Modeling and Simulating Modeling and Simulating the Wireless Channelsthe Wireless Channels

Prof. Li PinganProf. Li Pingan

Wuhan University of Tech.Wuhan University of Tech.

Page:2

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

Page:3

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)

Page:4

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

Page:5

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

Page:6

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

),[

Page:7

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

Page:8

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

??

Page:9

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

Page:10

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

Page:11

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

Page:12

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

Page:13

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

Page:14

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)

Page:15

Simulation Results by using MEDSimulation Results by using MED

瑞利过程仿真 对数正态过程仿真器

Page:16

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 :

Page:17

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 :

Page:18

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）

Page:19

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

Page:20

Questions!