EE576_Lec_01

EE 576 – Error Control Coding (142)1 Lec 01

King Fahd University of Petroleum and MineralsElectrical Engineering Department

Dr. Suhail Al-DharrabOffice: 59-0012-05Office Hours: Sun, Tue: 13:00-13:50

Mon,Wed: 11:00-11:50 or by appointmentE-mail: [email protected], Tel: 8210

EE 576 – Error Control Coding (142)2

Lecture Outline

• Basics of error control coding

• Coding gain

• Hamming bound

• Gilbert bound

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Communication System

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Error Control Coding

• What is error control?

Detecting/Correcting errors in digital data

• How?

By adding redundancy to the information sequence (encoding)

and utilize it to catch the errors (decoding)

• Why?

Enhance the reliability of the system

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Binary Repetition Codes

Ex: Consider a (3,1) Binary Repetition Code over BinarySymmetric Channel (BSC)

Coding Rule: 0 => 000 1 => 111If received sequence: 011. What was transmitted?

Scenario A: 111 with one error in 1st locationScenario B: 000 with two errors in 2nd & 3rd locations.

Maximum Likelihood Decoding (MLD):ScenarioA 1ScenarioB 1ScenarioA ScenarioB (for 0.5)

Decoding decision: 011 => 111Lec 01


Post-Decoding Probability of Error

• (3,1) repetition code can correct single errors.

Block Error Probability 32 1 3

3 1 1 1 3 1

• In general for a -error correcting code:

1

Bit error probability [for the (3,1) code, ]

Improvement: For a BSC with 10 , 3 10 .

Cost: Expansion in bandwidth.

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Coding Gain

• Coding gain is the reduction in / required to achieve aspecific BER for a coded system compared to uncodedsystem.

• For fair comparison, / of the coded system must bescaled by the rate of the code (or equivalently reduced by10 log 1/ dB), to ensure the same energy per informationbit for both systems.

Ex: For DPSK 0.5 exp / .When coding is introduced, / drops by a factor of

We should calculate 0.5 exp / , then evaluateas a function of ′. Now it is fair to compare and .

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Graphical Illustration

• Plot the curve of the uncodedsystem (A)

• Shift it to the right by 10log 1/ .(B). This is the BER of the codedsystem before decoding, at thesame energy per information bit.

• Use the values of BER on (B) tofind error rate after decoding (C).

• Compare A and C for code gain.• Asymptotic gain: gain at large

/• 4-5 dB gain is typical. Some codes

provide up to 9 dB.

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A

BC

Eb/N0

BER

G


Hamming Distance

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Def: The Hamming distance between two codewords and

, denoted by , , is the number of components (bits) at

which they differ.

011,000 2

011,111 1

Therefore 011 is closer to 111.

• Maximum Likelihood Decoding (MLD) reduces to Minimum

Distance Decoding.


Geometrical Illustration

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000

111

001

011

101100

110

010


Error Correction and Detection

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• Consider a code consisting of two codewords withHamming distance .

How many errors can be detected?How many errors can be corrected?

• Number of errors that can be detected 1

• Number of errors that can be corrected

• In other words, for -error correction, we must have

2 1


Error Correction and Detection

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Ex: A binary code of two codewords and 5

• It can correct two errors

or

• It can detect four errors

or

• It can correct one error and detect two more errors.

• In general: 2 1


Minimum Distance of a Code

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Def: The minimum distance of a code C is the minimum

Hamming distance between any two different codewords.

min , ∀ and in

• A code with minimum distance can correct all error

patterns up to and including -error patterns, where

2 1

It may be able to correct some higher error patterns, but not

all.


(7,4) Binary Code

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No. Message Codeword No. Message Codeword0 0000 0000000 8 0001 10100011 1000 1101000 9 1001 01110012 0100 0110100 10 0101 11001013 1100 1011100 11 1101 00011014 0010 1110010 12 0011 01000115 1010 0011010 13 1011 10010116 0110 1000110 14 0111 00101117 1110 0101110 15 1111 1111111


Coding: Gain and Cost

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• Given an , code.

- Gain is proportional to the error correction capability, .

- Cost is proportional to the number of check digits, .

• Given a sequence of information digits, it is desired to

add as few check digits ( ) as possible to correct as many

errors ( ) as possible.

What is the relation between these code parameters?


Hamming Bound

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• For an , code, there are 2 codewords and 2 possible

received words.

• Think of the 2 codewords as centers of spheres in an -

dimensional space.

• All received words that differ from codeword in or less

positions lie within the sphere of center and radius .

• For the code to be -error correcting (i.e. all t-error pattern

for any codeword transmitted can be corrected), all

spheres , 1, . . , 2 , must be disjoint.


Hamming Bound

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• In other words, when a codeword is selected, none of the

-bit sequences that differ from that codeword by or less

locations can be selected as a codeword.

• Consider the all-zero codeword. The number of words thatdiffer from this codeword by locations (bits) is

• The total number of words in any sphere (including thecodeword at the center) is

1


Hamming Bound

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• The total number of -bit sequences that must be availablefor the code to be a -error correcting code is

2

• But, the total number of sequences is 2n. Therefore,

2 2

or

2


Hamming Bound

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• This bound is known as the Hamming (sphere-packing)Bound. It provides a necessary, but not a sufficient,condition for the existence of an , -error correctingcode.

Ex: Is it theoretically possible to design a (10,7) single-errorcorrecting code?

100

101 1 10 11 2

It is not possible.

• A code for which the equality is satisfied is called a perfectcode.


Gilbert Bound

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• It provides the sufficient condition for the existence of a, code with .

12 2

Ex: What are the values of that are guaranteed to exist for asingle-error-correcting code of 10.Gilbert bound requires 5. A (10,5) code with 3does exist.Hamming bound requires 6. A (10,7) code with 3does not exist. The (10,6) code with 3, according to

the two bounds,may or may not exist. It is found to exist.


The Encoding Problem

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• How to select 2k codewords of the code C from the 2n

sequences such that some specified (or possibly themaximum possible) minimum distance of the code isguaranteed?

Ex: How were the 16 codewords of the (7,4) codeconstructed? Exhaustive search is impossible, except for veryshort codes (small k and n).

• How would the 2k n-bit codewords be assigned to the 2k k-bit information sequences?

Are we going to store the whole table of 2k(n+k) entries?!• A constructive procedure for encoding is necessary.


The Decoding Problem

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Standard Array

0000000 1101000 0110100 1011100 1110010 0011010 1000110 0101110 1010001 0111001 1100101 0001101 0100011 1001011 0010111 1111111

0000001 1101001 0110101 1011101 1110011 0011011 1000111 0101111 1010000 0111000 1100100 0001100 0100010 1001010 0010110 1111110

0000010 1101010 0110110 1011110 1110000 0011000 1000100 0101100 1010011 0111011 1100111 0001111 0100001 1001001 0010101 1111101

0000100 1101100 0110000 1011000 1110110 0011110 1000010 0101010 1010101 0111101 1100001 0001001 0100111 1001111 0010011 1111011

0001000 1100000 0111100 1010100 1111010 0010010 1001110 0100110 1011001 0`10001 1101101 0000101 0101011 1000011 0011111 1110111

0010000 1111000 0100100 1001100 1100010 0001010 1010110 0111110 1000001 0101001 1110101 0011101 0110011 1011011 0000111 1101111

0100000 1001000 0010100 1111100 1010010 0111010 1100110 0001110 1110001 0011001 1000101 0101101 0000011 1101011 0110111 1011111

1000000 0101000 1110100 0011100 0110010 1011010 0000110 1101110 0010001 1111001 0100101 1001101 1100011 0001011 1010111 0111111

Exhaustive decoding is impossible!!

Well-constructed decoding methods are required.


The Course is About …

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• Design/Find good codes, driven by the “Noisy Channel

Coding Theorem”

There exist codes of rate R that have arbitrarily small decodingerror probability (provided that R is less than the channelcapacity)

• Propose encoder and decoder circuits and algorithms

• Evaluate the performance of the code

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EE576_Lec_01