Chapter 5: Applications using S.D.E.’s
Channel state-estimation
State-space channel estimation using Kalman filtering
Channel parameter identificationa
Nonlinear filtering
Power control for flat fading channels
Convex optimization and predictable strategies
Channel capacity
Optimal encoding and decoding
Chapter 5: Linear Channel State-Estimation
The various terms of the state-space description are:
 X I (t ) 
 AI (t )
X (t )  
, A(t )  

 X Q (t ) 
 0
0 
 BI (t ) 
, B(t )  
,


AQ (t ) 
 BQ (t ) 
BI (t )  BQ (t )   0 k (t ) 
T
G (t ,t )   cos  c t
D   cos  c t
0  sin  c t
0 s (t  t )
 sin  c t 
 f I (t ) 
F t   
: specular components

 f Q (t ) 
 I (t )  1 0 0 0 
Q(t )   0 0 1 0  X (t )

 

Note that the parameters depend on the propagation
environment represented by t
Chapter 5: Channel Simulations
First must find model parameters for a given structure
Method 1: Approximate the power spectral density (see
short-term fading model)
Method 2: From explicit equations and data we have
X (t )  e X (0)   e A(t t ) BW (t )dt ; E  X (t )   e At E  X (0) 
t
At
0
r (t )  I 2 (t )  Q 2 (t );
E  r (t )    2 (t )
 E  I 2 (t ) 


(t )  E  X (t ) X T (t )   
 E  I (t ) I (t ) 

 
E  I (t ) I (t )  

2
E  I (t )  
From data: lim E  r (t )    2 X   2 lim E  I 2 (t ) 
t 
Obtain {k,z,n} parameters
t 
Chapter 5: Channel Simulations
Flat-fading channel state-space realization in statespace
+
dWI
ABCD
X
cos ct
dWQ
+
ABCD
Flat-fading channel
X
sin ct
Chapter 5: Linear Channel State-Estimation
State-Space Channel Estimation using Kalman filtering
Considering flat-fading
X (t )  A(t ) X (t )  F (t )  B(t )W (t ), X (0)
N (  , )
y (t )  G (t ) X (t )  D(t )v(t )
W (t ) N (0, Q),  (t ) N (0, R),
W (t ), (t ) independent and also independent with X (0)
Kalman filter for the state estimate is given by

Xˆ (t )  A(t ) Xˆ (t )  F (t )  K (t ) y (t )  G (t ) Xˆ (t )
K (t )  standard Kalman filter gain

Chapter 5: Linear Channel State-Estimation
State-Space Channel Estimation using Kalman Filtering
Inphase and Quadrature estimates
Iˆ(t ) E  I (t )  y ( s);0  s  t  CI Xˆ (t )
Qˆ (t ) E Q(t )  y ( s );0  s  t  CQ Xˆ (t )
Square-Envelop Estimate
rˆ 2 (t )  Iˆ 2 (t )  Qˆ 2 (t )  e 2 (t )  e 2 (t )
I

Q



2
2
2



ˆ
ˆ
e I (t ) E I (t )  I (t ) , eQ (t ) E Q(t )  Q(t ) 




Possible generalization to multi-path fading channel
2
N
y (t )   Gi (t )X i (t )  D(t )v(t )
i 1
Chapter 5: Channel State-Estimation: Simulations
Flat Fast Rayleigh Fading Channel, SNR = 10 dB, v = 60 km/h
-14
2
2
1
1
-18
0
0
-20
-1
-1
SD(f)
H(j)
S (f), H(j)2 in dB
D
2
-16
-22
-5
-4
-3
-2
-1
0
1
2
3
4
-2
5
0
50
100
150
200
[a] inphase. Time [ms]
[a] v = 5 km/h, fc = 910 MHz, m = 400 ; Frequency Hz.
-42
-2
250
10
0
50
100
150
200
[b] quadrature. Time [ms]
250
0
50
250
2
SD(f)
H(j)
-44
0
1
tan-1 [Q(t)/I(t)]
S (f), H(j)2 in dB
D
2
[dB]
-10
-46
-20
-48
-50
-150
-1
-30
-100
-50
0
50
100
-40
150
0
50
100
150
200
[c] envelope. Time [ms]
[b] v = 120 km/h, fc = 910 MHz, m = 400 ; Frequency Hz.
-2
100
150
200
[d] phase. Time [ms]
inphase MSE
quadrature MSE
-10
[dB]
1
0
-1
inphase
estimate
-20
-30
-40
-50
0
50
100
150
[a] Time [ms]
200
250
0
0.5
1
1.5
2
2.5
[a] Time [ms]
3
3.5
4
20
2
0
[dB]
1
0
-1
-2
250
0
2
-2
0
quadrature
estimate
-20
-40
-80
0
50
100
150
[b] Time [ms]
200
r2
-60
250
r2 estimate
0
50
100
150
[b] Time [ms]
200
250
Chapter 5: Channel State-Estimation: Simulations
Frequency-Selective Slow Fading, SNR=20dB, v=60km/h
6
6
signal
noiseless signal
4
4
2
2
0
0
1
2
[a] Time [ s]
3
8
0
0
1
2
[b] Time [ s]
3
6
noisy signal
signal
estimate
6
4
4
2
2
0
0
1
2
[c] Time [ s]
3
0
0
1
2
[d] Time [ s]
3
Chapter 5: Channel state-estimation: Conclusions
For flat slow fading, I(t), Q(t), r2(t) show very good tracking
at received SNR = -3 dB.
For flat fast fading, I(t), Q(t), r2(t) show very good tracking
when the received SNR = 10 dB.
For frequency-selective slow fading, I(t), Q(t), r2(t) of each
path show very good tracking, w.r.t. MSE, when the
received SNR = 20 dB.
Chapter 5: Channel state-estimation: References
J.F. Ossanna. A model for mobile radio fading due to building reflections:
Theoretical and experimental waveform power spectra. Bell Systems
Technical Journal, 43:2935-2971, 1964.
R.H. Clarke. A statistical theory of mobile radio reception. Bell Systems
Technical Journal, 47:957-1000, 1968.
M.J Gans. A power-spectral theory of propagation in the mobile-radio
environment. IEEE Transactions on Vehicular Technology, VT-21(1):27-38,
1972.
T. Aulin. A modified model for the fading signal at a mobile radio channel.
IEEE Transactions on Vehicular Technology, VT-28(3):182-203, 1979.
C.D. Charalambous, A. Logothetis, R.J. Elliott. Maximum likelihood
parameter estimation from incomplete data via the sensitivity equations.
IEEE Transactions on AC, vol. 5, no. 5, pp. 928-934, May 2000.
C.D. Charalambous, N. Menemenlis. A state-state approach in modeling
multi-path fading channels: Stochastic differential equations and OrnsteinUhlenbeck Processes. IEEE International Conference on Communications,
Helsinki, Finland, June 11-15, 2001.
Chapter 5: Channel state-estimation: References
K. Miller. Multidimensional Gaussian Distributions. John Wiley & Sons,
1963.
M.S. Grewal, A.P. Andrews. Kalman filtering – Theory and Practice,
Prentice Hall, Englewood Cliffs, New Jersey 07632, 1993.
D. Parsons. The mobile radio Propagation channel. John Wiley & Sons,
1995.
R.G. Brown, P.Y.C. Hwang. Introduction to random signals and applied
Kalman filtering: with MATLAB exercises and solutions, 3rd ed. John Wiley,
1996.
G. L. Stuber. Principles of Mobile Communication. Kluwer Academic
Publishers, 1997.
P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential
Equations. Springer-Verlag, New York, 1999.
Chapter 5: Channel Parameter Identification
Consider the quasi-static multi-path fading channel
model
y (t )   ri cos  c t   i (t;t i )  sl (t  t i )   (t )
K
 (t )
i 1
: noise
 i (t ;t )  i  ( c   di )t   di t
Given the observation process for each path find
estimates for the channel parameters:
r , 
i
di

, i : channel gain, Doppler spread and phase
Chapter 5: Non-Linear Filtering-Sufficient Statistic
Methodology:
Use concept of sufficient statistics in designing nonlinear channel parameter estimator.
Sufficient statistic: any quantity that carries the same
information as the observed signal, i.e. conditional
distribution.
Chapter 5: Bayes’ Decision Criteria
Detection criteria
Chapter 5: Non-Linear Filtering
Sketch of continuous-time non-linear filtering for parameter
estimation.
Derive a sufficient statistic and obtain the incomplete
data likelihood ratio of multipath fading parameters (for
flat and frequency selective channels)
One parameter at a time while keeping others fixed,
All parameters simultanously
Consider the band-pass representation of the received signal
y (t )   ri (t ) cos  ( c   di (t ))(t  t i (t ))   i  S (t  t i )   (t )
K
i 1
h(t , x(t ))   (t )
Chapter 5: Non-Linear Filtering
Sketch of continuous-time non-linear filtering approach
dx(t )  f  t , x(t )  dt    t , x(t )  dw(t ), x (0)  x0 : state process
dy (t )  h  t , x(t )  dt  N  t  dv (t ) : observation process
w(t ) and v(t ) independent Brownian motions
ˆ  x(t )   E   x(t )      ( z ) p  t , z dt

0,t 
N


pN  t , z  : normalized conditional density of x(t ) given
 0,t
  y ( s );0  s  t : observable events
ˆ  x(t )  : Least-squares estimate of   x(t )  

0,t
Non-linear filtering theory relies on successful computation of pN(.,.)
Chapter 5: Non-Linear Filtering
Continuous-time non-linear filtering
Radon-Nikodym derivative (complete data likelihood ratio)
dP
dP
Ft
t

exp  hT ( s, x( s ))( N (t ) N T (t ))dy ( s )
 0
1 t T

  h ( s, x( s))( N (t ) N T (t )) h( s, x( s)) ds 
2 0

 (t )
where

dx(t )  f  t , x(t )  dt    t , x(t )  dw(t ), x (0)  x0
(, Ft , P ) 

dy (t )  N  t  dv(t )
 (t ) : Complete data likelihood function
Chapter 5: Non-Linear Filtering
Continuous-time non-linear filtering; Bayes’ rule

dP

E   x(t ) 
 0,t 
dP

E   x(t )   0,t   
 dP

E
 0,t 
 dP


n
 ( z ) p (t , z )dz

n
p (t , z )dz
E  : expectation under P
p (, ) : unnormalized conditional density : sufficient statistic
satisfies the forward Kolmogorov equation
dp(t , x)  L(t ) p (t , x)dt  hT (t , x)( N (t ) N T (t )) 1 dy (t ), (t , x)   0, T  
p (0, x)  p0 ( x), x 
n
where
 
1  2
L(t ) (t )  Tr  2 ( (t , x) T (t , x)(t ))   ( f (t , x)(t ))
2  x
 x
n
Chapter 5: Phase Estimation
Problem 1: Flat-fading; phase estimation
Given the observation process
dy (t )  r (t ) cos  ( c   d (t ))(t  t (t ))    S (t  t (t ))  N  t  dv(t )
h  t , ,  (t )  dt  N  t  dv(t )
A fixed sample path  (t )
r ( s,  ),  d ( s,  );0  s  t
1
, find
2
1. pN (t , ) and p(t , ) normalized and unnormalized conditional densities
and  :  0, 2  , with a priori density p0 ( ) 
given the sample path  (t )  r ( s,  ),  d ( s,  );0  s  t ;
ˆ (t )
2. 
dP
dP
, for a fixed sample path  (t );
0,t
3. hˆ(t , )  E  h(t , ,  (t ))  0,t  , for a fixed sample path  (t );
4. ˆ(t )  E   0,t  , for a fixed sample path  (t ).
Chapter 5: Phase Estimation
Defintion: Flat-fading; phase estimation problem
For t  [0, Ts ], let z (t )
Vc (t )
Vs (t )
d
y (t ) and define
dt
 r (s) cos  ( s   (s))(s  t (s))  S ( s  t ( s))z ( s)ds
 r (s) sin  ( s   ( s))( s  t ( s))  S ( s  t ( s))z( s)ds
t
c
0
d
t
c
0
d
 Vs (t ) 
V (t )  V (t )  V (t ),  (t )   tan 

V
(
t
)
 c 
1 t 2
Wc (t )   r ( s ) cos  2( c s   d ( s ))( s  t ( s ))  S 2 ( s  t ( s))N 2 ( s)ds
4 0
1 t 2
Ws (t )   r ( s ) sin  ( c s   d ( s))( s  t ( s))  S 2 ( s  t ( s))N 2 ( s) ds
4 0
2
2
1  Ws (t ) 
W (t )  Wc (t )  Ws (t ),  (t )   tan 

W
(
t
)
 c 
2
c
2
s
1
Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation problem
The unnormalized conditional density of  given  0,t and the
sample path  (t )  r ( s,  ),  d ( s,  );0  s  t is given by
 1 t 2

p (t , )  p0 ( ) exp   r ( s ) S 2 ( s  t ( s )) N 2 ( s )ds 
 4 0

 exp W (t ) cos(2   (t )) exp V (t ) cos(   (t ))
1
where p0 ( ) is the a priori density of  , p0 ( ) 
,   [0, 2 ]
2
and W (t ), V (t ),  (t ),  (t ) as above.
Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation problem
ˆ (t ), is given by
The incomplete data likelihood ratio, 
2
ˆ
 (t )   p (t , )d
0
 1 t 2

 exp   r ( s) S 2 ( s  t ( s)) N 2 ( s)ds   exp W (t ) cos  (t )
 4 0


(1) j 2 j  k 1 h i
 
V (t )W j  k (t ) cos h  (t ) sin i  (t ) cos j  (t ) sin k  (t )
h ,i , j , k  0 h !i ! j ! k !
(h  k )!(i  2 j  k )!

 h  i  2 j  2k   i  2 j  k   h  k  h  i  2 j  2 k

 !
 !
 !2
2
2


 2 
and W (t ), V (t ),  (t ),  (t ) as above.
Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation problem
The normalized conditional density, pN (t , ), is given by
1
 1 t 2

pN (t , ) 
exp   r ( s )S 2 ( s  t ( s )) N 2 ( s )ds 
ˆ (t )
2
 4 0

 exp W (t ) cos(2   (t )) exp V (t ) cos(   (t ))
 1 t 2

 exp   r ( s ) S 2 ( s  t ( s )) N 2 ( s) ds   exp W (t ) cos  (t )
 4 0

and W (t ), V (t ),  (t ),  (t ) as above.
The conditional least-squares estimate hˆ(t , ) is given by
Chapter 5: Phase Estimation
hˆ(t , ) 

2
0
h(t , ) p (t , ) d

2
0

p (t , ) d
1
r (t ) cos  c t  S (t  t (t )) exp W (t ) cos  (t )
ˆ
2 (t )
(1) j 2 j  k 1 h i
V (t )W

h ,i , j , k  0 h !i ! j ! k !

j k
(t ) cos h  (t ) sin i  (t ) cos j  (t ) sin k  (t )
  ( h  k  2) 2    (i  2 j  k  1) 2 

  ( h  i  2 j  2k  3) 2 
 r (t ) sin  c t  S (t  t (t )) exp W (t ) cos  (t )
(1) j 2 j  k 1 h i
V (t )W

h ,i , j , k  0 h !i ! j ! k !


j k
(t ) cos h  (t ) sin i  (t ) cos j  (t ) sin k  (t )
  ( h  k  1) 2    (i  2 j  k  2) 2 
  ( h  i  2 j  2k  3) 2 
and W (t ), V (t ),  (t ),  (t ) as above.
Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation problem
The least-squares estimate ˆ(t ) is given by
2
2
0
0
  p(t , )d    p(t , )d

ˆ (t )

p
(
t
,

)
d



ˆ
where (t )   p(t , ) d is computed as in theorem 2
ˆ(t ) 
2
0
2
0
Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation
Neglecting double frequency terms
The unnormalized conditional density of  given  0,t and the
sample path  (t )  r ( s,  ),  d ( s,  );0  s  t is given by
 1 t 2

p (t , )  p0 ( ) exp   r ( s ) S 2 ( s  t ( s )) N 2 ( s )ds 
 4 0

 exp V (t ) cos(   (t ))
1
where p0 ( ) is the a priori density of  , p0 ( ) 
,   [0, 2 ]
2
and W (t ), V (t ),  (t ),  (t ) as above.
Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation
Neglecting double frequency terms
ˆ (t ), is given by
The incomplete data likelihood ratio, 
2
ˆ
 (t )   p(t , )d
0
 1 t 2

 exp   r ( s ) S 2 ( s  t ( s )) N 2 ( s )ds   I 0 V (t ) 
 4 0

where V (t ), as above
and I 0   is the modified zero order Bessel function,
1
I0  x  
2

  exp  x cos  d .

Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation
Neglecting double frequency terms
The normalized conditional density of  given  0,t and the
sample path  (t )  r ( s,  ),  d ( s,  ); 0  s  t is given by
pN (t , ) 
exp V (t ) cos(   (t ))
I 0 V (t ) 
The conditional least-squares estimate hˆ(t , ,  (t )) is given by
I1 V (t ) 
ˆ
cos  ( c   d (t ))(t  t (t ))   (t )  S (t  t (t ))
h(t , ,  (t ))  r (t )
I 0 V (t ) 
where I 0   is the modified zero order Bessel function, and
I1   is the modified first order Bessel function,
1
I1  x  
2

  cos  exp  x cos  d .

Chapter 5: Phase Estimation
Solution of Problem 1: Flat-fading; phase estimation
Neglecting double frequency terms
The conditional least-squares estimate ˆ(t ) is given by
ˆ(t )  
2
0
3


1
2

( 1 2  k )
 (t ) 
2k
 pN (t , )d 
V (t ) 



2
2
k

1


I 0 (V (t )) k 0
k !(2k )!
(k !) 2


Chapter 5: Channel Estimation
Same procedure for
Gain
Doppler Spread
Joint Estimation of Phase, Gain, Doppler Spread
Frequency Selective Channels
Chapter 5: Simulations of Phase Estimation
Phase estimation in continuous-time
Chapter 5: Nonlinear Filtering Conclusions
Conditional density is a sufficient statistic.
Explicit but complicated expressions can be found for the
various parameters of the channel.
These estimations are very useful in subsequent design
of various functions of a communications system.
Chapter 5: Channel parameter estimation: References
T. Kailath, V. Poor. Detection of stochastic processes. IEEE Transactions on
Information theory, vol. IT-15, no. 3, pp. 350-361, May 1969.
T. Kailath. A General Likelihood-ration formula for random signals in
Gaussian noise. IEEE Transactions on Information theory, vol. 44, no. 6, pp.
2230-2259, October 1998.
C.D. Charalambous, A. Logothetis, R.J. Elliott. Maximum likelihood
parameter estimation from incomplete data via the sensitivity equations.
IEEE Transactions on AC, vol. 5, no. 5, pp. 928-934, May 2000.
S. Dey, C.D. Charalambous. On assymptotic stability of continuous-time risk
sensitive filters with respect to initial conditions. Systems and Control
Letters, vol. 41, no. 1, pp. 9-18, 2000.
C.D. Charalambous, A. Nejad. Coherent and noncoherent channel estimation
for flat fading wireless channels via ML and EM algorithm. 21st Biennial
symposium on communications, Queen’s University, Kingston, Canada, June,
2002.
C.D. Charalambous, A. Nejad. Estimation and decision rules for multipath
fading wireless channels from noisy measurements: A sufficient statistic
approach. Centre for information, communication and Control of Complex
Systems, S.I.T.E., University of Ottawa, Technical report: 01-01-2002, 2002.
Chapter 5: Channel parameter estimation: References
P.M. Woodward. Probability and Information Theory with Applications to
Radar. Oxford, U.K.: Pergamon, 1953.
A.D. Whalen. Detection of signals in noise, Academic Press, New York,
1971.
A. Leon-Garcia. Probability and Random Processes for Electrical
Engineering. Addison-Wesley, New York, 1994.
L.A. Wainstein, V.D. Zubakov. Extraction of signals from noise, Englewood
Cliffs, Prentice-Hall, New Jersey, 1962.
C.W. Helstrom. Statistical theory of signal detection. Pergamon Press, New
York, 1960.
M.S. Grewal, A.P. Andrews. Kalman filtering – Theory and Practice,
Prentice Hall, Englewood Cliffs, New Jersey 07632, 1993.
A.H. Jazwinski. Stochastic processes and filtering theory, Academic Press,
New York, 1970.
V. Poor. An Introduction to signal detection and estimation, Springer-Verlag,
New York, 2000.
Chapter 5: Stochastic power control for wireless networks:
Probabilistic QoS measures
Review of the Power Control Problem
Probabilistic QoS Measures
Stochastic Optimal Control
Predictable Strategies
Linear Programming
Chapter 5: Power Control for Wireless Networks
QoS Measures
Review of the Power Control Problem
M
min
( p1  0, , pM  0)
p;
i 1
i

pn g nn
M
j 1
p j g nj  n
n
Chapter 5: Power Control for Wireless Networks
QoS Measures
Vector Form [Yates 1981]
Then
M

min
pi ; GI P    GP    


 p j , j 1, , M 0  i 1

Chapter 5: Power Control for Wireless Networks
QoS Measures
Probabilistic QoS Measures
Define
n
I ( p)

M
j 1
p j g nj  n 
1
n
pn g nn ,
The Constraints are equivalent to
I n ( p)  0,
n  1,
,M
n  1,
,M
Chapter 5: Power Control for Wireless Networks
QoS Measures
Decentralized Probabilistic QoS Measures
Chapter 5: Power Control for Wireless Networks
QoS Measures
Chapter 5: Power Control for Wireless Networks
Centralized Probabilistic QoS Measures
Chapter 5: Power Control for Wireless Networks
QoS Measures
Stochastic optimal control
Received signal
yn (t )   j 1 u j (t ) s j (t ) exp kX nj (t )  d n (t )
M
State-space representation
yn (t )   j 1 u j (t ) s j (t )S nj (t )  d n (t );
M
S nj (t )  exp kX nj (t )
dX nj (t ,t )   nj (t ,t )  nj (t ,t )  X nj (t ,t )  dt   nj (t , t ) dBnj (t )
X nj (t0 ,t )  PL d (d )  dB nj
Chapter 5: Power Control for Wireless Networks
QoS Measures
Pathwise QoS and Predictable Strategies
define
then
where
pi (t )dt;
s j (t ) S
S nk
s j (t ) S nj (t )
Power control for short-term flat fading
Pathwise QoS Measures and Predictable Strategies
Mobile
Base Station
calculates
t1
Mobile
implements
observe => calculate
t-1
pm(t) Sm (t-1
pm(t)
S(t1/2pm(t-1)
t1/2
S(t1/2pm(t)
pm(t+1)
t
t1/2
pm(t-1) Sm (t-1 => pm(t)
t1
pm(t) Sm (t => pm(t+1)
pm(t+1) Sm (t
Power control for short-term flat fading
Pathwise QoS Measures and Predictable Strategies
Base
Mobile
Station
Base
Mobile
implements
calculates
Station
pm(t) Sm (t1
pm(t+1) Sm (t
Value of
signal
pm(t+2) Sm (t1
desired
pm(t-1) Sm (.
pm(t)
t1
pm(t+1) Sm (.
pm(t) Sm (.
pm(t+1)
t
Observe pm(t)Sm (t => calculate pm(t+1)
t1
Chapter 5: Power Control for Wireless Networks
QoS Measures
Define
sni (t )Sni (t )
Chapter 5: Power Control for Wireless Networks
QoS Measures
Predictable Strategies over the interval
[tk , tk 1 ]
M
min  pi (tk 1 );
p ( tk 1 )U ad
 i 1
p(tk 1 )  GI1 (tk , tk 1 )  G (tk , tk 1 ) p(tk 1 )   (tk 1 ) S nk (t )t[t
Predictable Strategies
Linear Programming
k
,tk 1 ]

Chapter 5: Power Control for Wireless Networks
QoS Measures
Chapter 5: Power Control for Wireless Networks
QoS Measures
(t k 1 )
sn (t )S n (t )
s j (t ) S j (t )
sn (t )S n (t )
sn S n
Chapter 5: Power Control for Wireless Networks
QoS Measures
Generalizations
Linear Programming
S nk (t )
Stochastic Optimal Control with
Integral/Exponential-of-Integral Constraints
Snk (t )
Chapter 5: Power Control for Wireless Networks:
Conclusions
Predictable strategies and dynamic models
linear programming
Probabilistic QoS measures
Stochastic optimal control
linear programming
References
J. Zandler. Performance of optimum transmitter power control in cellular
radio systems. IEEE Transactions on Vehicular Technology, vol. 41, no. 1,
pp. 57-62, Feb. 1992.
J. Zandler. Distributed co-channel interference control in cellular radio
systems. IEEE Transactions on Vehicular Technology, vol. 41, no. 1, pp. 305311, Aug. 1992.
R. Yates. A framework for uplink power control in cellular radio systems.
IEEE Journal on Selected Areas in Communications, vol. 13, no. 7, pp.
1341-1347, Sept. 1995.
S. Ulukus, R. Yates. Stochastic Power Control for cellular radio systems.
IEEE Transaction on Communications, vol. 46, no. 6, pp. 784-798, Jume
1998.
P. Ligdas, N. Farvadin. Optimizing the transmit power for slow fading
channels. IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 565576, March 2000.
References
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.
C.D. Charalambous, N. Menemenlis. Dynamical spatial log-normal
shadowing models for mobile communications. Proceedings of XXVIIth
URSI General Assembly, Maastricht, August 2002.
C.D. Charalambous, S.Z. Denic, S.M. Djouadi, N. Menemenlis. Stochastic
power control for short-term flat fading wireless networks: Almost Sure QoS
Measures. Proceedings of 40th IEEE Conference on Decision and Control,
volm. 2, pp. 1049-1052, December 2001.
Chapter 5: Capacity, Optimal Encoding, Decoding
The channel capacity is the most important concept of
any communication channel because it gives the
maximal theoretical data rate at which reliable data
communication is possible
We show an efficient method for computing the channel
capacity of a single user time-varying wireless fading
channels by means of stochastic calculus.
We consider an encoding, and decoding strategy with
feedback that is optimal in the sense that it achieves the
channel capacity.
Although the feedback does not increase the channel
capacity, it is a tool for achieving the channel capacity
Chapter 5: Channel Model and Mutual Information in
Presence of Feedback
 , F , P  is a complete probability space with
filtration  Ft t 0 and finite time t  0, T  , T  
on which all random processes are defined
X   X t t 0 source signal
  t t 0 , t   r (t ,t k ),  d (t ,t k ),  (t ,t k )  ,
is state channel process
N   N t t 0 Wiener process independent of X,
representing thermal noise
Chapter 5: Channel Model and Mutual Information in
Presence of Feedback
The received signal can be modeled as
M
dYt   Z t( k )    At t k  X , Y ,   dt  dN t , Y0  0,
(1)
k 1
Z t
k
    r (t ,t k ) cos ct   d (t )  t  t k    (t ,t k ) 
t k : is a delay,  d : is a Doppler shift, r is an amplitude,
 : is a phase, M is a number of resolvable paths, and
 c : is a carrier frequency
At  X , Y ,   : is the non-anticipatory functional representing
encoding
Chapter 5: Channel Model and Mutual Information in
Presence of Feedback
Also, we define the following measurable spaces
associated with stochastic processes dX , Y , 
 X , BX    C  0, T  ; R  , B  C  0, T  ; R   
Y , BY    C  0, T  ; R  , B  C  0, T  ; R   


3
3

,
B

C
0,
T
;
R
,
B
C
0,
T
;
R

  
  


their filtrations





F0,XT  B C  0, T  ; R  , F0,YT  B C  0, T  ; R  ,

F0,T  B C  0, T  ; R 3
and truncations of filtrations
F0,Xt , F0,Yt , F0,t .
Chapter 5: Channel Model and Mutual Information in
Presence of Feedback
The following assumptions are made
2
T M




(k )
Pr    Z t    At t k  X , Y ,   dt     1


 0 k 1

(1) has the unique strong solution
Definition 1: The set of admissible encoders is defined as
follows
Aad   A : 0, T   C  0, T  ; R   C  0, T  ; R     R;
A   At  X , Y ,   ; t   0, T  is progressively measurable,

T

2

E   At  X , Y ,   dt    

0


Chapter 5: Channel Model and Mutual Information in
Presence of Feedback
Theorem 1. Consider the model (1). The mutual information
between the source signal X and received signal Y over the
interval [0,T], conditional on the channel state , IT(X,Y|F Q),
is given by the following equivalent expressions


dPX ,Y |  x, y |  
(i ) E X ,Y , log
( x, y ) 
dPX |  x |   dPY |  y |  


2
M

1


(k )
(ii) E  E   Z t    At t k  X , Y ,    
2 0  k 1

T



(k )
ˆ
  Z t   At t k Y ,   |     dt ,
 k 1


M
2
Aˆt t k Y ,   E  At t k  X , Y ,   | F0,Yt ,    
Chapter 5: Channel Model and Mutual Information in
Presence of Feedback
(iii )
 I  X , Y |     dP  ,
T


IT  X , Y |      E X ,Y


dPX ,Y |  x, y |  
( x, y ) 
log
dPX |  x |   dPY |  y |  


Definition 2. Consider the model (1). The Shannon
capacity of (1) is defined by
1
C  sup
IT  X , Y | F  
( X , A )X  Aad T
subject to the power constraint on the transmitted signal
E  At2  X , Y ,   |      P
(2)
Chapter 5: Upper Bound on Mutual Information
Theorem 2. Consider the model (1). Suppose the channel is
flat fading. The conditional mutual information between the
source signal X, and the received signal Y is bounded above
by
T
1
IT  X , Y | F   P  E  Z t2     dt
2 0

(3)
It can be proved that this upper bound is indeed the channel
capacity, by observing that there exists a source signal with
Gaussian distribution
dX t   X t dt  2 PdWt , X 0  0
such that the mutual information between that signal X and
received signal Y is equal to the upper bound (3).
Chapter 5: Upper Bound on Mutual Information
The capacity is
T


P
2
C
E Z t  dt

2T 0
Chapter 5: Optimal Encoding/Decoding Strategies for
Non-Stationary Gaussian Sources
We assume that the channel is flat fading (M=1), that it is
known to both transmitter and receiver, that a source is
Gaussian nonstationary, and can be described by the
following differential
dX t  Ft X t dt  Gt dWt
(4)
Ft and Gt are Borel measurable and bounded functions,
Integrable and square integrable, respectively, Gt Gttr>0,
t[0,T], W is a Wiener process independent of Gaussian
random variable X0~ N X ,V 
Chapter 5: Optimal Encoding/Decoding Strategies for
Non-Stationary Gaussian Sources
Decoding. The optimal decoder in the case of mean square error
criteria is the conditional expectation
Xˆ t Y , Z      E[ X t | F0,Yt ,    ]
while the error covariance is
2
ˆ
Vt Y , Z      E[( X t  X t Y , Z     | F0,Yt ,    ]
Encoding. The optimal encoder is derived by using equation for
optimal decoder, and equation for power constraint (6).
Chapter 5: Optimal Encoding/Decoding Strategies for
Non-Stationary Gaussian Sources
Definition 3. The set of linear admissible encoders Lad , where Lad
 Aad , is the set of linear non-anticipative functionals A with
respect to source signal X, which have the form
At  X , Y ,    At0 (Y , )  At1 (Y , ) X t
The received signal is then
dYt  Z t    ( At0 (Y , )  At1 (Y , ) X t )dt  dN t
(5)
The processes W, and N are independent, and the power constraint
(2) becomes
E[( At0 (Y , )  At1 (Y , ) X t ) 2 |    ]  P
(6)
Chapter 5: Optimal Encoding/Decoding Strategies for
Non-Stationary Gaussian Sources
Theorem 3 (Coding theorem for stochastic source). If the received
signal is defined by the equation (5), the source by (4), then the
encoding reaching the upper bound, C  P  E Z dt
2T
optimal decoder, and corresponding error covariance are
respectively given by
T

2
t
0
At*  X , Y * ,   

P
ˆ * Y * , Z    
X

X
t
t
Vt* Y * , Z    

dXˆ t* Y * , Z      Ft Xˆ t* Y * , Z     dt  Z t    PVt* Y * , Z    dYt*
Xˆ 0* Y * , Z      X
t
t
t
t
t




*
*
2
2
2
Vt Y , Z      V exp  2 Fs ds   Z s    Pds    Gs exp  2 Fu du   Z u    Pdu  ds
 0
 s
0
s
 0

V0* Y * , Z      V .
Chapter 5: Optimal Encoding/Decoding Strategies for
Random Variable Sources
Theorem 4 (Coding theorem for random variable source). If a
source signal X, which is Gaussian random variable N X ,V  is
transmitted over a flat fading wireless channel, then the optimal
encoding and decoding with feedback reaching the channel
capacity are
t


P
P
*
*
2
At  X , Y ,   
exp   Z s    ds  X  Xˆ t* Y * , Z    
V
2 0


dXˆ t* Y * , Z      Z t   
Xˆ 0* Y * , Z      X
 Pt 2
 *
PV exp    Z s    ds  dYt
 20

t


*
*
2
Vt Y , Z      V exp   P  Z s    ds 
 0

V0* Y * , Z      V

On Channel capacity: Conclusions
We can use the new stochastic dynamical models
developed to compute new results and get better insight
on various computations of channel capacity which is a
very important measure for transmission of information
More information in the session, friday, nov. 13th.
Chapter 5: Capacity, Optimal Encoding-Decoding: References
C. Shannon. Channel with side information at the transmitter. IBM Journal,
pp. 289-293, Oct. 1958.
A. Goldsmith, P. Varaia. Capacity of Fading Channels with channel side
information. IEEE Transactions on Information Theory, vol. 43, no. 6, pp.
1986-1992, Nov. 1997.
G. Caire, S. Shamai. On the capacity of some channels with channel state
information. IEEE Transactions on on Information Theory, vol. 45, no. 6, pp.
2007-2019, Sept. 1999.
E. Bigliery, J. Proakis, S. Shamai. Fading Channels: Information theoretic
and communication aspects. IEEE Transactions on Information Theory, vol.
44, no. 6, pp. 2619-2692, Oct. 1998.
T. Cover. Elements of information theory.
Future Work
Robust Modeling
Receiver Design
Optimal Coding Decoding
Joint Source and Channel Coding for Wireless Channels
Computation of the Channel Capacity for MIMO Channels
and Joint Source and Channel Coding
Power Control for Wireless Networks