A Mathematical Theory of Communication

Jin Woo Shin

Sang Joon Kim

Paper Review

By C.E. Shannon

Contents

Introduction Summary of Paper Discussion

Introduction

This paper opened the information theory.

Before this paper, people believed the only way to make the err. Prob. smaller is to reduce the data rate.

This paper revealed that there is an achievable positive data rate with negligible errors.

C.E. Shannon

Summary of Paper

Preliminary Discrete Source & Discrete Channel Discrete Source & Cont. Channel Cont. Source & Cont. Channel

[Summary of Paper]

Preliminary

Entropy

Ergodic source

Irreducible, aperiodic property Capacity

n

iii ppH

1

log

dxxpxph )(log)(

T

TNC

T

)(loglim

[Summary of Paper]

Disc. Source & Disc. Channel

Capacity Theory (Theorem 11 at page 22)

-The most important result of this paper

If the discrete source entropy H is less than or equal to the channel capacity C then there exists a code that can be transmitted over the channel with arbitrarily small amount of errors. If H>C then there is no method of encoding which gives equivocation less than H-C.

[Summary of Paper]

Disc. Source & Cont. Channel Domain size of input and output channel becomes i

nfinity. The capacity of a continuous channel is:

Tx rate does not exceed the channel capacity.

))|()((max)(

yxhxhCxp

)|()()|()( vuHuHyxhxh

u x

[Discrete]message

Encode

[Cont.]Tx signal

y v

[Cont.]Rx Signal

Decode

[Discrete]Recovered message

Channel

[Cont.]Noise

[Summary of Paper]

Cont. Source & Cont. Channel Continuous source needs an infinite number of binary digits for

exact specification. Fidelity: the measurement of how much distortion we allow Rate with fidelity constraint D of Cont. source P(X) is :

with For given fidelity constraint D,

);(min)|(

YXIRyxp

achievableCR

u’ x

[Discrete]Binary digits

Encode

[Cont.]Tx signal

y v’

[Cont.]Rx Signal

Decode

[Discrete]Recovered

Binary digits

Channel

[Cont.]Noise

u

[Cont.]message

Mapping

w\ fidelityv

[Cont.]Recovered message

Remapping

allow distortion

dxdyyxdyxPD ),(),(

Discussion

Ergodic source Practical approach Rate distortion

[Discussion]

Ergodic source Ergodic Source assumption is the essential one in th

e paper. Source is ergodic -> AEP holds -> capacity theorem Finding a source that is not ergodic and holds AEP i

s a meaningful work. One example:

10

2/12/1P A B

1/2

1/2 1

100

[Discussion]

Practical approach -1 This paper provides the upper bound of achievable

data rate. Finding a good encoding scheme is another problem. Turbo code, LDPC code are most efficient codes. Block size, rate, BER, decoding complexity are

important factors when choosing a code for a specific system.

[Discussion]

Practical approach -2

Year Rate ½ Code SNR Required for BER < 10-5

1948 SHANNON 0dB

1967 (255,123) BCH 5.4dB

1977 Convolutional Code 4.5dB

1993 Iterative Turbo Code 0.7dB

2001 Iterative LDPC Code 0.0245dB

0 1 2 3 4 5 6

10

-3

10

10

10

Turbo

LDPC-4

10

-2

-1

0

SNR

BE

R

SNR vs. BER for rate 1/2 codes

Code

Conv. Code ML decoding

Uncoded

Bound

4 dB

** This graph and chart are modified from the presentation data of Engling Yeo at Jan 15 2003

C. Berrou and A. Glavieux, "Near Optimum Error Correcting Coding And Decoding: Turbo-Codes," IEEE Trans. Comms., Vol.44, No.10, Oct 1996.

[Discussion]

Rate distortion The ‘Fidelity’ concept motives ‘Rate Distortion’ theo

ry. Rate with D distortion(fidelity) of Discrete source P

(x) is defined as: subject to

H(Entropy) is the rate with 0 distortion. (The Rate Distortion Theory) We can compress a D

isc. source P(x) up to ratio when allowing D distortion.

);(min)()|(

)( YXIDRyxp

I yx

yxdEyxdyxPD,

)],([),(),(

)()( DR I