2. Information Theory -Channel Capacity Ying Cui Department of - - PowerPoint PPT Presentation

1896 1920 1987 2006

Computing and Communications

• 2. Information Theory
• Channel Capacity

Ying Cui Department of Electronic Engineering Shanghai Jiao Tong University, China 2018, Autumn

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Outline

• Communication system
• Examples of channel capacity
• Symmetric channels
• Properties of channel capacity
• Definitions
• Channel coding theorem
• Source-channel coding theorem

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Reference

• Elements of information theory, T. M. Cover and J. A.

Thomas, Wiley

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CHANNEL CAPACITY

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Communication System

• Map source symbols from finite alphabet into some sequence of

channel symbols, i.e., input sequence of channel

• Output sequence of channel is random but has a distribution

depending on input sequence of channel

– two different input sequences may give rise to same output sequence, i.e., inputs are confusable – choose a “nonconfusable” subset of input sequences so that with high probability there is only one highly likely input that could have caused the particular output

• Attempt to recover transmitted message from output sequence of

channel

– reconstruct input sequences with a negligible probability of error

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Channel Capacity

– I(X;Y)=H(X)-H(X|Y)= H(Y)-H(Y|X)

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conditioned on the input at that time relative entropy between p(x,y) and p(x)p(y)

EXAMPLES OF CHANNEL CAPACITY

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Noiseless Binary Channel

• Binary input is reproduced exactly at output
• C = max I(X; Y) = 1 bit, achieved using p(x) = (1/2, 1/2)

– one error-free bit can be transmitted per channel use

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I(X;Y)=H(X)-H(X|Y)=H(X)

Noisy Channel with Nonoverlapping Outputs

• Two possible outputs corresponding to each of the two

inputs

– appear to be noisy, but really not

• C = max I(X; Y) = 1 bit, achieved using p(x) = (1/2, 1/2)

– input can be determined from the output – every transmitted bit can be recovered without error

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I(X;Y)=H(X)-H(X|Y)=H(X)

Noisy Typewriter

• Channel input is either

unchanged with probability 1/2

• r is transformed into the next

letter with probability 1/2

• If the input has 26 symbols and

we use every alternate input symbol, we can transmit one of 13 symbols without error with each transmission

• C = max I(X; Y)

= max (H(Y) – H(Y|X)) = max H(Y) – 1 = log 26 – 1 = log 13 achieved using p(x) = (1/26,…, 1/26)

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Y uniform dist. p(y)=(1/26,…,1/26)

• Input symbols are complemented with probability p
• \\\

equality achieved using p(x)=(1/2,1/2)

Binary Symmetric Channel

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equality achieved when Y follows uniform dist. p(y)=(1/2,1/2)

Binary Erasure Channel

• Two inputs and three outputs, a fraction of bits are

erased

• Xx
• Recover at most 1-α of bits, as α of bits are lost

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𝜌=1/2 achieved when

SYMMETRIC CHANNELS

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Symmetric

– example of symmetric channel

• x
• binary symmetric channel

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(x,y)-th element: p(y|x)

Proof

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PROPERTIES OF CHANNEL CAPACITY

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Properties of Channel Capacity

• is a continuous function of p(x)
• is a concave function of p(x)
• Problem for computing channel capacity is a convex

problem

– maximization of a bounded concave function over a closed convex set – maximum can then be found by standard nonlinear optimization techniques such as gradient search

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0 since ( ; ) C I X Y   log since max ( ; ) max ( ) log C C I X Y H X  =  = ( ; ) I X Y ( ; ) I X Y log since max ( ; ) max ( ) log C C I X Y H Y  =  =

DEFINITIONS

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Discrete Memoryless Channel (DMC)

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Code

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Probability of Error

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Rate and Capacity

– write (2𝑜𝑆, 𝑜) codes to mean ( 2𝑜𝑆 , 𝑜) codes to simplify the notation

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CHANNEL CODING THEOREM (SHANNON’S SECOND THEOREM)

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Basic Idea

• For large block lengths, every channel has a subset of

inputs producing disjoint sequences at the output

• Ensure that no two input X sequences produce the

same output Y sequence, to determine which X sequence was sent

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Basic Idea

• Total number of possible output Y sequences is ≈

2𝑜𝐼(𝑍)

• Divide into sets of size 2𝑜𝐼(𝑍|𝑌) corresponding to the

different input X sequences

• Total number of disjoint sets is less than or equal to

2𝑜(𝐼 𝑍 −𝐼(𝑍|𝑌)) = 2𝑜𝐽(𝑌;𝑍)

• Send at most ≈ 2𝑜𝐽(𝑌;𝑍) distinguishable sequences of

length n

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Channel Coding Theorem

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New Ideas in Shannon’s Proof

• Allowing an arbitrarily small but nonzero probability of

error

• Using the channel many times in succession, so that the

law of large numbers comes into effect

• Calculating the average of the probability of error over a

random choice of codebooks

– symmetrize the probability and can then be used to show the existence of at least one good code

• Shannon’s proof outline was based on idea of typical

sequences, but was not made rigorous until much later

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Current Proof

• Use the same essential ideas

– random code selection, calculation of the average probability of error for a random choice of codewords, and so on

• Main difference is in the decoding rule-decode by joint

typicality

– look for a codeword that is jointly typical with the received sequence – if find a unique codeword satisfying this property, declare that word to be the transmitted codeword – properties of joint typicality

• with high probability the transmitted codeword and the received sequence are

jointly typical, since they are probabilistically related

• probability that any other codeword looks jointly typical with the received

sequence is 2−𝑜𝐽

• thus, if we have fewer then 2𝑜𝐽 codewords, then with high probability there

will be no other codewords that can be confused with the transmitted codeword, and the probability of error is small

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SOURCE–CHANNEL SEPARATION THEOREM (SHANNON’S THIRD THEOREM)

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Two Main Basic Theorems

• Data compression: R>H
• Data transmission: R<C
• Is condition H < C necessary and sufficient for sending a

source over a channel?

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Example

• Consider two methods for sending digitized speech
• ver a discrete memoryless channel

– one-stage method: design a code to map the sequence of speech samples directly into the input of the channel – two-stage method: compress the speech into its most efficient representation, then use the appropriate channel code to send it over the channel

• Lose something by using the two-stage method?

– data compression does not depend on the channel – channel coding does not depend on the source distribution

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Joint vs. Separate Channel Coding

• Joint source and channel coding
• Separate source and channel coding

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Source–Channel Coding Theorem

– consider the design of a communication system as a combination of two parts

• source coding: design source codes for the most efficient representation of the

data

• channel coding: design channel codes appropriate for the channel encodes

(combat the noise and errors introduced by the channel)

– the separate encoders can achieve the same rates as the joint encoder

• hold for the situation where one transmitter communicates to one receiver

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Summary

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Summary

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cuiying@sjtu.edu.cn iwct.sjtu.edu.cn/Personal/yingcui

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