10 ( 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 
  L0
E { L p }  10  log 10 

d
 0
dB
Reference distance
• indoor 1-10m
Path loss
exponent
E  L p   L0
• outdoor 10-100m
10 
20dB
10
2
10
1
10
0
10
log 10 ( d / d 0 )
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;
Transmit
x ( t )   ( t  t0 )
t0
y (t ) 
 h k  ( t  t 0   k )  ...
t0
time
Example for 100m path difference we have a time delay
 
100
c

10
2
3  10
8

1
3
 sec
1  2
3
x ( t )   ( t  t0 )
channel
t0
Indoor
10  50 n sec
S uburbs
2  10
U rban
1  3  sec
H illy
3-10  sec
1
 2  sec
t0
1  2
 M AX
b. Spreading in Frequency: motion causes frequency shift (Doppler)
x (t )  X T e
j 2  Fc t
Transmit
y (t )  YR e
j 2   Fc   F  t
time
v
for each path
Doppler Shift
fc
Fc   F
Frequency (Hz)
time
Put everything together
Transmit
x (t )
time
v
time
y (t )
channel
x (t )
w (t )
y (t )
g T (t )
g R (t )
h (t )
Re{.}
LPF
LPF
e
j 2  FC t
Each path has …
e
… attenuation…
 j 2  FC t
…shift in time …

j 2  ( Fc   F
y (t )  R e   a (t ) x (t   ) e

paths
)( t   )



…shift in frequency …
(this causes small scale time variations)
2.1 Statistical Models of Fading Channels
Several Reflectors:
x (t )
1
t
Transmit
2
v
y (t )
t

For each path with NO Line Of Sight (NOLOS):
y (t )
average time delay 
v
t
v cos( )
• each time delay
  k
• each doppler shift  F  FD
t

j 2  ( Fc   F
y (t )  R e    a k e
 k
)( t     k )

x (t     k )  

Some mathematical manipulation …

j 2   F t  j 2  ( Fc   F
y (t )  R e    a k e
e
 k
)(    k )
 j 2  Fc t 
x (t     k )  e




    


j 2   F t  j 2  ( Fc   F
r (t )    a k e
e
 k
)    k


 x (t   )

Assume: bandwidth of signal << 1 /  k
x (t )  x (t   k )


y (t )  R e r (t ) e
j 2  Fc t

r (t )  c (t ) x (t   )
with
c (t ) 

k
ak e
j 2  F t
e
 j 2  ( Fc   F )    k

random, time varying
Statistical Model for the time varying coefficients
M
c (t ) 
a
k
e
j 2  F t
e
 j 2  ( Fc   F )    k

k 1
random
By the CLT c ( t ) is gaussian, zero mean, with:
E  c ( t ) c ( t   t )  P J 0 (2  F D  t )
*
with
FD 
v
c
FC 
v

the Doppler frequency shift.
Each coefficient c ( t ) is complex, gaussian, WSS with autocorrelation
E  c ( t ) c ( t   t )   P J 0 (2  F D  t )
*
and PSD
 2
 F
S ( F )  F T  J 0 (2  F D  t )  
D

0
with
FD
1
1  ( F / FD )
2
if | F | F D
otherw ise
maximum Doppler frequency.
S (F )
This is called Jakes
spectrum.
FD
F
Bottom Line. This:
x (t )
y (t )
time
time
v
1

N
time
… can be modeled as:
1
c1 ( t )
x (t )


y (t )
c (t )
time
time
N
delays
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
sec
P2

PL 
dB
Summary of Channel Model:
y (t ) 
c

(t ) x (t    )

c (t ) 
P  c (t )
c (t ) WSS with Jakes PSD
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
Power through LOS
PLOS 
Power through NOLOS
PNOLOS 
1 K
PTotal
1
1 K
PTotal
M-QAM Modulation
Bernoulli
Binary
Rectangular
QAM
Bernoulli Binary
Generator
Rectangular QAM
Modulator
Baseband
Channel
Transmitter
Attenuation
Gain
Multipath Rayleigh
-KB-FFT
Spectrum
Scope
Bit Rate
Parameters
-K-
Gain
-K-
Rayleigh
Set Numerical Values:
velocity
carrier freq.
Recall the Doppler Frequency:
Easy to show that:
FD 
v
c
 FD Hz
FC
3  10 m / sec
8
 v km / h  FC GHz
modulation
power
channel
Channel Parameterization
1. Time Spread and Frequency Coherence Bandwidth
3. Doppler Frequency Spread and Time Coherence
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  t i )
We obtain a number of received pulses
y i (t ) 
c
(t ) p (t  ti   )
c

0
transmitted
1
c1 ( t i   1 )
0

2
t  t1

c 2 (ti   2 )

1
(ti   ) p (t  ti   )
c (ti   )
2


t  ti



0
1
2


t  tN
Take the average received power at time   t  t i
P1
P2

1
0
P  E | c  ( t ) |
P
2



More realistically:
0
10
20
 RM S
 M EAN
time
2

This defines the Coherence Bandwidth.
Take a complex exponential signal
the channel is:
y (t ) 

x (t ) with frequency F . The response of
c  (t ) e
j 2  F ( t   MEAN     )

If | F |   RMS  1

 j 2  F ( t  M EAN )
then y ( t )    c ( t )  e


i.e. the attenuation is not frequency dependent
Define the Frequency Coherence Bandwidth as
Bc 
1
5   RM S
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
Signal Bandwidth
<
>
1
frequency
5   RM S
Just attenuation, no distortion
Frequency Coherence
Frequency Selective
Distortion!!!
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
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:

0
transmitted
1
c1 ( t i   1 )
2
0
t  t1

c 2 (ti   2 )

1

c (ti   )
2


t  ti



0
1
2


R2 (  t )
R (t )
R1 (  t )
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 F D
Within the Time Coherence the channel can be considered Time Invariant.
Summary of Time/Frequency spread of the channel
Time
Coherence
TC 
9
16 F D
S (t , F )
F
FD
t
 mean
 RMS
Frequency
Coherence
Bc 
1
5   RM S
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
```