1 on the channel capacity of wireless fading channels c. d. charalambous and s. z. denic school of...

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On the Channel Capacity of Wireless Fading Channels

C. D. Charalambous and S. Z. Denic

School of Information Technology and Engineering,University of Ottawa,

Ottawa, Ontario, K1N 6N5, Canada.

E-mail:{chadcha, sdenic}@site.uottawa.ca

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Introduction

•The channel capacity is one of the most important parameters of any communication channel because it gives the maximal theoreticalrate at which one can transmit data.

• Present an encoding and decoding strategy with feedback, which achieves the channel capacity of additive Gaussian wireless fading channels

• Although the feedback does not increase the channel capacity, itcan be used to reach the channel capacity.

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Introduction cont.

• The approach considered, which involves computation of channelcapacity, optimal encoding and and decoding gives an interestinginsight into simultaneous source and channel coding for wirelesschannels.

• The results are applicable to problems in control Engineering which employ wireless communication links

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Introduction cont.

• Assumption: The channel is modeled as a random process, where the attenuation, phases, and delays are random processes.

• The derivation, employs the mutual information in a general setting, defined through the Radon-Nikodym derivative.

• Results: Optimal encoding/decoding strategies when theinformation sources are

Non-stationary Gaussian source; Random variable source.

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Introduction cont.

• It turns out that the encoder, which reaches the channel capacity uses linear feedback and that the optimal decoder minimizes mean square error.

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Channel Model and Mutual Information in Presence of Feedback

• PF ,, is a complete probability space with filtration 0ttFand finite time TTt ,,0 on which all random processes

are defined

- 0 ttXX source signal

- ,),(),,(),,(,0 kkdkttt tttr

is state channel process

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0 ttNN- Wiener process independent of X,representing thermal noise

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• The received signal can be modeled as

M

ktt

ktt YdNdtYXAZdY

k1

0)( ,0,,,

k is a delay, d is a Doppler shift, r is an amplitude,

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

),()(cos),( kkdckk

t tttttrZ

-

,,YXAt- is the non-anticipatory functional representing encoding

c is a carrier frequency

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(1)


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• Also, we defined the following measurable spaces associatedwith stochastic processes ,,YX

XBX ,

YBY ,

B,

RTCBRTC ;,0,;,0

RTCBRTC ;,0,;,0

33 ;,0,;,0 RTCBRTC

, their filtrations

3,0,0,0 ;,0,;,0,;,0 RTCBFRTCBFRTCBF T

YT

XT

, and truncations of filtrations .,, ,0,0,0

t

Yt

Xt FFF

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• The following assumption is make

1,,Pr0

2

1

)(

T M

kt

kt dtYXAZ

k

• (1) has the unique strong solution.

• Definition 1. The set of admissible encoders is defined as follows

adA

T

t

t

dtYXAE

andmeasurableelyprogressivisTtYXAARRTCRTCTA

0

2,,

,,0;,,;;,0;,0,0:

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• Theorem 1. Consider the model (1). The mutual informationbetween the source signal X and received signal Y over the interval [0,T], conditional on the channel state , IT(X,Y|F ,is given by the following equivalent expressions

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),(

|||,

log||

|,,, yx

ydPxdPyxdP

EYX

YXYX

2( )

10

2( )

1

0,

1 , ,2

ˆ , | ,

ˆ , , , | ,

k

k

k k

T Mk

t tk

Mk

t tk

Yt t t

E E Z A X Y

Z A Y dt

A Y E A X Y F

i)

ii)


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,|,

dPYXITiii)

),(||

|,log|,

||

|,, yx

ydPxdPyxdP

EYXIYX

YXYXT


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

FYXI

TC T

AXAX ad

|,1sup),(

subject to the power constraint on the transmitted signal

PYXAE t |,,2 (2)

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Upper Bound on Mutual Information• Theorem 2. Consider the model (1). Suppose the channel isflat fading. The conditional mutual information between the source signal X, and the received signal Y is bounded above by

T

tT dtZEPFYXI0

2

21|,

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

(3)

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

0,2 0 XdWPdtXdX ttt

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Upper Bound on Mutual Information

• The capacity is

T

t dtZETPC

0

2

2

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Optimal Encoding/Decoding Strategies forNon-Stationary Gaussian Sources

• We assume that the channel is flat fading (M=1), it isknown to both transmitter and receiver, and the source is Gaussian, nonstationary given by

ttttt dWGdtXFdX

• Ft and Gt are Borel measurable and bounded functions, Gt Gt

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

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(4)

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Optimal Encoding/Decoding Strategies for Non-Stationary Gaussian Sources

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• Definition 3. The set of linear admissible encoders Lad , whereLad Aad , is the set of linear non-anticipative functionals A with respect to source signal X, which have the form

tttt XYAYAYXA ),(),(,, 10

• The received signal is then

tttttt dNdtXYAYAZdY )),(),(( 10

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

PXYAYAE ttt ]|)),(),([( 210 (6)

(5)

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• Decoding. The optimal decoder in the case of mean square errorcriteria is the conditional expectation

],|[,ˆ,0 Yttt FXEZYX

, while the error covariance is

],|,ˆ[(, ,02 Y

tttt FZYXXEZYV

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

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• 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 dtZETPC

0

2

2

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ZYXXZYV

PYXA ttt

t ,ˆ,

,, ****

**

XZYX

dYZYPVZdtZYXFZYXd tttttt

,ˆ,,ˆ,ˆ

**0

*******

.,

2exp2exp,

**0

2

0

2

0

2

0

**

VZYV

dsPduZduFGPdsZdsFVZYVt

su

t

su

t

s

t

s

t

st

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Optimal Encoding/Decoding Strategies forRandom Variable Sources

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

VXN ,

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ZYXXdsZP

VPYXA t

t

st ,ˆ2

exp,, **

0

2**

XZYX

dYdsZPPVZZYXd t

t

stt

,ˆ2

exp,ˆ

**0

*

0

2**

VZYV

dsZPVZYVt

st

,

exp,

**0

0

2**

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Conclusions

• In the case of non-stationary Gaussian source- encoder, and decoder depend of channel process Z and receivedsignal Y- error covariance on the statistics of the source through functionsFt and Gt

- the encoding consists of two terms, the first plays the role of power control, and the other is the decoding error- this result is consistent with water pouring argument- the result tells us that the optimal way of communications is byonly sending the error and not the whole signal- the fact that error covariance depends on the source suggeststhat the power of transmitted signals for different types of sourcesshould be adjust in order to have the same performance for different sources

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• In the case of random variable source- If we set Ft= Gt=0, in the result for error covariance of

the non-stationary source, the result will be the same as for therandom variable source, and that was expected- For the flat fading channel, it can be proved that the errorcovariance is meaning that the error decreases exponentially with time and the transmittedpower but also increases if the attenuation is large.

Conclusions

2/Prexp, 2** tVZYVt

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Future Work

• Future work can include the following topics

- Joint source and channel coding for wireless channels- Computation of the channel capacity for multipath channels- Application of the results to some problems in control and estimation

1 on the channel capacity of wireless fading channels c. d. charalambous and s. z. denic school of...

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