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Modeling and Simulating Modeling and Simulating the Wireless Channelsthe Wireless Channels
Prof. Li PinganProf. Li Pingan
Wuhan University of Tech.Wuhan University of Tech.
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Flat fading channel modelFlat fading channel model
Two classical simulation model earlyR.H. Clarke, Bell Syst. Tech. J., 1968W.C. Jakes, Microwave Mobile Comm., 1974
Conventional flat fading modelJakes modelClarke Gaussian noise filter modelM. Patzold fading model in cell mobile
communication system
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Jakes modelJakes model The sum of a set of sinusoids with uniformly distribution on phase
)2cos( 1tf
)2cos( tfM
)2cos(2
1tfm
X
X
X
+
1cos2
Mcos2
1cos2 M
X
N
2
)2cos( 1tf
)2cos( tfM
)2cos(2
1tfm
X
X
X
+
1sin2
Msin2
1sin2 M
X
N
2
)(tTc
)()()( tjTtTtT sc
(N=4M+2)
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Jakes modelJakes model
MnM
n
Mnn
,,2,1
14
MnN
nf
Mnff
m
m
n ,,2,12
cos
1
mfwhere denotes the maximal Doppler frequency
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Improved Jakes modelImproved Jakes model
)2cos( 11 tf
)2cos(2
11 Mmtf
X
X
X
+
1cos2
Mcos2
1cos2 M
X
N
2
)2cos( 11 tf
)2cos( MM tf
)2cos(2
11 Mmtf
X
X
X
+
1sin2
Msin2
1sin2 M
X
N
2
)(tTc
)()()( tjTtTtT sc
(N=4M+2)
)2cos( MM tf
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Improved Jakes modelImproved Jakes model
MnM
n
Mnn
,,2,1
14
MnN
nf
Mnff
m
m
n ,,2,12
cos
1
mfwhere denotes the maximal Doppler frequency, are independent random variables
uniformly distributed over for all n.)11( Mnn
),[
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Gaussian noise filterGaussian noise filterThe sum of two quadrature independent Gaussian random variables
Independent complex Gaussian white noise source
12
Ng*
12
Ng
2
Ng*
2
Ng
mf mf
12
Ng*
12
Ng
2
Ng*
2
Ng
mf mf
)( fSD
mf mf
)( fSD
mf mf
IFFT
IFFT
2
2
o90
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Suzuki processSuzuki process
Suzuki process is a product process of a Rayleigh process and log-normal process
)()()()( 22
21 tt、tt
)()()( 21 tjtt
)2cos()( ,,1
, nini
N
nnii tfct
i
)(3)( tsmet
Complex Gaussian
Gaussian
Rayleigh
Log-normal
)(t)(t )(t
)()()( ttt
??
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PDF and auto-correlation for PDF and auto-correlation for
Jakes power spectral density function
The autocorrelation function
The mean power of
max
max2
maxmax
2
0
)/(1)(0
ff
ffffffS
i
2,1),2()( max02
0 itfJtr
i
i
dffSi
)(2
0
2,1ii
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Method of equal distances: Rayleigh-Method of equal distances: Rayleigh-distributiondistribution
2
1
2,
2
1,,
,
02/)0(
,,2,
,,1),12(2
)12(2
i
ii
i
i
N
ini
ininif
ii
m
fni
cr
Nnff
NnnN
f
nf
24,25,2,1
)]}arcsin()[arcsin({
21
2/114
,
20
NNi
cii N
nNn
ni
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PDF and auto-correlation forPDF and auto-correlation for 3
)2
exp(2
1)(
2
2
3
cc
ffS
2ln2
cc
f
Power spectral density function
The autocorrelation function3dB cutoff
F
2
3
)(2)( tcetr
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Method of equal distances: Log-normal Method of equal distances: Log-normal distributiondistribution
4.., ccc gef
25;,,1
)]}2
)1(()
2([2{
33
2/1
33
,3
NNn
N
nerf
N
nerfc cc
n
)12(2 3
,3
nN
f ccn
To limit the F-region
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Suzuki Process SimulatorSuzuki Process Simulator1,1c
)2cos( 1,11,1 tf
)2cos( 2,12,1 tf
)2cos(11 ,1,1 NN tf
X
X
X
+
2,1c
1,1 Nc
)2cos( 1,21,2 tf
)2cos( 2,22,2 tf
)2cos(11 ,2,2 NN tf
X
X
X
+
1,2c
2,2c
1,2 Nc
)(~t
)(~1 t
)(~2 t
)(~ t
)2cos( 1,31,3 tf
)2cos( 2,32,3 tf
)2cos(11 ,3,3 NN tf
X
X
X
+
1,3c
2,3c
1,3 Nc
X + e
)(~3 t
s m
X
)(~ t
)(~
)(~)(~ ttt
RayleighProcess
Log-normalProcess
SuzukiProcess
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Simulation Approaches for Suzuki ProcessSimulation Approaches for Suzuki Process
等距离法 Method of Equal Distances (MED)
等面积法 Method of Equal Area (MEA)
蒙特卡罗方法 Monte Carlo Method (MCM)
均方误差法 Mean Square Error Method (MSEM)
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Simulation Results by using MEDSimulation Results by using MED
瑞利过程仿真 对数正态过程仿真器
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理论值和仿真值比较理论值和仿真值比较
It can be seen that the simulation model coincides very closely to the theoretical model
0,2
exp)(2
2
2
xxx
xpii
理论模型 :
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理论值和仿真值比较理论值和仿真值比较
It can be seen that the simulation model coincides very closely to the theoretical model
0,
2
lnexp
2
1)(
2
x
s
mx
sxp
xx
理论模型 :
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理论值和仿真值比较(相关函数)理论值和仿真值比较(相关函数)
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Multi-paths Channel Model
0
TDL (Tap-Delay-Line) ModelTDL (Tap-Delay-Line) Model
time
tap1 tap2tap3
tap4
X X X X
RayleighFading
Simulator
RayleighFading
Simulator
RayleighFading
Simulator
RayleighFading
Simulator
1
2
3
Tx
Rx
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Questions!