Jakes’ Fading Channel Simulator

指導教授：黃文傑 老師學　　生：曾凱霖學　　號： M9121014

Outline Introduction & Problems Background Clarke’s Mathematical Reference Model Jakes’ Simulation Model Time-Average Analyses Statistics of the Reference Model and Jakes’

Fading Channel Simulator Conclusion

Introduction & Problems

1 、 Clarke’s Mathematical Model

2 、 Jakes’ Simulator Family

Background

Clarke’s Mathematical Reference Model (1/3) Received signal RD(t) is a superposition of waves

Normalize RD(t) to have

unit power as

N

nnnmcn

N

n

tjwnnmnD

AtwtwCE

eAtwjCEtR c

10

10

coscos

cosexp)(

N

nnnmcn AtwtwCtR

1

)coscos(2)(cmm

c

n

n

n

vww

w

A

C

E

/2 frequency,radian Dopper :

frequencyradian scosine' dTransmitte :

raynth by the undergoneshift Phase :

raynth theof arrival of Angle :

path nth theofn Attenuatio :

wavecosine ed transmitt theof Amplitude :0

Clarke’s Mathematical Reference Model (2/3)

N

nnnmcn

N

n

tjwnnmnD

AtwtwCE

eAtwjCEtR c

10

10

coscos

cosexp)(

Clarke’s Mathematical Reference Model (3/3)

Rayleigh flat fading narow-band signal

Properties of Rayleigh flat fading narow-band signalThe envelope pdf without LOS is

Phase pdf given by the uniform distribution

Autocorrelation function of the received signal of 2-D isotropic scattering and an omnidirectional receiving antenna

0,)( 2/2

rrerf rR

20 21

)( f

)()cos()( mocR wJw

Jakes’ Simulation Model (1/2)

122

1M

2

)(

form theinto rearranged becan )( then integer, oddan is 2/ If

,...,2,1,2

,)(

)ˆ()ˆ(

1

)cosˆ()cosˆ(0

)ˆ()ˆ(12/

1

)cosˆ()cosˆ(0

1

cos0

N

eeeeN

E

eeeeN

EtT

tTN

NnN

necEtT

tjtjM

n

tjtj

tjtjN

n

tjtj

n

N

n

tjn

mNmNnmnnmn

mNmNnmnnmn

nmn

])(Re[)( ti cetTtE

Jakes’ Simulation Model (2/2)

M

nnnmMs

M

nnnmMc

cscc

twtwX

twtwX

twtXtwtXtR

11

11

cossin2cossin2~

coscos2coscos2~

sin)(~

cos)(~

)(~

Time-Average Analyses (1/2)

Single sinusoid whit fixed amplitude and random phase is both ergodic and stationary. But, sums of fixed amplitude, random-phase sinusoids are not egodic and stationary.

Cn ,An ,n are RVs in the physical model but are f

ixed constants in the simulators.

Time-Average Analyses (2/2)

In Jakes’ simulator, in-phase and quadrature share common frequencies as seen in Fig. 1.

But, in fact, the in-phase and quadrature components share no common Doppler frequency shifts.

)(~ tX c )(~ tX s

Statistics of the Reference Model and Jakes’ Fading Channel Simulator (1/2) Autocorrelation of Reference model,

When N, autocorrelation of low frequency terms, shown in fig.3 becomes Bessel function.

Removing the constraint of (6a), the An becomes uniform I.I.d over [0,2), and

N

nmcR

N

nnmcRR

Nn

wwN

R

ttwttwN

ttR

1

1212121

2coscoscos

1)(

cos)()(cos1

),(

)()(cos),( 2102121 ttwJttwttR mcRR

Statistics of the Reference Model and Jakes’ Fading Channel Simulator (2/2)

From fig.4, the statistical variance of the simulator fading process is time variant. This means Jakes’ model does not present WSS.

Stochastic autocorrelation of the signal of Jakes’ simulator is time dependent with .1212 and tttt

Conclusion

Jakes’ Simulation Model is nonstationary and difficult to generate multiple uncorrelated fading waveforms.

Some model can improved Jakes’ Simulation Model.