GROUP03_AMAK:ERROR DETECTION AND CORRECTION PPT

MSRIT INFORMATION SCIENCE 1

ERROR DETECTION&

ERROR CORRECTION


AMAK

A-> ANKITA (1MS07IS133)

M-> MAYANK (1MS07IS047)

A-> ANSHUJ (1MS07IS011)

K-> KRISH (1MS07IS038)


INTRODUCTION TO ERROR

REDUNDANCY

CODING

• LINEAR BLOCK CODING

• HAMMING CODE

• CYCLIC REDUNDANCY CHECK(CRC)

• IMPLEMENTATION OF HAMMING CODE

TABLE OF CONTENTS:


What is an error???Unpredictable change of bits from 1->0 or 0-

>1.

Typeso Single bit erroro Burst error(Multiple)

Redundancy: Correction or detection of errors.

INTRODUCTION


Forward error correction: Method of Guessing the actual message using the redundant bits.

Retransmission: Repeated sending of message until error free.

Modulo Arithmetic:• Modulo 2- Remainder after division can be either 0 or1.• Modulo 3- Remainder after division can be either 0,1 or

2....

• Modulo n- Remainder after division can be either 0,1,2….n-1.


Adding : 0+0=0 0+1=1 1+0=1 1+1=0

Subt: 0-0=0 0-1=1 1-0=1 1-1=0

XOR operation:

1 0 1 1 0 1 1 1 0 0 0 1 0 1 0


CODING: Redundancy is achieved through coding.

BLOCK CODING: Message divided into blocks.

• k bit datawords• r redundant bits• n= k+r, n>k,n bit codeword. • 2k total datawords(equal to number of

valid codewords.)• 2n total codewords• 2n -2k invalid codewords


ERROR DETECTIONo If resulting codeword is invalid.o If multiple errors in the codeword result in

valid codeword.

ERROR CORRECTIONo More difficult.o Need more number of redundant bits than

for detection.o Involves error detection as well as finding

the position(s) where error has occurred.


Generation of codewords for each dataword: C(7,4) n ,k

Codeword is generated by the generator which appends 3 redundant bits at the end of the dataword.

Ro =a2 + a1 + a0 Modulo-2

R1= a3 + a2 + a1 Modulo-2

R2= a1+ a0 + a3 Modulo-2

HAMMING CODE GENERATION


DATAWORD

CODEWORD

DATAWORD

CODEWORD

0000 0000000 1000 1000110

0001 0001101 1001 1001011

0010 0010111 1010 1010001

0011 0011010 1011 1011100

0100 0100011 1100 1100101

0101 0101110 1101 1101000

0110 0110100 1110 1110010

0111 0111001 1111 1111111

A CRC CODE WITH C(7,4)


checker on the receiver side will generate a 3bit syndrome by the formulae given below:

s0 = b2 + b1 + b0 + q0 modulo-2

s1 = b3 + b2 + b1 + q1 modulo-2

s2 = b1 + b0 + b3 + q2 modulo-2

If the syndrome i.e s2s1s0 is 000 then data is error free…or undetected.

Otherwise, received codeword has an error(s).


MAGIC TABLE

Depending upon the value of syndrome we can find the position of occurrence of error and then the bit position where error has occurred is flipped.

SYNDROME

000 001 010 011 100 101 110 111

ERROR None

q0 q1 b2 q2 b0 b3 b1


CODEWORD NOTATION ON SENDER’S AND RECEIVER’S SIDE

a3 a2 a1 a0 R2 R1 R0

b3 b2 b1 b0 q2 q1 q0

RECEIVED CODEWORD

0 1 0 1 0 0 0

CODEWORD SENT

1 1 0 1 0 0 0


Number of bit change occurring between two codewords.

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

0 1 0 0 1 0 1

The total number of ones is equal to the number

of bit changes between two codewords.

HAMMING DISTANCE


Smallest Hamming Distance between all sets of codewords.

Ex- d(0000000,0001101) = 3d(0001100,0111001) = 4d(0110100,0111001) =3d(11111111,0000000) =7…. & so on..

Dmin= 3 for the above set of codewords.

MINIMUM HAMMING DISTANCE


Suppose ‘s’ errors are to be detected, then dmin should be s+1.

for the example taken, it can detect upto a maximum of 2 errors.

Suppose ‘t’ errors are to be corrected, then the dmin should be 2t+1.

in the above example, it can correct upto only 1 error.


Linear block code?

Linear block code with an extra property: code word is cyclically rotated that generates another codeword.

1010110 is a codeword on rotating 0101101 which is another codeword.

CYCLIC CODE


Type of linear block code which only detects errors.

Its computation resembles a long division operation in which the quotient is discarded and the remainder becomes the result.

CYCLIC REDUNDANCY CHECK(CRC)


CRC ENCODER AND DECODER


Encoder on sender’s side generates codeword.

Dataword size is k bits.

Desired codeword is n bits.

Augment dataword by appending n-k 0’s.

Divisor (predefined) of size n-k+1, divides augmented dataword in generator.

Obtained remainder is appended to dataword.


The generated codeword is sent to receiver via some transmission medium.

Decoder on receiver’s side checks for errors.

The checker divides the codeword by the same divisor.

This generates a remainder which is called a syndrome.

If the syndrome is 0 then there is no error or the error is undetected.

If syndrome is non zero, error has been detected and data is discarded.


Cyclic codes can be represented using polynomials.

Special polynomials in which co-efficient can be either 0 or 1.

The bit position of dataword indicates power of the polynomial.

Ex:- 1 0 0 1 1 0 1 1

x7 x6 x5 x4 x3 x2 x1 x0

equivalent polynomial expression x7+x4+x3+x+1.

POLYNOMIAL REPRESENTATION


The given dataword can be represented in polynomial terms.

Multiply the dataword with xn-kto generate augmented dataword.

The augmented dataword is divided by the generator polynomial g(x) and the resulting remainder is added to the augmented dataword.

Note in division when we subtract we actually perform XOR operation.

CODEWORD GENERATION


The divisor on the receiving side divides the received code word and generates a remainder.

Remainder is also called as a syndrome. If the syndrome generated is 0 then there is

no error in transmission or undetected error. Non zero syndrome means that error has

been detected. No error correction is possible using CRC.

ERROR DETECTION


Received codeword can be represented asReceived codeword=c(x)+e(x) where c(x) is original codeword e(x) is the error.

The error is detected if received codeword=c(x)+e(x) is not divisible.

g(x) If e(x) is divisible by g(x) then error goes

undetected. Single bit error:

e(x)=xi.

xi should not be divisible by g(x).x0 term should be 1 so that we can catch

the error.


Two Isolated bit errors: e(x)=xi+xj.

e(x)=xi(1+xj-i) where i<j. let j-i=t

so, e(x)=xi(1+xt) To catch xi the generator should have x0=1. To catch error of 1+xt the generator

polynomial should not divide 1+xt for 0<t<n-1.

Odd number of errorsGenerator polynomial should be a factor of

x+1.


Polynomial should contain more than one term.

Polynomial should have the x0 term equal to 1.

Polynomial should contain x+1 as a factor. Polynomial should not divide 1+xt for

0<t<n-1.

PROPERTIES OF GOOD GENERATOR POLYNOMIAL


ACKNOWLEDGEMENT

•We would like to thank Mydhili Ma’m for giving us an opportunity to present this presentation and for the support extended by her.

•We would also like to thank Mr. Mohan Kumar our project incharge.


QUESTIONS

???

GROUP03_AMAK:ERROR DETECTION AND CORRECTION PPT

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