Error Detection And Correction

Rutvi ShahRutvi Shah 11

ERROR CORRECTION

& ERROR

DETECTION


Data can be corrupted during Data can be corrupted during transmission. For reliable transmission. For reliable communication, errors must be communication, errors must be detected and corrected.detected and corrected.

Error detection and correction are Error detection and correction are implemented either at data link layer implemented either at data link layer or the transport layer of the OSI model.or the transport layer of the OSI model.


TYPES OF ERRORSTYPES OF ERRORS Single bit error :-Single bit error :-

-- Only one bit in the data unit has Only one bit in the data unit has changed.changed.

Burst error :-Burst error :--- It means that two or more bits in It means that two or more bits in the the data unit has changed. data unit has changed.


Single bit ErrorSingle bit Error

00 00 00 00 11 00 11 00 00 00 00 00 00 00 11 00

Burst ErrorBurst Error00 11 00 00 00 11 00 00 00 11 00 00 00 00 11 11

00 11 00 11 11 11 00 11 00 11 00 00 00 00 11 11

0 changed to 1

Received Sent

Sent

Received

Bits corrupted by Burst Error


ERROR DETECTIONERROR DETECTION Error detecting code is to include only Error detecting code is to include only

enough redundancy to allow the enough redundancy to allow the receiver to deduce that an error receiver to deduce that an error occurred, but not which error, and occurred, but not which error, and have it request a re-transmission.have it request a re-transmission.

Error detection uses the concept of Error detection uses the concept of redundancy, which means adding redundancy, which means adding extra bits for detecting error at the extra bits for detecting error at the destination.destination.


RedundancyRedundancy Instead of repeating the entire data Instead of repeating the entire data

stream, a shorter group of bits may stream, a shorter group of bits may be appended to the end of each unit. be appended to the end of each unit. This technique is called Redundancy This technique is called Redundancy because the extra bit are redundant because the extra bit are redundant to the information. They are to the information. They are discarded as soon as the accuracy of discarded as soon as the accuracy of the transmission has been the transmission has been determined.determined.



There are basically four types of There are basically four types of redundancy checks. They are:redundancy checks. They are:

1.1. VRCVRC (Vertical Redundancy Check). (Vertical Redundancy Check).2.2. LRCLRC (Longitudinal Redundancy (Longitudinal Redundancy Check).Check).3.3. CRCCRC (Cyclical Redundancy Check). (Cyclical Redundancy Check).


ERROR DETECTIONERROR DETECTION

VERTICAL REDUNDUNCY CHECKVERTICAL REDUNDUNCY CHECK

LONGITUDINAL REDUNDANCY CHECKLONGITUDINAL REDUNDANCY CHECK

CYCLIC REDUNDANCY CHECKCYCLIC REDUNDANCY CHECK


VERTICAL REDUNDANCY CHECK

It is also known as parity checkIt is least expensive mechanism for error detectionIn this technique,the redundant bit called parity bit is appended to every data unit so that the total number of 1s in the unit becomes even (including parity bit)


VERTICAL REDUNDANCY CHECK

Checking function

Is total number of 1s even ?

Receiver

1100001 | 1 Even – parity generator

1100001

1

Data

VRC

Sender


Example :Example :

1110110 1101111 1110010 1110110 1101111 1110010

- After adding the parity bit- After adding the parity bit

1110110111011011 1101111 110111100 1110010 111001000

VERTICAL REDUNDANCY VERTICAL REDUNDANCY CHECKCHECK


VRC can detect all single – bit errorsVRC can detect all single – bit errors It can detect burst errors if the total It can detect burst errors if the total

number of errors in each data unit is number of errors in each data unit is odd.odd.

VRC can not detect errors where the VRC can not detect errors where the total number of bits changed is even. total number of bits changed is even.

VERTICAL REDUNDANCY VERTICAL REDUNDANCY CHECKCHECK


LONGITUDINAL REDUNDANCY CHECK(LRC)

In this method , a block of bits is organized in table(rows and columns)calculate the parity bit for each column and the set of this parity bit is also sending with original data.From the block of parity we can check the redundancy.


LRC ExampleLRC Example

11100111 1101101 00111001 10101001 10101010

11100111110111010011100110101001

11100111 11011101 00111001 1010100111100111 11011101 00111001 10101001

LRC 10101010

Original data plus LRC


LRC ExampleLRC Example

Suppose the following block is sent :Suppose the following block is sent :10101001 00111001 11011101 11100111 10101001 00111001 11011101 11100111 1010101010101010(LRC)(LRC)

However,it is hit by burst of length eight and some However,it is hit by burst of length eight and some bits are corrupted (Yellow bits are changed)bits are corrupted (Yellow bits are changed) : :101010100000111 1 11000001001 11011101 11100111 1010101001001 11011101 11100111 10101010(LRC)(LRC)

When the receiver checks the LRC,some of the bits When the receiver checks the LRC,some of the bits are not are not

follow even parity rule and whole block is follow even parity rule and whole block is discardeddiscarded

(the non matching bits are shown in red ) :(the non matching bits are shown in red ) :10100011 10001001 11011101 11100111 10100011 10001001 11011101 11100111 1100110011001100


Advantage :Advantage :-> LRC of n bits can easily detect burst -> LRC of n bits can easily detect burst

error of n bits.error of n bits.

Disadvantage :Disadvantage :-> If two bits in one data units are -> If two bits in one data units are

damaged and two bits in exactly same damaged and two bits in exactly same position in another data unit are also position in another data unit are also damaged , the LRC checker will not damaged , the LRC checker will not detect the error.detect the error.


CYCLIC REDUNDANCY CYCLIC REDUNDANCY CHECK (CRC)CHECK (CRC)

In this method , a sequence of redundant bits ,In this method , a sequence of redundant bits , called the CRC or the CRC remainder, is appended to the end of the called the CRC or the CRC remainder, is appended to the end of the

unit so that the resulting data unit become exactly divisible by a unit so that the resulting data unit become exactly divisible by a second, predetermined binary number. At its destination , the second, predetermined binary number. At its destination , the incoming data unit is divided by the same number. If at this step incoming data unit is divided by the same number. If at this step there is no remainder ,the data unit assume to be correct and is there is no remainder ,the data unit assume to be correct and is accepted, otherwise it indicate that data unit has been damaged in accepted, otherwise it indicate that data unit has been damaged in transmission and therefore must be rejected.transmission and therefore must be rejected.

The redundancy bits is used by CRC are derived by dividing the data The redundancy bits is used by CRC are derived by dividing the data

unit by a predetermined divisor. The remainder is the CRC.unit by a predetermined divisor. The remainder is the CRC.


CRC generator and checkerCRC generator and checker

DATA CRC

DIVISOR

REMAINDER CRC

DATA CRC DIVIS0R

DATA 00…0

Zero accept

Nonzero reject

N bits

N+1 bits

N bits

Receiver Sender


DivisorDivisor The divisor is determined according to the The divisor is determined according to the

algebraic polynomial.algebraic polynomial. for e.g.for e.g. A polynomial isA polynomial is X^7 + x^5 + x^2 + x + 1X^7 + x^5 + x^2 + x + 1 generation of divisor from polynomialgeneration of divisor from polynomial

X^7 + X^5 + X^2 + X + 1

1 0 1 0 0 1 1 1

X^4 X^3X^6


A polynomial should be selected A polynomial should be selected according to the following rule:-according to the following rule:-

1.1. It should not be divisible by x.It should not be divisible by x.2.2. It should be divisible by x+1.It should be divisible by x+1.


Example :-Example :- The CRC generator at sender end :The CRC generator at sender end :

1 1 0 11 1 1 1 0 1

1 0 0 1 0 0 0 0 01 1 0 1

1 0 0 0 1 1 0 1

1 0 1 0 1 1 0 1

1 1 1 0 1 1 0 1

0 1 1 0 0 0 0 0

1 1 0 01 1 0 1

0 0 1


The CRC checker at receiver end :The CRC checker at receiver end :

1 1 0 11 1 1 1 0 1

1 0 0 1 0 0 0 0 11 1 0 1

1 0 0 0 1 1 0 1

1 0 1 0 1 1 0 1

1 1 1 0 1 1 0 1

0 1 1 0 0 0 0 0

1 1 0 1 1 1 0 1

0 0 0


ERROR CORRECTIONERROR CORRECTION Error correcting code is to include enough Error correcting code is to include enough

redundant information along with each block of redundant information along with each block of data sent to enable the receiver to deduce data sent to enable the receiver to deduce what the transmitted character must have what the transmitted character must have been.been.

Error Correction must be handled in two ways :Error Correction must be handled in two ways :-- When an error is discovered, the receiver When an error is discovered, the receiver can have the sender retransmit the entire can have the sender retransmit the entire data unit.data unit.-- Receiver can use an error correcting code, Receiver can use an error correcting code, which automatically corrects certain errors.which automatically corrects certain errors.


There are two types of Error There are two types of Error Correcting techniques :Correcting techniques :

1. 1. Single bit error correction.Single bit error correction.2. 2. Burst error correction.Burst error correction.

Error Correction can be done with the Error Correction can be done with the help help

of HAMMING CODE.of HAMMING CODE.


HAMMING CODEHAMMING CODE It is a technique developed by It is a technique developed by

R.W.Hamming.R.W.Hamming. Hamming code can be applied to Hamming code can be applied to

data units of any length and uses the data units of any length and uses the relationship between data and relationship between data and redundancy bits. For eg.redundancy bits. For eg.


A 7 bit ASCII code requires 4 A 7 bit ASCII code requires 4 Redundancy bits that can be added Redundancy bits that can be added to the end of the data unit or to the end of the data unit or interspersed with the original data interspersed with the original data bits.bits.

These bits are placed in positions These bits are placed in positions 1,2,4 and 8. We refer to these bits as 1,2,4 and 8. We refer to these bits as r1,r2,r4 and r8.r1,r2,r4 and r8.


dd dd dd rr dd dd dd rr dd rr rr

Redundancy Bits

Positions of Redundancy Bits in Hamming Code

11 10 9 8 7 6 5 4 3 2 1


In the Hamming code, each r bit is the VRC In the Hamming code, each r bit is the VRC bit for one combination of data bits :bit for one combination of data bits :-- r1 is the one combination of data bits.r1 is the one combination of data bits.-- r2 is another combination of data bits.r2 is another combination of data bits.

and so on.and so on.

The combination used to calculate each of The combination used to calculate each of the four values for a 7 bit data sequence are the four values for a 7 bit data sequence are as follows :as follows :-- r1 : bits 1,3,5,7,9,11.r1 : bits 1,3,5,7,9,11.-- r2 : bits 2,3,6,7,10,11.r2 : bits 2,3,6,7,10,11.-- r4 : bits 4,5,6,7.r4 : bits 4,5,6,7.-- r8 : bits 8,9,10,11.r8 : bits 8,9,10,11.


11 00 00 11 11 00 11

11 00 00 11 11 00 11 11

11 00 00 11 11 00 11 00 11

11 00 00 11 1 1 00 00 1 1 00 11

1 1 00 00 11 11 11 00 00 11 00 11

Data : 1 0 0 1 1 0 1

Data

Adding r1

Adding r2

Adding r4

Adding r8

Code : 1 0 0 1 1 1 0 0 1 0 1

11 10 9 8 7 6 5 4 3 2 1


1 1

00

00

11

11

11

00

00

11

00

11

1 1

00

00

11

00

11

00

00

11

00

11

Sent

Received

Error


1 1 00 00 11 00 11 00 00 11 00 1111 10 9 8 7 6 5 4 3 2 1

11 00 00 11 00 11 00 00 11 00 1111 10 9 8 7 6 5 4 3 2 1

1 1 00 00 11 00 11 00 00 11 00 1111 10 9 8 7 6 5 4 3 2 1

1 1 00 00 11 00 11 00 00 11 00 1111 10 9 8 7 6 5 4 3 2 1

0 1 1 1The bit in position 7 is in error

Error Detection And Correction

Documents

longitudinal redundancy check

data unit

hamming code

error correction

redundancy bits

error detection

parity bit

data bits