Error Detection (1)(4)

ERROR DETECTION & ERROR CORRECTION

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AMAK A-> ANKITA (1MS07IS133) M-> MAYANK (1MS07IS047) A-> ANSHUJ (1MS07IS011) K-> KRISH (1MS07IS038)

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TABLE OF CONTENTS: 

INTRODUCTION TO ERROR



REDUNDANCY



CODING

• LINEAR BLOCK CODING • HAMMING CODE • CYCLIC REDUNDANCY CHECK(CRC) • IMPLEMENTATION OF HAMMING CODE •

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INTRODUCTION  •

What is an error???

Unpredictable change of bits from 1->0 or 0->1.

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

Types

o Single bit error o Burst error(Multiple) o ØRedundancy: Correction or detection of errors.  MSRIT INFORMATION SCIENCE

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

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. 

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

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

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

CODING: Redundancy is achieved through coding.

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qBLOCK CODING: Message divided into blocks. •

k bit datawords

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r redundant bits

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n= k+r, n>k,n bit codeword.

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2k total datawords(equal to number of valid codewords.)

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2n total codewords

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2n -2k invalid codewords MSRIT INFORMATION SCIENCE

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

ERROR DETECTION

o If resulting codeword is invalid. o If multiple errors in the codeword result in valid codeword. •



ERROR CORRECTION

o 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. MSRIT INFORMATION SCIENCE

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HAMMING CODE GENERATION  •

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. •R Modulo-2 o =a2 + a1 + a0 

R1= a3 + a2 + a1

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R2= a1+ a0 + a3

• •

Modulo-2

Modulo-2 MSRIT INFORMATION SCIENCE

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A CRC CODE WITH C(7,4) DATAWO RD 0000 0001 0010 0011 0100 0101 0110 0111

CODEWOR DATAWOR CODEWORD D D 0000000 1000 1000110 0001101 1001 1001011 0010111 1010 1010001 0011010 1011 1011100 0100011 1100 1100101 0101110 1101 1101000 0110100 1110 1110010 0111001 1111 1111111

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 

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

•

s0 = b2 + b1 + b0 + q0

modulo-2

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s1 = b3 + b2 + b1 + q1

modulo-2

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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). MSRIT INFORMATION SCIENCE

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•

MAGIC TABLE

• •

SYNDRO 000 001 010 011 100 101 • ME ERROR None q0 q1 b2 q2 b0

110 111 b3 b1

• • •

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.

•

• • • •

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CODEWORD NOTATION ON SENDER’S AND RECEIVER’S SIDE 

CODEWO 1 RD SENT

1 0 1 0 0 0

a3 a2 a1 a0 R2 R1 R0

RECEIVED 0 CODEWO RD

1 0 1 0 0 0

b3 b2 b1 b0 q2 q1 q0

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HAMMING DISTANCE 

• • •

Number of bit change occurring between two codewords. 0111010 10111 11 0100101

• •

The total number of ones is equal to the number of bit changes between two codewords.

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• •

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MINIMUM HAMMING DISTANCE 

Smallest Hamming Distance between all sets of codewords.

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Ex-

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d(0000000,0001101) = 3

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d(0001100,0111001) = 4

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d(0110100,0111001) =3

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d(11111111,0000000) =7…. & so on..

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Dmin = 3 for the above set of codewords. MSRIT INFORMATION SCIENCE

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

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

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

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

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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.

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CYCLIC CODE 

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.

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CYCLIC REDUNDANCY CHECK (CRC)  

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.

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CRC ENCODER AND DECODER

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

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.

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

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Obtained remainder is appended to dataword. MSRIT INFORMATION SCIENCE

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

 

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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.

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POLYNOMIAL REPRESENTATION 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. MSRIT INFORMATION SCIENCE

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CODEWORD GENERATION 

The given dataword can be represented in polynomial terms.

 

Multiply the dataword with xn-k to generate augmented dataword.

 

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

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Note in division when we subtract we actually perform XOR operation. MSRIT INFORMATION SCIENCE

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ERROR DETECTION 

 



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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. MSRIT INFORMATION SCIENCE

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Received codeword can be represented as • Received 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. ØSingle 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. 

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 •

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)

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§ 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
Generator polynomial should be a factor of x+1.

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PROPERTIES OF GOOD GENERATOR POLYNOMIAL

Polynomial term.  Polynomial 1.  Polynomial  Polynomial 0
should contain more than one should have the x0 term equal to should contain x+1 as a factor. should not divide 1+xt for

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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. MSRIT INFORMATION SCIENCE

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QUESTIONS

??? MSRIT INFORMATION SCIENCE

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