Chapter 5: Applications using S.D.E.’s Channel state-estimation State-space channel estimation using Kalman filtering Channel parameter identificationa

Chapter 5: Applications using S.D.E.’s

Channel state-estimationState-space channel estimation using Kalman filtering

Channel parameter identificationaNonlinear filtering

Power control for flat fading channelsConvex optimization and predictable strategies

Channel capacityOptimal encoding and decoding

Chapter 5: Linear Channel State-Estimation

The various terms of the state-space description are:

Note that the parameters depend on the propagation environment represented by

( ) ( ) 0 ( )( ) , ( ) , ( ) ,

( ) 0 ( ) ( )

(

: spec

) ( ) 0 ( )

( , ) cos 0 sin 0 ( )

cos s

ular comp

in

( )

( )

( ) 1 0 0 0

( ) 0 0 1 0

onents

I I I

Q Q Q

T

I Q

c c

c c

I

Q

X t A BX t A B

X t A B

B B k

G t t t s t

D t t

f t

f t

I t

Q

F t

t

( )X t

Chapter 5: Channel Simulations

First must find model parameters for a given structureMethod 1: Approximate the power spectral density (see short-term fading model)

Method 2: From explicit equations and data we have

Obtain {k,,n} parameters

( )

0

2 2

2

2

( ) (0) ( ) ; ( ) (0)

( ) ( ) ( ); ( ) 2 ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

From data: lim ( )

tAt A t At

T

t

X t e X e BW d E X t e E X

r t I t Q t E r t t

E I t E I t I tt E X t X t

E I t I t E I t

E r t

22 2 lim ( )Xt

E I t

Chapter 5: Channel Simulations

Flat-fading channel state-space realization in state-space

dWI

cos ct

ABCD X

dWQ

sin ct

ABCD X

++

-

Flat-fading channel


State-Space Channel Estimation using Kalman filteringConsidering flat-fading

( ) ( ) ( ) ( ) ( ) ( ), (0) ( , )

( ) ( ) ( ) ( ) ( )

( ) (0, ), ( ) (0, ),

( ), ( ) independent and also independent with (0)

Kalman filter for the state estimate is given by

ˆ ( )

X t A X t F t B W t X N

y t G t X t D t v t

W t N Q t N R

W t t X

X t

ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) standard Kalman filter gain

A X t F t K t y t G t X t

K t


State-Space Channel Estimation using Kalman Filtering

2 2 2 2 2

22 2

Inphase and Quadrature estimates

ˆ ( )

ˆ ( )

Squ

ˆ( ) ( ) ( );0

ˆ ( ) ( ) ( );0

ˆˆˆ (

are-Envelop Estimate

) ( ) ( ) ( ) ( )

ˆˆ( ) ( ) ( ) , ( ) ( ) (I

I

Q

I Q

Q

I t E X tI t y s s t C

Q t E Q t y s s t C

r t I t Q t e t e

X t

t

e t E I t I t e t E Q t Q t

1

2

)

Possible generalization to multi-path fading channel

( ) ( ) ( ) ( ) ( )N

i ii

y t G t X t D t v t

Chapter 5: Channel State-Estimation: Simulations

Flat Fast Rayleigh Fading Channel, SNR = 10 dB, v = 60 km/h

-5 -4 -3 -2 -1 0 1 2 3 4 5-22

-20

-18

-16

-14

SD(f

),

H(j

)2 in

dB

[a] v = 5 km/h, fc = 910 MHz,

m = 400; Frequency Hz.

SD(f)

H(j) 2

-150 -100 -50 0 50 100 150-50

-48

-46

-44

-42

[b] v = 120 km/h, fc = 910 MHz,

m = 400; Frequency Hz.

SD(f

),

H(j

)2 in

dB

SD(f)

H(j) 2

0 50 100 150 200 250-2

-1

0

1

2

[a] inphase. Time [ms]0 50 100 150 200 250

-2

-1

0

1

2

[b] quadrature. Time [ms]

0 50 100 150 200 250-40

-30

-20

-10

0

10

[c] envelope. Time [ms]

[dB

]

0 50 100 150 200 250-2

-1

0

1

2

[d] phase. Time [ms]

tan-1

[Q(t

)/I(

t)]

0 50 100 150 200 250-2

-1

0

1

2

[a] Time [ms]

inphase estimate

0 50 100 150 200 250-2

-1

0

1

2

[b] Time [ms]

quadratureestimate

0 0.5 1 1.5 2 2.5 3 3.5 4-50

-40

-30

-20

-10

0

[a] Time [ms][d

B]

inphase MSE quadrature MSE

0 50 100 150 200 250-80

-60

-40

-20

0

20

[b] Time [ms]

[dB

]

r2

r2 estimate

Chapter 5: Channel State-Estimation: Simulations

Frequency-Selective Slow Fading, SNR=20dB, v=60km/h

0 1 2 3

0

2

4

6

[a] Time [s]

signal

0 1 2 30

2

4

6

[b] Time [s]

noiseless signal

0 1 2 30

2

4

6

8

[c] Time [s]

noisy signal

0 1 2 30

2

4

6

[d] Time [s]

signal estimate

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.

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 Ornstein-Uhlenbeck 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 state-estimation: References

Chapter 5: Channel Parameter Identification

Consider the quasi-static multi-path fading channel model

Given the observation process for each path find estimates for the channel parameters:

1

( ) cos ( ; ) ( ) ( )

( ) : noise

( ; ) ( )

, , : channel gain, Doppler spread and phase

i i

i

K

i c i i l ii

i i c d d

i d i

y t r t t s t t

t

t t

r

Chapter 5: Non-Linear Filtering-Sufficient Statistic

Methodology:

Use concept of sufficient statistics in designing non-linear 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

1

Consider the band-pass representation of the received signal

( ) ( ) cos ( ( ))( ( )) ( ) ( )

( , ( )) ( )

i

K

i c d i i ii

y t r t t t t S t t

h t x t t


Sketch of continuous-time non-linear filtering approach

Non-linear filtering theory relies on successful computation of pN(.,.)

0

0,

( ) , ( ) , ( ) ( ), (0) : state process

( ) , ( ) ( ) : observation process

( ) and ( ) independent Brownian motions

ˆ ( ) ( ) ( ) ,

, : normalized conditiona

t N

N

dx t f t x t dt t x t dw t x x

dy t h t x t dt N t dv t

w t v t

x t E x t z p t z dt

p t z

0,

0,

l density of ( ) given

( );0 : observable events

ˆ ( ) : Least-squares estimate of ( )

t

t

x t

y s s t

x t x t


Continuous-time non-linear filteringRadon-Nikodym derivative (complete data likelihood ratio)

0

0

0exp ( , ( ))( ( ) ( )) ( )

1 ( , ( ))( ( ) ( )) ( , ( )) ( )

2

wher

( )

e

( ) : Complete data likel

, ( ) , (

ihood fu

) ( ), (0)( , , )

( ) ( )

nct

tF

t

t T T

t T T

dPh s x s N t N t dy s

dP

h s x s N t N t

dx t f t x t dt t x t dw t x xF P

dy t N t dv t

h s x s ds t

t

ion


Continuous-time non-linear filtering; Bayes’ rule

0,

0,

0,

( ) ( ) ( , )( )

( , )

: expectation under

( , ) : unnormalized conditional density : sufficient statistic

satisfies the forward Kol

n

n

t

t

t

E x t z p t z dzE x t

p t z dzE

E P

p

dPdP

dPdP

1

0

2

2

mogorov equation

( , ) ( ) ( , ) ( , )( ( ) ( )) ( ), ( , ) 0,

(0, ) ( ),

1( ) ( ) ( ( , ) ( , ) ( )) ( ( , ) ( ))

2

T T n

n

T

dp t x L t p t x dt h t x N t N t dy t t x T

p x p x x

where

L t t Tr t x t x t f t x tx x

Chapter 5: Phase Estimation

Problem 1: Flat-fading; phase estimationGiven the observation process

0

( ) ( ) cos ( ( ))( ( )) ( ( )) ( )

, , ( ) ( )

A fixed sample path ( ) ( , ), ( , );0

1and : 0,2 , with a priori density ( ) , find

21. ( , ) and ( , ) normalized a

c d

d

N

dy t r t t t t S t t N t dv t

h t t dt N t dv t

t r s s s t

p

p t p t

0,

0,

nd unnormalized conditional densities

given the sample path ( ) ( , ), ( , );0

ˆ ( ) , for a fixed sample path ( );

, for a fixed s

;

2.

ˆ3. ( , ) ( , , ( ))

4.

ample path ( );

t

d

t

dPt t

dP

t r s s s t

h t E h tt t

0, , for a fixed sample path ( )ˆ ( ) .tt tE


Defintion: Flat-fading; phase estimation problem

c 0

s 0

2 2 1 s

c

2c

For [0, ], let ( ) ( ) and define

V ( ) ( ) cos ( ( ))( ( )) ( ( )) ( )

V ( ) ( )sin ( ( ))( ( )) ( ( )) ( )

V ( )( ) V ( ) V ( ), ( ) tan

V ( )

1W ( ) (

4

s

t

c d

t

c d

c s

dt T z t y t

dt

t r s s s s s S s s z s ds

t r s s s s s S s s z s ds

tV t t t t

t

t r

2 2

0

2 2 2s 0

2 2 1 s

c

) cos 2( ( ))( ( )) ( ( )) ( )

1W ( ) ( )sin ( ( ))( ( )) ( ( )) ( )

4

W ( )( ) W ( ) W ( ), ( ) tan

W ( )

t

c d

t

c d

c s

s s s s s S s s N s ds

t r s s s s s S s s N s ds

tW t t t t

t


Solution of Problem 1: Flat-fading; phase estimation problem

0,

2 2 20 0

The unnormalized conditional density of given and the

sample path ( ) ( , ), ( , );0 is given by

1( , ) ( ) exp ( ) ( ( )) ( )

4

exp ( )cos(2 ( )) exp ( )co

t

d

t

t r s s s t

p t p r s S s s N s ds

W t t V t

0 0

s( ( ))

1where ( ) is the a priori density of , ( ) , [0,2 ]

2and ( ), ( ), ( ), ( ) as above.

t

p p

W t V t t t



2

0

2 2 2

0

1

, , , 0

ˆThe incomplete data likelihood ratio, ( ), is given by

ˆ ( ) ( , )

1 exp ( ) ( ( )) ( ) exp ( )cos ( )

4

( 1) 2 ( ) ( ) cos ( )sin

! ! ! !

t

j j kh i j k h i

h i j k

t

t p t d

r s S s s N s ds W t t

V t W t th i j k

2 2

( ) cos ( )sin ( )

( )!( 2 )!

2 2 2! ! !2

2 2 2

and ( ), ( ), ( ), ( ) as above.

j k

h i j k

t t t

h k i j kh i j k i j k h k

W t V t t t



2 2 2

0

2 2 2

0

The normalized conditional density, ( , ), is given by

1 1( , ) exp ( ) ( ( )) ( )

ˆ 42 ( )

exp ( )cos(2 ( )) exp ( )cos( ( ))

1 exp ( ) ( ( )) ( )

4

N

t

N

t

p t

p t r s S s s N s dst

W t t V t t

r s S s s N s ds

exp ( )cos ( )

and ( ), ( ), ( ), ( ) as above.

ˆThe conditional least-squares estimate ( , ) is given by

W t t

W t V t t t

h t


2

02

0

1

, , , 0

( , ) ( , )ˆ( , )

( , )

1 ( ) cos ( ( )) exp ( )cos ( )

ˆ2 ( )

( 1) 2( ) ( ) cos ( )sin ( ) cos ( )sin ( )

! ! ! !

( 2) 2 ( 2 1) 2

(

c

j j kh i j k h i j k

h i j k

h t p t dh t

p t d

r t t S t t W t tt

V t W t t t t th i j k

h k i j k

h

1

, , , 0

2 2 3) 2

( )sin ( ( )) exp ( )cos ( )

( 1) 2( ) ( ) cos ( )sin ( ) cos ( )sin ( )

! ! ! !

( 1) 2 ( 2 2) 2

( 2 2 3) 2

and ( ), ( ), ( ), ( )

c

j j kh i j k h i j k

h i j k

i j k

r t t S t t W t t

V t W t t t t th i j k

h k i j k

h i j k

W t V t t t

as above.



2 2

0 02

0

2

0

ˆThe least-squares estimate ( ) is given by

( , ) ( , )ˆ( )

ˆ ( )( , )

ˆwhere ( ) ( , ) is computed as in theorem 2

t

p t d p t dt

tp t d

t p t d


Solution of Problem 1: Flat-fading; phase estimationNeglecting double frequency terms

0,

2 2 20 0

0

The unnormalized conditional density of given and the


1( , ) ( ) exp ( ) ( ( )) ( )

4

exp ( )cos( ( ))

where (

t

d

t

t r s s s t

p t p r s S s s N s ds

V t t

p

0

1) is the a priori density of , ( ) , [0, 2 ]

2and ( ), ( ), ( ), ( ) as above.

p

W t V t t t



2

0

2 2 200

0

ˆThe incomplete data likelihood ratio, ( ), is given by

ˆ ( ) ( , )

1 exp ( ) ( ( )) ( ) ( )

4

where ( ), as above

and is the modified zero order Bessel function,

t

t

t p t d

r s S s s N s ds I V t

V t

I

I

0

1exp cos .

2x x d



0,

0

The normalized conditional density of given and the


exp ( )cos( ( ))( , )

( )

ˆThe conditional least-squares estimate ( , , ( )) is given

t

d

N

t r s s s t

V t tp t

I V t

h t t

1

0

0

1

1

by

( )ˆ( , , ( )) ( ) cos ( ( ))( ( )) ( ) ( ( ))( )

where is the modified zero order Bessel function, and

is the modified first order Bessel function,

1cos exp cos .

2

c d

I V th t t r t t t t t S t t

I V t

I

I

I x x d



3 12 2 22 2 10

00

ˆThe conditional least-squares estimate ( ) is given by

1 2 ( ) ( )ˆ( ) ( , ) ( )( ( )) !(2 )! ( !) 2

kN k

k

t

k tt p t d V t

I V t k k k

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.

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: Channel parameter estimation: References

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 MeasuresReview of the Power Control Problem

1( 0, , 0)1

1

min ; M

Mn nn

i nMp pi j nj nj

p gp

p g


QoS MeasuresVector Form [Yates 1981]

Then

, 1, , 0 1

min ; j

M

i Ip j M i

p G P GP


QoS MeasuresProbabilistic QoS Measures

Define

The Constraints are equivalent to

1

1( ) , 1, ,

( ) 0, 1, ,

Mnj nj n n nnj

n

n

I p p g p g n M

I p n M


QoS MeasuresDecentralized Probabilistic QoS Measures


QoS Measures


Centralized Probabilistic QoS Measures


QoS Measures Stochastic optimal control

Received signal

State-space representation

1

1

0

( ) ( ) ( ) exp ( ) ( )

( ) ( ) ( ) ( ) ( ); ( ) exp ( )

( , ) ( , ) ( , ) ( , ) ( , ) ( )

( , ) ( )

M

n j j nj nj

M

n j j nj n nj njj

nj nj nj nj nj nj

dnj nj

y t u t s t kX t d t

y t u t s t S t d t S t kX t

dX t t t X t dt t dB t

X t PL d dB


QoS Measures Pathwise QoS and Predictable Strategies

define

then

where

Sts j )( )()( tSts njj

nkS;)( dttpi

Power control for short-term flat fading

t

t

t-1

t

t

t

Base Stationcalculates

Mobile

S(tpm(t-1)pm

(t)

pm(t) Sm (t => pm(t+1)

pm(t+1)

observe => calculate

Send backpm(t-1) Sm (t-1 => pm(t)

pm(t

)p

m(t

+1)

pm(t+1) Sm (t

pm(t) Sm (t-1

S(tpm(t)

Mobileimplements

Pathwise QoS Measures and Predictable Strategies

Power control for short-term flat fading

t

Base Station

Mobile

Observe pm(t)Sm (t => calculate pm(t+1)

pm(t) Sm (t

tt

pm(t+1) Sm (tpm(t+2) Sm (t

pm(t-1) Sm (.pm(t) Sm (. pm(t+1) Sm (.

Mobileimplements

Base Station

calculates

pm(t+1)

Value ofsignal

desired

pm(t)

Pathwise QoS Measures and Predictable Strategies


QoS Measures Define

)()( tSts nini


QoS Measures Predictable Strategies over the interval

Predictable Strategies Linear Programming

],[ 1kk tt

1

1

1( )

1

11 1 1 1 1 [ , ]

min ( );

( ) ( , ) ( , ) ( ) ( ) ( )

k ad

k k

M

i kp t U

i

k I k k k k k k nk t t t

p t

p t G t t G t t p t t S t


QoS Measures


QoS Measures

)()( tSts nn

)()( tSts nn

)()( tSts jj nnSs

)( 1kt


QoS MeasuresGeneralizations

Linear Programming

Stochastic Optimal Control with Integral/Exponential-of-Integral Constraints

)(tSnk

)(tSnk

Chapter 5: Power Control for Wireless Networks: Conclusions

Predictable strategies and dynamic models linear programming

Probabilistic QoS measures

Stochastic optimal control linear programming

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. 305-311, 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. 565-576, 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.

References

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

0

0

0

, , is a complete probability space with

filtration and finite time 0, ,

on which all random processes are defined

source signal

, ( , ), ( , ), ( , ) ,

is state channel pr

t t

t t

t t k d k kt

F P

F t T T

X X

r t t t

0

ocess

Wiener process independent of X,

representing thermal noise

t tN N


( )0

1

, , , 0, (1)

( , ) cos ( ) ( , )

: is a delay, : is a Doppler shift, is an amplitude,

: is a phase, is a number of resolvable paths, and

: is

k

Mk

t t t tk

kt k c d k k

k d

c

dY Z A X Y dt dN Y

Z r t t t t t

r

M

a carrier frequency

, , : is the non-anticipatory functional representing

encodingtA X Y

The received signal can be modeled as


3 3

0, 0,

30,

, ,

, 0, ; , 0, ;

, 0, ; , 0, ;

, 0, ; , 0, ;

their filtrations

0, ; , 0, ; ,

0, ;

and truncations of f

X

Y

X YT T

T

dX Y

X B C T R B C T R

Y B C T R B C T R

B C T R B C T R

F B C T R F B C T R

F B C T R

0, 0, 0,

iltrations

, , .X Yt t tF F F

Also, we define the following measurable spaces associated with stochastic processes


2

( )

10

Pr , , 1

(1) has the unique strong solution

Definition 1: The set of admissible encoders is defined as

follows

: 0, 0, ; 0, ; ;

, , ; 0, is pr

k

T Mk

t tk

ad

t

Z A X Y dt

A A T C T R C T R R

A A X Y t T

2

0

ogressively measurable,

, ,T

tE A X Y dt

The following assumptions are made


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

, |, ,

| |

2

( )

10

2

( )

1

0,

, |( ) log ( , )

| |

1(ii) , ,

2

ˆ , | ,

ˆ , , , | ,

k

k

k k

X YX Y

X Y

T Mk

t tk

Mk

t tk

Yt t t

dP x yi E x y

dP x dP y

E E Z A X Y

Z A Y dt

A Y E A X Y F


Definition 2. Consider the model (1). The Shannon capacity of (1) is defined by

, |,

| |

( , )

2

( ) , | ,

, |, | log ( , )

| |

1sup , |

subject to the power constraint on the transmitted signal

, , | (2)

ad

T

X YT X Y

X Y

TX A X A

t

iii I X Y dP

dP x yI X Y E x y

dP x dP y

C I X Y FT

E A X Y P

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

It can be proved that this upper bound is indeed the channel capacity, by observing that there exists a source signal with Gaussian distribution

such that the mutual information between that signal X and received signal Y is equal to the upper bound (3).

2

0

0

1, | (3)

2

2 , 0

T

T t

t t t

I X Y F P E Z dt

dX X dt PdW X

Chapter 5: Upper Bound on Mutual Information

The capacity is

T

t dtZET

PC

0

2

2

Chapter 5: Optimal Encoding/Decoding Strategies forNon-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

Ft and Gt are Borel measurable and bounded functions, Integrable and square integrable, respectively, Gt Gt

tr>0, t[0,T], W is a Wiener process independent of Gaussian random variable X0~

(4)t t t t tdX F X dt G dW

VXN ,


Decoding. The optimal decoder in the case of mean square error criteria is the conditional expectation

while the error covariance is

Encoding. The optimal encoder is derived by using equation for optimal decoder, and equation for power constraint (6).

0,

2

0,

ˆ , [ | , ]

ˆ, [( , | , ]

Yt t t

Yt t t t

X Y Z E X F

V Y Z E X X Y Z F


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

The received signal is then

The processes W, and N are independent, and the power constraint (2) becomes

0 1

0 1

0 1 2

, , ( , ) ( , )

( ( , ) ( , ) ) (5)

[( ( , ) ( , ) ) | ] (6)

t t t t

t t t t t t

t t t

A X Y A Y A Y X

dY Z A Y A Y X dt dN

E A Y A Y X P


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, optimal decoder, and corresponding error covariance are respectively given by

T

t dtZET

PC

0

2

2

* * * *

* *

* * * * * * *

* *0

* * 2 2 2

0 0 0

* *0

ˆ, , ,,

ˆ ˆ, , ,

ˆ ,

, exp 2 exp 2

, .

t t t

t

t t t t t t

t t t t t

t s s s u u

s s

PA X Y X X Y Z

V Y Z

dX Y Z F X Y Z dt Z PV Y Z dY

X Y Z X

V Y Z V F ds Z Pds G F du Z Pdu ds

V Y Z V

Chapter 5: Optimal Encoding/Decoding Strategies forRandom Variable Sources

Theorem 4 (Coding theorem for random variable source). If a source signal X, which is Gaussian random variable is transmitted over a flat fading wireless channel, then the optimal encoding and decoding with feedback reaching the channel capacity are

VXN ,

* * 2 * *

0

* * 2 *

0

* *0

* * 2

0

* *0

ˆ, , exp ,2

ˆ , exp2

ˆ ,

, exp

,

t

t s t

t

t t s t

t

t s

P PA X Y Z ds X X Y Z

V

PdX Y Z Z PV Z ds dY

X Y Z X

V Y Z V P Z ds

V 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.

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.

Chapter 5: Capacity, Optimal Encoding-Decoding: References

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

Documents

Chapter 5: Applications using S.D.E.’s Channel state-estimation State-space channel estimation using Kalman filtering Channel parameter identificationa