Error Control Coding -Introduction

ERROR CONTROL CODING -

INTRODUCTION

ER. FARUK BIN POYEN, Asst. Professor

DEPT. OF AEIE, UIT, BU, BURDWAN, WB, INDIA

[email protected]

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Contents:

Introduction

Axioms

Types

Data Compression (Source Coding)

Error Correction Codes (Channel Coding)

Classification of Codes

Classification of Errors

Error Detection Techniques

Error Correcting Techniques

Drawbacks of coding Techniques

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Contents: Contd….

Classification of Error – Correcting Codes

Types of Error Control

Types of Linear Block Codes

Definitions related to codes

Overview of Error Control Coding Techniques

Automatic Repeat Request (ARQ)

Forward Error Correction (FEC) Technique

Transmission Errors

Power and Bandwidth Channels

Error Detection Method

Cyclic Redundancy Check

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Introduction:

Coding theory is one of the most important and direct applications ofinformation theory.

It can be subdivided into source coding theory and channel coding theory.

Using a statistical description for data, information theory quantifies thenumber of bits needed to describe the data, which is the information entropy ofthe source.

Coding theory, sometimes called algebraic coding theory, deals with the designof error-correcting codes for the reliable transmission of information acrossnoisy channels.

It makes use of classical and modern algebraic techniques involving finitefields, group theory, and polynomial algebra.

It has connections with other areas of discrete mathematics, especially numbertheory and the theory of experimental designs.

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Axioms:

1. Closure of addition: if x, y are in F, then x+ y is in F.

2. Closure of multiplication: if x, y are in F, then x*y is in F.

3. Associative Law of Addition: if x, y, z are in F, then (x+ y)+z=x+(y+ z)

4. Associative Law of Multiplication: if x, y, z are in F, then (x*y)z=x(y*z)

5. Distributive Law: if x, y, z are in F, then x(y+ z)=x*y + y*z

6. Existence of 0: an element of F satisfying x+0=x for all x in F

7. Existence of 1: an element of F satisfying x.1=x for all x in F

8. Existence of additive inverses (negatives):If x is in F, there exists y in F suchthat x+ y=0

9. Existence of multiplicative inverses (reciprocals), except for: If x in F is not azero element, then there exists an element y in F such that x*y=1.

10. Commutative Law of Addition: If x, y are in F, then x+ y = y+ x

11. Commutative Law of Multiplication: If x, y are in F, then x*y = y*x.

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Types:

There are essentially two aspects to coding theory:

1. Data compression (or SOURCE CODING)

2. Error correction (or CHANNEL CODING)

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Data Compression (Source Coding):

There are two formulations for the compression problem.

1. Loss less Data Compression: The data must be reconstructed exactly;

2. Lossy Data compression: Allocates bits needed to reconstruct the data,

within a specified fidelity level measured by a distortion function.

This subset of Information Theory is called Rate–Distortion Theory.

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Error-Correcting Codes (Channel Coding):

While data compression removes as much redundancy as possible, an

error correcting code adds just the right kind of redundancy (i.e., error

correction) needed to transmit the data efficiently and faithfully across a

noisy channel.

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Classification of Codes:

1. Error Detecting

2. Error Correcting

3. Error Correction Techniques

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Classification of Errors:

1. Content Error (errors in the content of message introduced

due to noise during transmission).

2. Flow integrity Error (missing blocks of data, data lost in

network or data delivered to wrong destination).

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Error Detection Techniques:

1. Parity Checking

2. Check Sum Error Detecting

3. Cyclic Redundancy Check (CRC)

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Error Correcting Techniques:

Based on generation of code words at the transmitter

Coded words contain the data bits and check bits

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Drawbacks of Coding Technique:

1. Increased transmission bandwidth because of addition of

extra bits.

2. Increased complexity of communication system.

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Classification of Error – Correcting Codes: Block Codes (No memory is required)

(n, k) block code is generate when the channel encoder accepts

information in successive k bit blocks. At the end of each such block, (n

- k) parity bit is added, which contains no information and termed as

redundant bits.

Convolutional Codes (Memory is required)

Here the code words are generated by discrete – time convolution of the

input sequence with impulse response of the encoder. Unlike block

codes, channel encoder accepts messages as a continuous sequence and

generates a continuous sequence of encoded bits at the output.

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Another classification:

Linear Codes

Non – linear codes

Linear codes have the unique property that when any two code

words of linear code are added in modulo – 2 adder, a third code

word is produced which is also a code, which is not the case for

non – linear codes.

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Types of Error Control:1. Automatic Request for Retransmission (ARQ): The receiver can request for

retransmission of complete or part of message, requiring additional channelcalled feedback channel.

2. Forward Error Correction (FEC): No feedback channel is available.

In coding theory, a linear code is an error-correcting code for which any linearcombination of code words is also a code word.

Linear codes are traditionally partitioned into block codes and convolutionalcodes, although turbo codes can be seen as a hybrid of these two types.

Linear codes allow for more efficient encoding and decoding algorithms thanother codes (cf. syndrome decoding).In coding theory, block codes comprisethe large and important family of error-correcting codes that encode data inblocks.

Examples of block codes are Reed–Solomon codes, Hammingcodes, Hadamard codes, Expander codes, Golay codes, and Reed–Mullercodes. These examples also belong to the class of linear codes, and hence theyare called linear block codes.

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Types of Linear Block Codes:

1. Cyclic codes (e.g. Hamming codes)

2. Repetition codes

3. Parity codes

4. Polynomial codes (e.g., BCH codes)

5. Reed–Solomon codes

6. Algebraic geometric codes

7. Reed–Muller codes

8. Perfect codes

9. Single Parity Check Bit Codes

10. Repeated Codes

11. Hadamard Code

12. Extended Code

13. Goley Code

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Important Definitions Related to Codes:

Code Word: The code word is the n bit encoded block of bits.

It contains message bits and parity or redundant bits.

Block Length: The number of bits ‘n’ after coding is known as block

length.

Code Rate: The code rate is defined as the ration of the number of

message bits (k) to the total number of bits (n) in a code word.

𝐶𝑜𝑑𝑒 𝑟𝑎𝑡𝑒 𝑟 = 𝑘/𝑛.

Code Vector: An ‘n’ bit code word can be visualized in an n –

dimensional space as a vector whose elements or coordinates are bits in

the code word.

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Important Definitions Related to Codes:

Hamming Distance: It is the distance between the two codes expressed

in the number of locations in which their respective elements differ.

Hamming Weight of a Code Word [w(x)]: It is defined as the number of

non – zero elements in the code word.

Code Efficiency: It is defined as the ratio of message bits to the number

of transmitted bits per block.

Code efficiency is equal to that of code rate.

Minimum Distance dmin: It is defined as the smallest Hamming distance

between any pair of code vectors in the code.

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Overview of Error Control Coding Techniques 20

Error Detection and Correction Capabilities

Sl. No. Description Expression

1. Detect up to s errors per word 𝑑𝑚𝑖𝑛 ≥ (𝑠 + 1)

2. Correct up to t errors per word 𝑑𝑚𝑖𝑛 ≥ (2𝑡 + 1)

3. Correct up to t errors and detect s > t errors per word 𝑑𝑚𝑖𝑛 ≥ (𝑡 + 𝑠 + 1)

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Automatic Repeat Request (ARQ):

Here, when an error is detected, a request is made for retransmission of

signal.

A feedback channel is necessary for this retransmission.

It differs from the FEC system in the following three aspects.

1. Less number of check bits (parity bits) are required increasing the (k/n)

ration for (n, k) block code.

2. Additional hardware to implement feedback path is required.

3. But rate at forward transmission needs to make allowance for backward

repeat transmission.

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Operation of ARQ:

For each message signal at the input, the encoder produces code words

which are stored temporarily at encoder output and transmitted over

forward transmission channel.

At the destination, decoders decode the signals and give a positive

acknowledgement (ACK) and negative acknowledgement (NAK)

respectively in case no error and error is detected.

On receipt of NAK, the controller retransmits the appropriate word

stored in the input buffer.

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Operation of ARQ:

The bit rate of return transmission involving ACK/NAK transmission is

lower compared to bit rate of forward transmission decreasing the

probability error in return transmission to such a small value that it may

be neglected.

There exists three types of ARQ systems viz.

i) Stop and Wait ARQ

ii) Go back N ARQ

iii) Selective Repeat ARQ

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Forward Error Correction (FEC) Technique:

It is a digital modulation system where discrete source generates

information in binary form.

The channel encoder accepts these message bits and add redundant bits

to them leaving higher bit rate for transmission.

Channel decoders uses the redundant bits to check for actually and

erroneous transmitted messages.

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Transmission Errors:

There are two types of transmission errors viz. random error and burst

error.

Random errors are those that occur in a purely random manner.

BCH codes are useful in dealing with this sort of error.

Burst errors occur in forms of bunches and are not independent.

Convolution codes are not effective for this sort of errors.

Fire codes which form a subclass of cyclic codes are effective in

mitigating these types of errors.

Interleaving is a technique which is used as an alternative for correcting

burst errors.

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Power and Bandwidth Channels:

Let the input data rate be b bits/sec.

The FEC encoder converts this into coded word at a rate of b/ R where

R stands for information rate.

The M symbol modulator converts the encoder output into M possible

symbol constellation having the symbol rate r_s=(b/kR) baud at the

output.

Minimum system bandwidth required for successful transmission is

𝐵 = 𝑟𝑠 =𝑏

𝑘𝑅𝐻𝑍

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Power and Bandwidth Channels:

The bandwidth efficiency is

ƞ =𝑏

𝑟𝑠= 𝑘𝑅 = 𝑅 log2𝑀𝑏𝑖𝑡𝑠/𝐻𝑧;

M = number arrays for PSK.

Band limited Channel: They have finite fixed bandwidth. Signals

requiring larger bandwidths therefore cannot be transmitted over these

channels without distortion. Telephonic lines come under band limited

channels.

Power limited Channel: They have limited power associated with them

but have large bandwidth viz. satellite channels. It is therefore possible

to accommodate FEC even with increased data rate.

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Error Detection Method:

Error correction is only possible if errors are detected in the

code words. There exists many methods for error detection,

the most popular ones are

i) Parity Checking

ii) Check sum Error Detection

iii)Cyclic Redundancy Check (CRC)

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Parity Checking:

Here an additional bit is appended with the existing message bits,

known as the parity bit.

As a result of addition of this extra bit, the resultant word now will have

either even or odd parity i.e. number of 1s in the code word will be

either even or odd.

If it is known that the parity of the received message is always going to

be even or odd as the case may be and if the received signal does not

tally with the expected result, the presence of an error is detected.

The limitation of this method is that it can only detect odd number of

errors and also it is unable to locate the position of the error.

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Check Sum Error Detection:

For burst errors, parity check method is not useful.

Here then the check sum method is applied.

Here the check sum is transmitted along with every block of data bytes (8 bits).

Here an 8 bit accumulator is used to add 8 bit of a block of data to find thecheck sum and the carries in the MSB are discounted while finding the checksum byte.

Transmission of data byte is followed by transmission of checksum byte whichis regenerated at the receiver separately by adding the received bytes.

After comparison with the transmitted byte, if results are identical it isconcluded that no error has occurred otherwise there exist errors.

As byte of checksum is transmitted, there is 255 to 1 chance of detectingrandom error.

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Cyclic Redundancy Check:

The concept of parity checking can be extended from detection to

correction of single error by arranging the data block in rectangular

matrix.

This will lead to two set of parity bits, viz.

Longitudinal Redundancy Check (LRC) and

Vertical Redundancy Check (VRC).

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Longitudinal Redundancy Check:

In Longitudinal Redundancy Check, one row is taken up at a time and

counting the number of 1s, the parity bit is adjusted to achieve even

parity.

Here for checking the message block, a complete character known as

Block Check Character (BCC) is added at the end of the block of

information, which may be of even or odd parity.

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Characters C O M P U T E R LRC bits

7 bit

ASCII

Codes

(Message

Bits)

B1 1 1 1 0 1 0 1 0 1

B2 1 1 0 0 0 0 0 1 1

B3 0 1 1 0 1 1 1 0 1

B4 0 1 1 0 0 0 0 0 0

B5 0 0 0 1 1 1 0 1 0

B6 0 0 0 0 0 0 0 0 0

B7 1 1 1 1 1 1 1 1 0

VRC bits 1 1 0 0 0 1 1 1 1

Vertical Redundancy Check:

In VRC, the ASCII code for individual alphabets are considered

arranged vertically and then counting the number of 1s, the parity bit is

adjusted to achieve even parity.

A single error in any bit will result in a non – correct LRC in the row

and a non – correct VRC in the column. The bit which is common to

both the row and column is the bit in error. The limitation is though it

can detect multiple errors but is capable to correct only a single error as

for multiple errors it is not suitable to locate the position of the errors.

1 in the square box in the next table, is the bit in error as it is evident

from the erroneous results both in the LRC and the VRC columns

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CRC:

Characters C O M P U T E R LRC

bits

7 bit

ASCII

Codes

(Message

Bits)

B1 1 1 1 0 0 1 0 1 (W)

B2 1 1 0 0 0 0 0 1 1

B3 0 1 1 0 1 1 1 0 1

B4 0 1 1 0 0 0 0 0 0

B5 0 0 0 1 1 1 0 1 0

B6 0 0 0 0 0 0 0 0 0

B7 1 1 1 1 1 1 1 1 0

VRC bits 1 1 0 0 0 (W) 1 1 1 1

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Advantages of LRC over VRC:

i) Single errors can be detected and corrected.

ii) Double and even triple errors can be detected at the cost of increased complexity.

Check Sum:

The bits is row b8 are character parity bits, which append to the data bits as an error checking group.

These bits combining are called Checksum.

References:

1. Digital Communications; Dr. Sanjay Sharma; Katson Books.

2. Communication Engineering; B P Lathi.

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