A Mathematical Theory of Communication

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A Mathematical Theory of Communication
Paper Review
By C.E. Shannon

• Jin Woo Shin
• Sang Joon Kim

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Contents

• Introduction
• Summary of Paper
• Discussion

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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
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Summary of Paper

• Preliminary
• Discrete Source Discrete Channel
• Discrete Source Cont. Channel
• Cont. Source Cont. Channel

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Summary of PaperPreliminary

• Entropy
• Ergodic source
• Irreducible, aperiodic property
• Capacity

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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 HgtC then there is no method of encoding which
  gives equivocation less than H-C.

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Summary of Paper Disc. Source Cont. Channel

• Domain size of input and output channel becomes
  infinity.
• The capacity of a continuous channel is
• Tx rate does not exceed the channel capacity.

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Summary of PaperCont. 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,

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Discussion

• Ergodic source
• Practical approach
• Rate distortion

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DiscussionErgodic source

• Ergodic Source assumption is the essential one in
  the paper.
• Source is ergodic -gt AEP holds -gt capacity
  theorem
• Finding a source that is not ergodic and holds
  AEP is a meaningful work.
• One example

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DiscussionPractical 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.

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DiscussionPractical approach -2
Year Rate ½ Code SNR Required for BER lt 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
C. Berrou and A. Glavieux, "Near Optimum Error
Correcting Coding And Decoding Turbo-Codes,"
IEEE Trans. Comms., Vol.44, No.10, Oct 1996.
This graph and chart are modified from the
presentation data of Engling Yeo at Jan 15 2003
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DiscussionRate distortion

• The Fidelity concept motives Rate Distortion
  theory.
• 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
  Disc. source P(x) up to ratio when
  allowing D distortion.