chapter 4: motivation for dynamic channel models short-term fading varying environment obstacles...
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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
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