1896 1920 1987 2006

Computing and Communications2. Information Theory

-Channel Capacity

Ying Cui

Department of Electronic Engineering

Shanghai Jiao Tong University, China

2017, 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

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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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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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Noisy Typewriter

• Channel input is either unchanged with probability 1/2 or 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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• Input symbols are complemented with probability p

• \\\

equality is achieved when the input distribution is uniform

Binary Symmetric Channel

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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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𝜋=1/2achieved when

SYMMETRIC CHANNELS

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Symmetric

– example of symmetric channel

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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 ( ; ) 0C I X Y

log since max ( ; ) max ( ) logC C I X Y H X

( ; )I X Y

( ; )I X Y

log since max ( ; ) max ( ) logC 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 (2𝑛𝑅, 𝑛) codes to mean ( 2𝑛𝑅 , 𝑛) 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 ≈

2𝑛𝐻(𝑌)

• Divide into sets of size 2𝑛𝐻(𝑌|𝑋) corresponding to the different input X sequences

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

2𝑛(𝐻 𝑌 −𝐻(𝑌|𝑋)) = 2𝑛𝐼(𝑋;𝑌)

• Send at most ≈ 2𝑛𝐼(𝑋;𝑌) 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−𝑛𝐼

• thus, if we have fewer then 2𝑛𝐼 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 over 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.cniwct.sjtu.edu.cn/Personal/yingcui

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