GROUP03_AMAK:ERROR DETECTION AND CORRECTION PPT

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GROUP03_AMAK:ERROR DETECTION AND CORRECTION PPT

  1. 1. ERROR DETECTION&ERROR CORRECTION<br />1<br />MSRIT INFORMATION SCIENCE<br />
  2. 2. AMAK<br />A-&gt; ANKITA (1MS07IS133)<br />M-&gt; MAYANK (1MS07IS047)<br />A-&gt; ANSHUJ (1MS07IS011)<br />K-&gt; KRISH (1MS07IS038)<br />2<br />MSRIT INFORMATION SCIENCE<br />
  3. 3. INTRODUCTION TO ERROR<br />REDUNDANCY<br />CODING<br /><ul><li>LINEAR BLOCK CODING
  4. 4. HAMMING CODE
  5. 5. CYCLIC REDUNDANCY CHECK(CRC)
  6. 6. IMPLEMENTATION OF HAMMING CODE</li></ul>TABLE OF CONTENTS:<br />3<br />MSRIT INFORMATION SCIENCE<br />
  7. 7. What is an error???<br />Unpredictable change of bits from 1-&gt;0 or 0-&gt;1.<br />Types<br /><ul><li>Single bit error
  8. 8. Burst error(Multiple)
  9. 9. Redundancy: Correction or detection of errors. </li></ul>INTRODUCTION <br />4<br />MSRIT INFORMATION SCIENCE<br />
  10. 10. Forward error correction: Method of Guessing the actual message using the redundant bits.<br />Retransmission: Repeated sending of message until error free.<br />Modulo Arithmetic:<br /><ul><li>Modulo 2- Remainder after division can be either 0 or1.
  11. 11. Modulo 3- Remainder after division can be either 0,1 or 2.</li></ul>.<br />.<br />.<br /><ul><li>Modulo n- Remainder after division can be either 0,1,2….n-1.</li></ul>5<br />MSRIT INFORMATION SCIENCE<br />
  12. 12. Adding : 0+0=0 0+1=1 1+0=1 1+1=0<br />Subt: 0-0=0 0-1=1 1-0=1 1-1=0<br />XOR operation:<br /> 1 0 1 1 0<br />1 1 1 0 0<br /> 0 1 0 1 0<br />6<br />MSRIT INFORMATION SCIENCE<br />
  13. 13. CODING: Redundancy is achieved through coding.<br /><ul><li>BLOCK CODING: Message divided into blocks.
  14. 14. k bit datawords
  15. 15. r redundant bits
  16. 16. n= k+r, n>k,n bit codeword.
  17. 17. 2k total datawords(equal to number of valid codewords.)
  18. 18. 2n total codewords
  19. 19. 2n -2k invalid codewords</li></ul>7<br />MSRIT INFORMATION SCIENCE<br />
  20. 20. ERROR DETECTION<br /><ul><li>If resulting codeword is invalid.
  21. 21. If multiple errors in the codeword result in valid codeword.</li></ul>ERROR CORRECTION<br /><ul><li>More difficult.
  22. 22. Need more number of redundant bits than for detection.
  23. 23. Involves error detection as well as finding the position(s) where error has occurred.</li></ul>8<br />MSRIT INFORMATION SCIENCE<br />
  24. 24. Generation of codewords for each dataword:<br /> C(7,4)<br /> n ,k<br />Codeword is generated by the generator which appends 3 redundant bits at the end of the dataword.<br />Ro =a2 + a1 + a0 Modulo-2<br />R1= a3 + a2 + a1 Modulo-2<br />R2= a1+ a0 + a3 Modulo-2<br />HAMMING CODE GENERATION<br />9<br />MSRIT INFORMATION SCIENCE<br />
  25. 25. A CRC CODE WITH C(7,4)<br />10<br />MSRIT INFORMATION SCIENCE<br />
  26. 26. checker on the receiver side will generate a 3bit syndrome by the formulae given below:<br /> s0 = b2 + b1 + b0 + q0 modulo-2<br /> s1 = b3 + b2 + b1 + q1 modulo-2<br /> s2 = b1 + b0 + b3 + q2 modulo-2<br />If the syndrome i.e s2s1s0 is 000 then data is error free…or undetected.<br />Otherwise, received codeword has an error(s).<br />11<br />MSRIT INFORMATION SCIENCE<br />
  27. 27. MAGIC TABLE<br />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.<br />12<br />MSRIT INFORMATION SCIENCE<br />
  28. 28. CODEWORD NOTATION ON SENDER’S AND RECEIVER’S SIDE<br />a3 a2 a1 a0 R2 R1 R0<br />b3b2 b1 b0 q2 q1 q0 <br />13<br />MSRIT INFORMATION SCIENCE<br />
  29. 29. Number of bit change occurring between two codewords.<br /> 0 1 1 1 0 1 0<br />1 0 1 1 1 1 1<br /> 0 1 0 0 1 0 1<br />The total number of ones is equal to the number of bit changes between two codewords. <br />HAMMING DISTANCE<br />14<br />MSRIT INFORMATION SCIENCE<br />
  30. 30. Smallest Hamming Distance between all sets of codewords.<br />Ex- <br />d(0000000,0001101) = 3<br />d(0001100,0111001) = 4<br />d(0110100,0111001) =3<br />d(11111111,0000000) =7…. & so on..<br />Dmin= 3 for the above set of codewords.<br />MINIMUM HAMMING DISTANCE<br />15<br />MSRIT INFORMATION SCIENCE<br />
  31. 31. Suppose ‘s’ errors are to be detected, then dmin should be s+1.<br /> for the example taken, it can detect upto a maximum of 2 errors.<br />Suppose ‘t’ errors are to be corrected, then the dmin should be 2t+1. <br /> in the above example, it can correct upto only 1 error.<br />16<br />MSRIT INFORMATION SCIENCE<br />
  32. 32. Linear block code?<br />Linear block code with an extra property: code word is cyclically rotated that generates another codeword.<br />1010110 is a codeword on rotating <br /> 0101101 which is another codeword.<br />CYCLIC CODE<br />17<br />MSRIT INFORMATION SCIENCE<br />
  33. 33. Type of linear block code which only detects errors.<br />Its computation resembles a long division operation in which the quotient is discarded and the remainder becomes the result.<br />CYCLIC REDUNDANCY CHECK(CRC)<br />18<br />MSRIT INFORMATION SCIENCE<br />
  34. 34. CRC ENCODER AND DECODER<br />19<br />MSRIT INFORMATION SCIENCE<br />
  35. 35. Encoder on sender’s side generates codeword.<br />Dataword size is k bits.<br />Desired codeword is n bits.<br />Augment dataword by appending n-k 0’s.<br />Divisor (predefined) of size n-k+1, divides augmented dataword in generator.<br />Obtained remainder is appended to dataword. <br />20<br />MSRIT INFORMATION SCIENCE<br />
  36. 36. The generated codeword is sent to receiver via some transmission medium.<br />Decoder on receiver’s side checks for errors.<br />The checker divides the codeword by the same divisor.<br />This generates a remainder which is called a syndrome.<br />If the syndrome is 0 then there is no error or the error is undetected.<br />If syndrome is non zero, error has been detected and data is discarded.<br />21<br />MSRIT INFORMATION SCIENCE<br />
  37. 37. Cyclic codes can be represented using polynomials.<br />Special polynomials in which co-efficient can be either 0 or 1.<br />The bit position of dataword indicates power of the polynomial.<br />Ex:- 1 0 0 1 1 0 1 1<br />x7 x6 x5 x4 x3 x2 x1 x0<br /> equivalent polynomial expression x7+x4+x3+x+1.<br />POLYNOMIAL REPRESENTATION<br />22<br />MSRIT INFORMATION SCIENCE<br />
  38. 38. The given dataword can be represented in polynomial terms.<br />Multiply the dataword with xn-kto generate augmented dataword.<br />The augmented dataword is divided by the generator polynomial g(x) and the resulting remainder is added to the augmented dataword.<br />Note in division when we subtract we actually perform XOR operation.<br />CODEWORD GENERATION<br />23<br />MSRIT INFORMATION SCIENCE<br />
  39. 39. The divisor on the receiving side divides the received code word and generates a remainder.<br />Remainder is also called as a syndrome.<br />If the syndrome generated is 0 then there is no error in transmission or undetected error.<br />Non zero syndrome means that error has been detected.<br />No error correction is possible using CRC.<br />ERROR DETECTION<br />24<br />MSRIT INFORMATION SCIENCE<br />
  40. 40. Received codeword can be represented as<br /> Received codeword=c(x)+e(x) where c(x) is original codeword <br /> e(x) is the error.<br />The error is detected if<br /> received codeword=c(x)+e(x) is not divisible.<br /> g(x)<br />If e(x) is divisible by g(x) then error goes undetected.<br /><ul><li>Single bit error:</li></ul>e(x)=xi.<br />xi should not be divisible by g(x).<br />x0 term should be 1 so that we can catch the error.<br />25<br />MSRIT INFORMATION SCIENCE<br />
  41. 41. Two Isolated bit errors:<br /> e(x)=xi+xj.<br /> e(x)=xi(1+xj-i) where i&lt;j.<br /> let j-i=t<br /> so, e(x)=xi(1+xt)<br /><ul><li>To catch xi the generator should have x0=1.
  42. 42. To catch error of 1+xt the generator polynomial should not divide 1+xt for 0<t<n-1.
  43. 43. Odd number of errors</li></ul>Generator polynomial should be a factor of x+1.<br />26<br />MSRIT INFORMATION SCIENCE<br />
  44. 44. Polynomial should contain more than one term.<br />Polynomial should have the x0 term equal to 1.<br />Polynomial should contain x+1 as a factor.<br />Polynomial should not divide 1+xt for 0&lt;t&lt;n-1.<br />PROPERTIES OF GOOD GENERATOR POLYNOMIAL<br />27<br />MSRIT INFORMATION SCIENCE<br />
  45. 45. MSRIT INFORMATION SCIENCE<br />28<br />ACKNOWLEDGEMENT<br /><ul><li>We would like to thank MydhiliMa’m for giving us an opportunity to present this presentation and for the support extended by her.
  46. 46. We would also like to thank Mr. Mohan Kumar our project incharge.</li></li></ul><li>MSRIT INFORMATION SCIENCE<br />29<br />QUESTIONS<br />???<br />

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