This document discusses error detection and correction techniques. It introduces concepts like redundancy, forward error correction, retransmission, linear block coding, Hamming codes, and cyclic redundancy checks (CRC). Specific error correction codes covered include Hamming codes, which use modulo-2 arithmetic to add redundant bits to detect and correct single bit errors. CRC codes are also discussed, which use cyclic codes and polynomial representations to detect errors by computing a syndrome value. The document provides information on generating and detecting errors for different codes.

1. ERROR DETECTION&ERROR CORRECTION1MSRIT INFORMATION SCIENCE

2. AMAKA-> ANKITA (1MS07IS133)M-> MAYANK (1MS07IS047)A-> ANSHUJ (1MS07IS011)K-> KRISH (1MS07IS038)2MSRIT INFORMATION SCIENCE

3. INTRODUCTION TO ERRORREDUNDANCYCODINGLINEAR BLOCK CODING

4. HAMMING CODE

5. CYCLIC REDUNDANCY CHECK(CRC)

6. IMPLEMENTATION OF HAMMING CODETABLE OF CONTENTS:3MSRIT INFORMATION SCIENCE

7. What is an error???Unpredictable change of bits from 1->0 or 0->1.TypesSingle bit error

8. Burst error(Multiple)

9. Redundancy: Correction or detection of errors. INTRODUCTION 4MSRIT INFORMATION SCIENCE

10. 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.

11. Modulo 3- Remainder after division can be either 0,1 or 2....Modulo n- Remainder after division can be either 0,1,2….n-1.5MSRIT INFORMATION SCIENCE

12. Adding : 0+0=0 0+1=1 1+0=1 1+1=0Subt: 0-0=0 0-1=1 1-0=1 1-1=0XOR operation: 1 0 1 1 01 1 1 0 0 0 1 0 1 06MSRIT INFORMATION SCIENCE

13. CODING: Redundancy is achieved through coding.BLOCK CODING: Message divided into blocks.

14. k bit datawords

15. r redundant bits

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

17. 2k total datawords(equal to number of valid codewords.)

18. 2n total codewords

19. 2n -2k invalid codewords7MSRIT INFORMATION SCIENCE

20. ERROR DETECTIONIf resulting codeword is invalid.

21. If multiple errors in the codeword result in valid codeword.ERROR CORRECTIONMore difficult.

22. Need more number of redundant bits than for detection.

23. Involves error detection as well as finding the position(s) where error has occurred.8MSRIT INFORMATION SCIENCE

24. Generation of codewords for each dataword: C(7,4) n ,kCodeword is generated by the generator which appends 3 redundant bits at the end of the dataword.Ro =a2 + a1 + a0 Modulo-2R1= a3 + a2 + a1 Modulo-2R2= a1+ a0 + a3 Modulo-2HAMMING CODE GENERATION9MSRIT INFORMATION SCIENCE

25. A CRC CODE WITH C(7,4)10MSRIT INFORMATION SCIENCE

26. 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-2If the syndrome i.e s2s1s0 is 000 then data is error free…or undetected.Otherwise, received codeword has an error(s).11MSRIT INFORMATION SCIENCE

27. MAGIC TABLEDepending 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.12MSRIT INFORMATION SCIENCE

28. CODEWORD NOTATION ON SENDER’S AND RECEIVER’S SIDEa3 a2 a1 a0 R2 R1 R0b3b2 b1 b0 q2 q1 q0 13MSRIT INFORMATION SCIENCE

29. 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 1The total number of ones is equal to the number of bit changes between two codewords. HAMMING DISTANCE14MSRIT INFORMATION SCIENCE

30. 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 DISTANCE15MSRIT INFORMATION SCIENCE

31. 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.16MSRIT INFORMATION SCIENCE

32. 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 CODE17MSRIT INFORMATION SCIENCE

33. 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)18MSRIT INFORMATION SCIENCE

34. CRC ENCODER AND DECODER19MSRIT INFORMATION SCIENCE

35. 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. 20MSRIT INFORMATION SCIENCE

36. 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.21MSRIT INFORMATION SCIENCE

37. 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 1x7 x6 x5 x4 x3 x2 x1 x0 equivalent polynomial expression x7+x4+x3+x+1.POLYNOMIAL REPRESENTATION22MSRIT INFORMATION SCIENCE

38. 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 GENERATION23MSRIT INFORMATION SCIENCE

39. 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 DETECTION24MSRIT INFORMATION SCIENCE

40. 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.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.25MSRIT INFORMATION SCIENCE

41. 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.

42. To catch error of 1+xt the generator polynomial should not divide 1+xt for 0

43. Odd number of errorsGenerator polynomial should be a factor of x+1.26MSRIT INFORMATION SCIENCE

44. 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 POLYNOMIAL27MSRIT INFORMATION SCIENCE

45. MSRIT INFORMATION SCIENCE28ACKNOWLEDGEMENTWe would like to thank MydhiliMa’m for giving us an opportunity to present this presentation and for the support extended by her.

46. We would also like to thank Mr. Mohan Kumar our project incharge.MSRIT INFORMATION SCIENCE29QUESTIONS???