Chapter 4: Motivation for Dynamic Channel Models

Short-term Fading

Varying environmentObstacles on/off

Area 2Area 1

Transmitter

Log-normalShadowing

Varying environmentObstacles on/off

Mobiles move

Complex low-pass representation of impulse response:

Chapter 4: Motivation for Dynamic Channel Models

( )( ; )

1

2 2

( ; ) ( ; ) ( ( ))

( ; ) ( ; ) ( ; ): Signal attenuation: random process

Stochastic Differential Equations (S.D.E.s) of specific type

a) for log-normal shadowing model

i

i

N tj t

l i ii

i i

C t r t e t t

r t I t Q t

1

s and

b) short-term fading models

( ; )( ; ) tan : Phase angle (random pr.)( ; )

( ), ( ) : Propagation delay (random) and number of

waves impinging on the receiver antenna

at time (a

ii

i

i

Q tt I t

t N t

t

counting process)

Chapter 3: S.D.E.’s for Short-Term Fading

Dynamics represent time-variations of environment Captured by Doppler power spectrum

From power spectral densities to S.D.E.’s

State-space realizations of In-phase and Quadrature components

Time-domain simulations flat-fading, frequency select.

Distributions of short-term dynamical channel Summary of distributions

Chapter 3: 3-Dimensional Scattering Model

n

n

x

z

y

nth inco

ming wave

E n=:{r n

, n, n

, n}; n

=1,…, N

O

O’(x0 ,y0 ,z0)

direction of motion of mobile on x-y plane

v

x0

z0

y0

O’’

3-Dimensional Model [Clarke 68, Aulin 79]

Chapter 3: Autocorrelation and Power Spectral Density

c 0

Time variations of channel: time-variations in

propagation environment described by

Doppler power spectrum: S ;

For specific case [Clarke '68, Aulin '79]:

(conditions on angles of arrival

t c fF t f

c

of the wave)

S

Explicit equation

D

t E t

S f

F R t F E E t t E t

Doppler Power Spectral Density

Factorization (Normalization, Approximation)

State-Space Model

Chapter 3: Doppler P.S.D. is Band-limited

Doppler Power Spectral Density

Factorization (Normalization, Approximation)

State-Space Model

Power Spectral Density: Syy()=FRyy(t)]

Chapter 4: Power Spectral Densities and S.D.E.’s

Linearsystem

h(t)

Gaussian process

Gaussian process

Sxx() Syy()|H()|2x =

x(t) y(t)WSS

Rxx(t) Ryy(t)LTI

Chapter 4: Paley-Wiener Factorization Condition

2-

2

Paley-Wiener condition: Given a non-negative integrable

logfunction such that , then there

1

exists a causal, stable, minimum-phase such that:

, implying that is factorizab

SS d

H s

H S S

0

0

le:

eg. satisfies P-W condition

,

: Orthogonal increments process

yy

xx xx

t

t

S s H s H s

S qS

R t q t S q

y t h w t d

w t

Chapter 4: Factorization of Approximated P.S.D.

2 1 1

1 1

2 2

2 2

Factorization is trivial if is an even rational function:

Re 0, Re 0, , , minimum phase

e.g. : 4th order2

n n

l l ll lm m

k k kk k

l l

n n

S s

s z s z s zS s K H s A

s p s p s p

z z A K n m H s

AS s H s

s s

x t

2

0

2 , 0 , 0 :

: white-noise process

n n

t

x t x t Aw t x x given

w t

Factorization

Chapter 4: Approximate P.S.D.

0 1 Normalization:

2 2

Approximation:

Factorization:

Nominal state-space model follows

D m Dm

D D

D

ES pu f f S f

f

S f S f

S s H s H s

Chapter 4: Approximate P.S.D.

2

4 2 2 2 4

2 2

maxmax max

2maxmax max2

( ) ( )2 1 2

2

011 10 0

2

, 01 2

D

n n

n n

DD D

DD D

n n D

KS s H s H s

s s

KH s

s s

SS S

SS S

K S

Approximation

Chapter 4: State-space realizations of Fading Process

2

0

( ;

( )

( )

,

00 1, , 1 0

2

, : specular components

, : Independent Brownian motion processes

( ))y co

I I I

Q Q Q

I I Q Q

n n

I Q

I Q t

dX t A X t dt B dW t

dX t A X t dt B dW t

I t CX t f t Q t CX t f t

A B CK

f t f t

W t

I t

W t

t t

s ( ; ( ))sin ( ( )) ( )c c lt Q t t t s t t n t

Nominal state-space model

Chapter 4: State-Space Realizations of Fading Process

2

( )

( ) ( )

( );

i i

ii

i T T Ti i

T Ti i i

Ti i i i

dM tA M t

dtdP t

A P t P t A B Bdt

P t E X t X t M t M t

m t CM t p t CP t C

Nominal state-space model: mean and variance

Chapter 4: T.-D. Simulations of Fading Process

Nominal state-space model

s(t) delay

dWI

cos ct

ABCD X

dWQ

sin ct

ABCD X

+ X

y(t)+

-

Flat-Fading Channel

Chapter 4: T.-D. Simulations of Fading Process

Simulation of Flat-Fading Channel using Matlab

Experimental Data

(Pahlavan)

Chapter 4: T.-D. Simulations of Fading Process

Simulation of frequency-selective Channel using Matlab

y

y(t)

s(t

-tau_3)

tau_3

s(t

-tau_2)

tau_2

s(t

-tau_1)

tau_1

s(t)

signal in

r_3b(t)

path 3

r_2b(t)

path 2

r_1b(t)

path 1

Chapter 4: T.-D. Simulations of Fading Process

Simulation of received signal through a frequency-selective channel using Matlab

Temporal simulations of received signal for a multipath channel:

From PSD obtain parameters of state-space model.Dynamics of channel gain obtained through solving state-space model (generate independent brownian motions for in-phase and quadrature components).Identify the parameters of the non-homogeneous Poisson process (t). This characterizes the obstacles in the environment.Generate points of non-homogeneous poisson process. This corresponds to generating the path arrival times.Associate each path with a gain computed using the state-space model.

Chapter 4: T.-D. Simulations of Fading Process

Temporal simulations of received signal:

Chapter 4: Shot-Noise Model Simulations

Chapter 4: Probability Distributions of Attenuations

( )

1

2 2

2 2 22

( ; ( )) cos ; ( ( ))

( ; ) ( ; ) ( ; )

( ; ), ( ; ) : Gaussian ( ; ) : Rayleigh, Ricean

Probability distribution

( ; )( ( ; ), ) exp ( ; ) ( ; ) / 2

Xi

N t

i i c i l ii

ii s i i

i

y r t t t t s t t

r t I t Q t

I t Q t r t

r tf r t t r t r t p t

p t

20

2 22 2

( ; ) ( ; ) / ,

( ; ) ( ) ( ) , ( ) ( )

i s i

s i

I r t r t p t

r t E I t E Q t p t Var I t Var I t

Chapter 4: Probability Distributions of Attenuations

, ,

2 2

1Generate using S.D.E.'s,

n j n j

m

n n jX t r t t X t

Xj(s)= 0,j = 0 j =1,…,nX1(s)= X2 (s)= 0

1= 2 =0

n=2)(tn

Non-StationaryRayleigh

StationaryRayleigh

n=2

n=2

Non-Stationary

Nakagami

Stationary Nakagami

t large

1= 2 =0

Non-Stationary Rician

Stationary Rician

and/or t large

n=2

CRC-TU/e-TU/d-U/Ottawa

Signature Functions & Parameters

Autocorrelation Power Spectral Density

maxsin4

v

Pave

maxmax cos v

f

Lag (S)

Cor

rela

tion

Rel

ati v

e P

ower

Frequency (Hz)

1

0

-0.1 0 0.1-1

-80 - 60 -40 -20 0 20 40 60 80

50

25

0

Re: Paper of R. Bultitude et al. from URSI G.A.

E. Wong, B. Hajek. Stochastic Processes in Engineering Systems. Springer-Verlag, New York, 1985.M.C. Jeruchim, P. Balaban, S. Shanmugan. Simulation of Communication Systems. Plenum, New York, 1994.P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York, 1999.C.D. Charalambous, N. Menemenlis. Stochastic models for short-term multipath fading: Chi-Square and Ornstein-Uhlenbeck processes. Proceedings of 38th IEEE Conference on Decision and Control, 5:4959-4964, December 1999.C.D. Charalambous, N. Menemenlis. Multipath channel models for short-term fading. 1999 International Workshop on mobile communications, pp 163-172, Creta, Greece, June 1999.C.D. Charalambous, N. Menemenlis. A state-space approach in modeling multipath fading channels via stochastic differntial equations. ICC-2001 International Conference on Communications, 7:2251-2255, June 2001.

Chapter 4: References