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Data and Telecommunication Problem

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University of Dhaka
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING,
1Data And Telecommunications
Presentation on:
CHAPTER 10 (ERROR DETECTION AND CORRECTION)
By: Md. Al – Zihad
Roll: 35
CSEDU 20th Batch
To: Professor Dr. Md. Abdur Razzaque
CSEDU
University of Dhaka
Contents
6/7/2016
2
• General Discussion of CRC
• Figure out the problems assigned
• Developing a perfect solution for each of them
Cyclic Redundancy Check (CRC)
 Error detection mechanism
 Simple to implement in binary hardware
 Calculated by performing a modulo 2 division of the data by a generator
polynomial and recording the remainder after division.
6/7/2016
3
Problem 27
27. Referring to the CRC-8 polynomial in Table 10.7, answer the following
questions:
 a. Does it detect a single error? Defend your answer.
 b. Does it detect a burst error of size 6? Defend your answer.
 c. What is the probability of detecting a burst error of size 9?
 d. What is the probability of detecting a burst error of size 15?
6/7/2016
4
Resources
6/7/2016
5
Problem 28
28. Referring to the CRC-32 polynomial in Table 10.7, answer the following
questions:
 a. Does it detect a single error? Defend your answer.
 b. Does it detect a burst error of size 16? Defend your answer.
 c. What is the probability of detecting a burst error of size 33?
 d. What is the probability of detecting a burst error of size 55?
6/7/2016
6

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Data and Telecommunication Problem

  • 1. University of Dhaka DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING, 1Data And Telecommunications Presentation on: CHAPTER 10 (ERROR DETECTION AND CORRECTION) By: Md. Al – Zihad Roll: 35 CSEDU 20th Batch To: Professor Dr. Md. Abdur Razzaque CSEDU University of Dhaka
  • 2. Contents 6/7/2016 2 • General Discussion of CRC • Figure out the problems assigned • Developing a perfect solution for each of them
  • 3. Cyclic Redundancy Check (CRC)  Error detection mechanism  Simple to implement in binary hardware  Calculated by performing a modulo 2 division of the data by a generator polynomial and recording the remainder after division. 6/7/2016 3
  • 4. Problem 27 27. Referring to the CRC-8 polynomial in Table 10.7, answer the following questions:  a. Does it detect a single error? Defend your answer.  b. Does it detect a burst error of size 6? Defend your answer.  c. What is the probability of detecting a burst error of size 9?  d. What is the probability of detecting a burst error of size 15? 6/7/2016 4
  • 6. Problem 28 28. Referring to the CRC-32 polynomial in Table 10.7, answer the following questions:  a. Does it detect a single error? Defend your answer.  b. Does it detect a burst error of size 16? Defend your answer.  c. What is the probability of detecting a burst error of size 33?  d. What is the probability of detecting a burst error of size 55? 6/7/2016 6
  • 7. Detecting Single Bit Error 6/7/2016 7 Lets Assume, Dataword = d(x) Codeword = c(x) Error = e(x) So, Received Codeword = c(x)+e(x) Generator = g(x)
  • 8. Detecting Single Bit Error Cont… 6/7/2016 8 • If a single-bit error is caught, then xi is not divisible by g(x). • If the generator has more than one term and the coefficient of x0 is 1, all single errors can be caught
  • 9. Detecting Single Bit Error Cont… 6/7/2016 9 Both In, The generator has more than one term and the coefficient of x0 is 1, all single errors can be caught and
  • 10. Detecting Burst Error 6/7/2016 10 e(x) = (x^j + . . . + x^i) Or, e(x) = x^i (x^ (j-i) + . . . + 1) If our generator can detect a single error (minimum condition for a generator), then it cannot divide xi.
  • 11. Detecting Burst Error Cont… 6/7/2016 11 So our concern is: (x^ (j-i) + . . . + 1) / ( x^r + . . . +1) If the division has non zero remainder, error can be detected
  • 12. Detecting Burst Error Cont… 6/7/2016 12 • If j - i < r, the remainder can never be zero. • all burst errors with length smaller than or equal to the number of check bits r will be detected.
  • 13. Detecting Burst Error Cont… 6/7/2016 13 Here in problem 27(b), • This is a 8 degree polynomial. • Number of check bit is 8 So, it will obviously detect burst error size of 6 as 6 < 8
  • 14. Detecting Burst Error Cont… 6/7/2016 14 Here in problem 27(b), • This is a 32 degree polynomial. • Number of check bit is 32 So, it will obviously detect burst error size of 16 as 16 < 32
  • 15. Probability of missing detection of Burst error 6/7/2016 15 • If, j - i = r • Then, L – 1 = r • So, L = r + 1 Here, r is polynomial size L is for burst error size In this case, Probability = (1/2)^(r-1)
  • 16. Detecting Burst Error Cont… 6/7/2016 16 Here in problem 27(c), So, L = r + 1 So, it will miss to detect (1/2)^(8-1) burst errors. • r = 8 • L = 9 Probability of detecting burst error of size 9 = 1 – (1/2)^7.
  • 17. Detecting Burst Error Cont… 6/7/2016 17 Here in problem 28(c), So, L = r + 1 So, it will miss to detect (1/2)^(32-1) burst errors. • r = 32 • L = 33 Probability of detecting burst error of size 33 = 1 – (1/2)^31.
  • 18. Probability of missing detection of Burst error 6/7/2016 18 • If, j - i > r • Then, L – 1 > r • So, L > r + 1 Here, r is polynomial size L is for burst error size In this case, Probability = (1/2)^(r)
  • 19. Detecting Burst Error Cont… 6/7/2016 19 Here in problem 27(d), So, L > r + 1 So, it will miss to detect (1/2)^(8) burst errors. • r = 8 • L = 15 Probability of detecting burst error of size 15 = 1 – (1/2)^8.
  • 20. Detecting Burst Error Cont… 6/7/2016 20 Here in problem 28(c), So, L > r + 1 So, it will miss to detect (1/2)^(32) burst errors. • r = 32 • L = 55 Probability of detecting burst error of size 33 = 1 – (1/2)^32.
  • 21. Lesson Learned 6/7/2016 21 • All g(x) having multiple terms and X0 = 1, all single bit error will be detected • All burst errors with L < = r will be detected. • All burst errors with L = r + 1 will be detected with probability 1 - (1/2)^(r-1) • All burst errors with L > r + 1 will be detected with probability 1- (1/2)^r