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Stochastic Models
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EidgenssischeTechnische Hochschule
Zrich
Stochastic Models 1
Lecture 4:
4. Stochastic Channel Models
Contents:
Radio channel simulation
Stochastic modelling
COST 207 model
CODIT model
Turin-Suzuki-Hashemi model
Saleh-Valenzuela model
EidgenssischeTechnische Hochschule
Zrich
Stochastic Models 2
4. Stochastic Channel Models
Contents (contd):
WAND model
Spencer-Jeffs-Jensen-Swindlehurst model
IEEE 802.11 model
3GPP Stochastic Channel Model
EidgenssischeTechnische Hochschule
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Stochastic Models 3
Main application fields of stochastic channel models:
Design and optimization of communication systems:- Design of the constituents of communication systems- Optimization of the behaviour and performance of these constituents- Analytical or simulation-based investigations of the performance of these
constituents
Monte Carlo simulations for system performance assessment- Link-level simulations (today)
- System-level simulations (today)
- Simulation of cooperative networks (coming soon)
in complex scenarios as close as possible to real operation conditions
Stochastic channel models (SCMs) are crucial and indispensable tools inthe design of communication systems.
The importance of stochastic channel modelling
EidgenssischeTechnische Hochschule
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Stochastic Models 4
The different approaches for channel simulations:
Stored Channel Models (ATDMA)
Measured space-variant or time-variant delay spread func-tions (SFs) are selected according to some specified cri-teria to form classes of so-called reference channels.These delay SFs are stored and can be read on demandfor simulation purposes.
Example:Reference space-variant delay SF inan atypical microcellular urbanenvironments:
Radio channel simulation
EidgenssischeTechnische Hochschule
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Stochastic Models 5
The different approaches for channel simulations (contd):
Deterministic Channel Models
These models employ ray optical techniques (ray tracing, raylaunching combined with UTD method) to compute the delay SFor the direction-delay SF using some more or less extensive geo-graphical information (buildinglayout, electric properties ofwalls and floors, etc.) about thepropagation environment underconsideration.
Radio channel simulation
EidgenssischeTechnische Hochschule
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Stochastic Models 6
The different approaches for channel simulations (contd):
Stochastic Channel Models (COST 207, CODIT)
- A parametric model for the delay SFs is derived.
- Realizations of the delay SF are thengenerated according to specifiedprobability distributions of the modelparameters.
- These probability distributions aregathered by means of statisticalanalyses of measurement datacollected during extensive measure-ment campaigns.
0
5
10
15
20 05
1015
2025
0
0.2
0.4
0.6
0.8
1
real time [ms]delay [mys]Delay [ s] Time [ms]t
g t ;( )Example: COST 207 HT
Radio channel simulation
EidgenssischeTechnische Hochschule
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Stochastic Models 7
Comparison of the different approaches:
Channelsimulationapproaches
Computationalexpense
Intrinsicvariability of the
models
Retained channelfeatures
Challenge
Referencechannels
High(storage)
Low(=number of
stored channels)
All Choice of theappropriate
selection criteria
Deterministicchannel models
High(identification of
the dominantpropagation
paths)
Medium(=number ofenvironmentsconsidered)
Part of them(ray optical
methods are notexact)
Identification ofthe dominantprop. paths +
accurate methodfor computing
the path weights
Stochasticchannel models
Low High(probability
distributions)
Part of them(depending on the
model used)
Incorporate allrelevant channel
features
Radio channel simulation
EidgenssischeTechnische Hochschule
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Stochastic Models 8
Features to be incorporated into SCMs:
Stochasticchannel model
Environment:Type of environment
Frequency range
Transceiver characteristics:Trajectory, velocity
BandwidthAntenna (array) types
Short-term fluctuations:Fast fading
Long-term fluctuations:Path loss
ShadowingTransitionsDelay drift
Drift in directionsBirth & death of paths
Channel dispersionDelay
Direction of departureDirection of incidence
Doppler frequencyPolarization
Short range/term (fast)fading
Stochastic modelling
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Stochastic Models 9
Requirements for SCMs:
Completeness
SCMs must reproduce all effects that impact on the performance of com-munication systems.
=> Guarantee simulation scenarios close to reality=> Full basis for system comparisons
Accuracy
SCMs must accurately describe these effects.=> Realistic results from analytical and/or simulation-based investigations
Simplicity/low complexity
Each effect must be described by a simple model.=> Enable theoretical study of some particular system aspects and performance=> Tractable computational effort to simulate the channel in Monte Carlo simulations
Stochastic modelling
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Stochastic Models 10
Approach for SCM:
Specification of the area type additional system parame-
ters (array characteristics,mobile velocity, etc.)
Statistical processing of data obtained fromextensive measurement campaigns in the
identified areasEstimation of the probability density func-
tions of the parameters occurring in theparametric system function
Realizations of the system function,e.g. g t ;( )
Software package generating a specific para-metric system function of the channel, e.g.
g t ;( ) gi t( ) i t( )( )i 1=
N
Stochastic modelling
EidgenssischeTechnische Hochschule
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Stochastic Models 11
Model for :gi t( )
giST( )
t( ) gi c, t( ) gi d, t( )+=
Short-term fluctuations:
Specular part:
: Doppler frequency
gi c, t( ) hi c, j2it( )exp=i
Diffuse part:is a WSS zero-mean circular symmetric com-
plex Gaussian process specified by its ACF
or equivalently its (Doppler) spectrum :
gi d, t( )Ri t( )
Pi ( )
Ri t( ) Pi ( )t
Stochastic modelling
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Stochastic Models 12
Model for (contd):
Long-term fluctuations (path loss & shadowing effect):
: time-dependent path loss computed from one of the modelspresented in Lecture 3.
: real zero-mean Gaussian process with ACF .
Usually,
gi t( )
gi t( ) 10L t( )10
----------
10
Li vt( )10
------------------
giST( )
t( )=
L t( )
Li d( ) RLi d( )
RLi d( ) 2 d2 2( )exp=
: standard deviation of
: decorrelation lengthSmall macrocells:
Li d( ) 6 8 dB=
5 8m=
Stochastic modelling
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Stochastic Models 13
Models for :
Short-term fluctuations:
where is a specified random point process.
Long-term fluctuations:
where is the above random point process and
is a sequence of random processes describing the drift of the components inthe time-variant SF on the delay axis.
i t( )
i t( ) i=1 2 , i , , ,{ }
i t( ) i i t( )+=1 2 , i , , ,{ } i t( ){ }
Stochastic modelling
EidgenssischeTechnische Hochschule
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Stochastic Models 14
Main characteristics:
Cell type Macrocell
Area Typical non-hilly urban (TU), bad hilly urban (BU) Non-hilly rural area (RA), hilly terrain (HT)
Frequency range Around 1 GHz
Time-variant SF
Input Area type Vehicle velocity Number of components in Delay and Doppler resolution
g t ;( ) PN---- j 2it i+( ){ }exp i( )
i 1=
N
=
g t ;( )
COST 207 model
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Stochastic Models 15
Normalized delay-Doppler scattering function:
We can decompose as
The COST 207 models are specified by the two functions:
Pn ,( )1P---P ,( ) Pn ,( ) d d 1=( )
Pn ,( )
Pn ,( ) Pn ( ) Pn ( )=Normalized delayscattering function
Delay-dependent normalizedDoppler scattering function
Pn ( ) Pn ,( ) d( )
Pn ( )Pn ( )
COST 207 model
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Stochastic Models 16
Normalized delay scattering function:
Typical urban non-hilly area (TU):
MSvLocal
scattering
Pn ( )( )exp ; 0 s[ ] 7
0 ; elsewhere
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
1.2Pn ( )
[s]
COST 207 model
EidgenssischeTechnische Hochschule
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Stochastic Models 17
Normalized delay scattering function (contd):
Pn ( )( )exp ; 0 s[ ] 5
0.5 5 ( )exp ; 5 s[ ] 10 0 ; elsewhere
Typical bad urban hilly area (BU):
MSv Local
scattering
Distantscattering
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7Pn ( )
[s]
COST 207 model
EidgenssischeTechnische Hochschule
Zrich
Stochastic Models 18
Normalized delay scattering function (contd):
Typical rural non-hilly area (RA):
Pn ( )9.2( )exp ; 0 s[ ] 0.7
0 ; elsewhere
Pn ( )
[s]
MSv Localscattering
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
6
7
8
9
10
COST 207 model
EidgenssischeTechnische Hochschule
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Stochastic Models 19
Normalized delay scattering function (contd):
Pn ( )3.5( )exp ; 0 s[ ] 2
0.1 15 ( )exp ; 15 s[ ] 20 0 ; elsewhere
Typical hilly terrain (HT):
Pn ( )
[s]
MSv
Localscattering
Distantscattering
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
COST 207 model
EidgenssischeTechnische Hochschule
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Stochastic Models 20
Normalized Doppler scattering function:
MSv
Many scattererswith nearly thesame featuresuniformly dis-tributed around
the MS
Pn ( )1
D---------- 1
1 v D( )2
---------------------------------- ; D The parameter settings in the COST 231 tables are not appropriate forinvestigating the performance of systems which exploit channelfrequency selectivity,e.g. GSM with frequency hopping.
Solution:Choose an irregularly spacing of the delays or randomly select them.
i mi= mi 0.2s=
COST 207 model
EidgenssischeTechnische Hochschule
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Stochastic Models 32
The COST 207 tables (contd):Frequency hopping:
Real channel frequency transfer function:
Transfer function of the COST 207 channel:
B1 B2 B2'B1'
B1 B2 B2'B1'
Bc 0.2=
f
f
H f( )
H f( )
1------
COST 207 model
EidgenssischeTechnische Hochschule
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Stochastic Models 33
Main characteristics:
Cell type Macro-, micro, picocell
Area Macrocell: urban, suburban, suburban hillyrural, rural hilly
Microcell: dense urban linear street, town square, industrial area Indoor: floor cell in buildings, corridor,
large and very large rooms
Frequency band 2 GHz range Up to 20 MHz signal bandwidth
Time-variant SF
Input Area type Vehicle velocity System bandwidth
g t ;( ) gi t( ) i t( )( )i 1=
N
CODIT model
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Stochastic Models 34
Short-term variations of :gi t( )
xxxxxxx
xxxxxxx
i c, i j,
xxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxx
Specular part Diffuse part
giST( )
t( ) hi c, j2v--- i c,( )cos t exp hi j, j2
v--- i j,( )cos t exp
j 1=
J
+=
,
uniformly distributed over
: mean power of the th component
: coherent to diffuse power ratio of the th component
hi c, i c,=
hi c,{ }arg 0 2 ),[
hi j, N 01J---i d,
2,
E gi t( )2[ ] i c,
2i d,
2+ i
2= i
i c, i d,( )2
i
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 35
Probability distribution of the azimuths and :i c, i j,
Scatterer i
v
i c,
i j,
MS0 2
pi j, ( )
i c,
i j, N i c, 0.15[radian]2
,( )mod2
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 36
Long-term variations of :
Shadowing effects:
gi t( )
gi t( ) zi t( ) gi ST( ) t( )=Term describing the
long-term fluctuations
zi t( ) 1 zi2qi------ v---t i+ cos+ 1 2zi
zi t( )
tqiv ( )--------------: Random variable uniformly distributed overi 0 2 ),[
CODIT model
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Stochastic Models 37
Long-term variations of (contd):Emergence of the components:
Fading of the components:
is a random variable uniformly distributed over .
gi t( )
1zi t( )1
1vt z
5------------- 6
+
------------------------------- ; vt z
1 ; vt z>
zi t( )
tz v
1zi t( )1
1vt z
5------------- 6
+
------------------------------- ; vt z
1 ; vt z
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Stochastic Models 38
Short-term variations of :
Long-term variations of :
i t( )
i t( ) i=
i t( )
i t( ) i i 12pi------ v---t i+ cos++= i
2i
i t( )
tpiv ( )--------------: Random variable uniformly distributed overi 0 2 ),[
CODIT model
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Stochastic Models 39
Selection of the parameters of :
The number of components depends on the area type but is fixed for agiven area, .
The parameters describing the
behaviour of the components of are random variables specified by
probability distributions.
g t ;( )N
N max 20=
J 100=
i2
i c, i d,( )2 i c, zi qi i i pi, , , , , , ,gi t( )
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 40
Setting Tables:
Example: Suburban hilly environment:
i2 zii i [s ] qi pi i [ns ]
E hi2[ ]2
Var hi2[ ]
--------------------------
Source: CODIT Report: Final Propagation Modeli
2
i
N
1=
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 41
Setting Tables (contd):
Example: Power delay spectrum generated with the previous table:
S
o
u
r
c
e
:
C
O
D
I
T
R
e
p
o
r
t
:
F
i
n
a
l
P
r
o
p
a
g
a
t
i
o
n
M
o
d
e
l
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 42
Example: Microcell LOS Area:
Contour plot of a realization of :g d ;( ) Evolution of the correspondingdelay scattering function
D
i
s
t
a
n
c
e
d
[
]
P
o
w
e
r
[
d
B
]
Delay [ns] Delay [ns]
Source: CODIT Report: Final Propagation Model
d 80=
d 0=
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 43
Example: Indoor Picocell LOS Area:
Contour plot of a realization of :g d ;( ) Evolution of the correspondingdelay scattering function
P
o
w
e
r
[
d
B
]
Delay [ns] Delay [ns]
Source: CODIT Report: Final Propagation Model
D
i
s
t
a
n
c
e
d
[
]
CODIT model
EidgenssischeTechnische Hochschule
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Stochastic Models 44
Main characteristics:
Cell type Macro-, micro- and picocell(by appropriately setting the probability densities of model parame-ters)
Area Urban
Frequency range 500MHz -1GHz
Time-invariant delay SF
Main features Time-invariant Clustering of the components in is reproduced by modelling
the sequence as a Poisson point process or a modificationthereof.
g t ;( ) h ( ) hi i( )i 1=
N
= =
h ( )i{ }
Turin-Suzuki-Hashemi model
EidgenssischeTechnische Hochschule
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Stochastic Models 45
Modelling as a Poisson point process:
is a Poisson point process with rate :
: Number of points in the time interval
i{ }i{ } ( )
( )
T
1 2 i N
T( )
i 1
T( ) ( ) dT
N T T
P N T n=[ ] T( )n
n!--------------- T( )( )exp= E N T[ ] Var N T[ ] T( )= =[ ]
Turin-Suzuki-Hashemi model
EidgenssischeTechnische Hochschule
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Stochastic Models 46
Modelling as a homogeneous Poisson point process:
In this case, the delay differences are independent randomvariables which are (identically) exponentially distributed with parameter :
Drawback of the Poisson model:It does not describe sufficiently accurately the clustering of the componentsas observed in measured delay SFs.
i{ } ( ) =i i 1
pi i 1 ( )
pi i 1 ( ) ( )exp=
Probability density of :i i 1
1
Turin-Suzuki-Hashemi model
EidgenssischeTechnische Hochschule
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Stochastic Models 47
Estimated rates in outdoor environments:
1ft 1ns
S
o
u
r
c
e
:
P
a
p
e
r
b
y
T
u
r
i
n
a
t
a
l
.
Turin-Suzuki-Hashemi model
EidgenssischeTechnische Hochschule
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Stochastic Models 48
Modelling as a modified Poisson point process:
The rate depends on the random sequence in the following way:
i{ } ( ) i{ }
The basic rate is multiplied by a factor
during a predetermined relaxation interval at
each :
: Clustering effect
: Poisson point process
: Isolated delay points
Values for and (macrocell):
(arbitrarily selected)
(estimated from measure-ments)
0 ( ) css
ics 1>
cs 1=
cs 1
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