Full error detection and correction
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This document discusses error detection and correction techniques used at the data link layer. It describes different types of errors that can occur like single-bit and burst errors. Error detection methods like parity checks, cyclic redundancy checks (CRC), and checksums are explained. Forward error correction codes like Hamming codes that allow for error correction are also covered. The document provides examples to illustrate how various error detection and correction schemes work.
Full error detection and correction
- 1. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Data Link LayerData Link Layer PARTPART IIIIII
- 2. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Data link layer duties
- 3. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Chapter 10 Error Detection and Correction
- 4. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Data can be corrupted during transmission. For reliable communication, errors must be detected and corrected. NoteNote::
- 5. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.1 Types of Error10.1 Types of Error Single-Bit Error Burst Error
- 6. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 In a single-bit error, only one bit in the data unit has changed. NoteNote:: Single-Bit Error
- 7. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 A burst error means that 2 or more bits in the data unit have changed. NoteNote:: Burst Error
- 8. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.2 Detection10.2 Detection Repetition Redundancy Parity Check Cyclic Redundancy Check (CRC) Checksum
- 9. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 2/28/03 9 Repetition The simplest form of redundancy is: Repetition! Sender Receiver “0” Did she say “1” ? I said “0” Sounded like “0” One more time: “0” Sounded like “0” again She was sending “0”
- 10. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 2/28/03 10 3-Repetition Code • Encoding rule: Repeat each bit 3 times Example: 1 . 0 . 1 . 111 . 000 . 111 . • Decoding rule: Majority vote! Examples of received codewords: 110 . 000 . 111 . 1 . 0 . 1 . Error-free! 111 . 000 . 010 . 1 . 0 . 0 . Error!
- 11. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 2/28/03 11 How good is this 3-repetition code? • The code can correct 1 bit error per 3-bit codeword. • The price we pay in redundancy is measured by the efficiency or rate of the code, denoted by R: R= #information bits / # bits in codeword • For the 3-repetition code: R=33%
- 12. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 2/28/03 12 How good is this 3-repetition code? Suppose that, on average, the noisy channel flips 1 code bit in 100 Then, on average, the 3-repetition code makes only 1 information bit error in 3333 bits!
- 13. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 2/28/03 13 Can we do better? How about repeat 5 times? On average, only 1 bit error in 100,000 bits. How about repeat 7 times? On average, only 1 bit error in 2,857,142 bits If we let the number of repetitions grow and grow, we can approach perfect reliability !
- 14. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 2/28/03 14 What’s the catch???? The catch is: As the number of repetitions grows to infinity, the transmission rate shrinks to zero!!! This means: slow data transmission / low storage density. Is there a better (more efficient) error correcting code?
- 15. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Error detection uses the concept of redundancy, which means adding extra bits for detecting errors at the destination. NoteNote::
- 16. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.3 Redundancy
- 17. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.4 Detection methods
- 18. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.5 Even-parity concept
- 19. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 In parity check, a parity bit is added to every data unit so that the total number of 1s is even (or odd for odd-parity). NoteNote::
- 20. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 1Example 1 Suppose the sender wants to send the word world. In ASCII the five characters are coded as 1110111 1101111 1110010 1101100 1100100 The following shows the actual bits sent 11101110 11011110 11100100 11011000 11001001
- 21. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 2Example 2 Now suppose the word world in Example 1 is received by the receiver without being corrupted in transmission. 11101110 11011110 11100100 11011000 11001001 The receiver counts the 1s in each character and comes up with even numbers (6, 6, 4, 4, 4). The data are accepted.
- 22. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 3Example 3 Now suppose the word world in Example 1 is corrupted during transmission. 11111110 11011110 11101100 11011000 11001001 The receiver counts the 1s in each character and comes up with even and odd numbers (7, 6, 5, 4, 4). The receiver knows that the data are corrupted, discards them, and asks for retransmission.
- 23. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Simple parity check can detect allSimple parity check can detect all single-bit errors. It can detect burstsingle-bit errors. It can detect burst errors only if the total number oferrors only if the total number of errors in each data unit is odd.errors in each data unit is odd. NoteNote::
- 24. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.6 Two-dimensional parity
- 25. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 4Example 4 Suppose the following block is sent: 10101001 00111001 11011101 11100111 10101010 However, it is hit by a burst noise of length 8, and some bits are corrupted. 10100011 10001001 11011101 11100111 10101010 When the receiver checks the parity bits, some of the bits do not follow the even-parity rule and the whole block is
- 26. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 In two-dimensional parity check, a block of bits is divided into rows and a redundant row of bits is added to the whole block. NoteNote::
- 27. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.7 CRC generator and checker
- 28. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.8 Binary division in a CRC generator
- 29. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.9 Binary division in CRC checker
- 30. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.10 A polynomial
- 31. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.11 A polynomial representing a divisor
- 32. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Table 10.1 Standard polynomialsTable 10.1 Standard polynomials Name Polynomial Application CRC-8CRC-8 x8 + x2 + x + 1 ATM header CRC-10CRC-10 x10 + x9 + x5 + x4 + x2 + 1 ATM AAL ITU-16ITU-16 x16 + x12 + x5 + 1 HDLC ITU-32ITU-32 x32 + x26 + x23 + x22 + x16 + x12 + x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1 LANs
- 33. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 5Example 5 It is obvious that we cannot choose x (binary 10) or x2 + x (binary 110) as the polynomial because both are divisible by x. However, we can choose x + 1 (binary 11) because it is not divisible by x, but is divisible by x + 1. We can also choose x2 + 1 (binary 101) because it is divisible by x + 1 (binary division).
- 34. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 6Example 6 The CRC-12 x12 + x11 + x3 + x + 1 which has a degree of 12, will detect all burst errors affecting an odd number of bits, will detect all burst errors with a length less than or equal to 12, and will detect, 99.97 percent of the time, burst errors with a length of 12 or more.
- 35. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.12 Checksum
- 36. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.13 Data unit and checksum
- 37. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 The sender follows these steps:The sender follows these steps: •The unit is divided into k sections, each of n bits.The unit is divided into k sections, each of n bits. •All sections are added using one’s complement to get the sum.All sections are added using one’s complement to get the sum. •The sum is complemented and becomes the checksum.The sum is complemented and becomes the checksum. •The checksum is sent with the data.The checksum is sent with the data. NoteNote::
- 38. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 The receiver follows these steps:The receiver follows these steps: •The unit is divided into k sections, each of n bits.The unit is divided into k sections, each of n bits. •All sections are added using one’s complement to get the sum.All sections are added using one’s complement to get the sum. •The sum is complemented.The sum is complemented. •If the result is zero, the data are accepted: otherwise, rejected.If the result is zero, the data are accepted: otherwise, rejected. NoteNote::
- 39. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 7Example 7 Suppose the following block of 16 bits is to be sent using a checksum of 8 bits. 10101001 00111001 The numbers are added using one’s complement 10101001 00111001 ------------ Sum 11100010 Checksum 00011101 The pattern sent is 10101001 00111001 00011101
- 40. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 8Example 8 Now suppose the receiver receives the pattern sent in Example 7 and there is no error. 10101001 00111001 00011101 When the receiver adds the three sections, it will get all 1s, which, after complementing, is all 0s and shows that there is no error. 10101001 00111001 00011101 Sum 11111111 Complement 00000000 means that the pattern is OK.
- 41. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Example 9Example 9 Now suppose there is a burst error of length 5 that affects 4 bits. 10101111 11111001 00011101 When the receiver adds the three sections, it gets 10101111 11111001 00011101 Partial Sum 1 11000101 Carry 1 Sum 11000110 Complement 00111001 the pattern is corrupted.
- 42. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.3 Correction10.3 Correction Retransmission Burst Error Correction Automatic-repeat-request (ARQ) Forward error correction (FEC)
- 43. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Table 10.2 Data and redundancy bitsTable 10.2 Data and redundancy bits Number of data bits m Number of redundancy bits r Total bits m + r 11 2 3 22 3 5 33 3 6 44 3 7 55 4 9 66 4 10 77 4 11
- 44. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.14 Positions of redundancy bits in Hamming code
- 45. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.15 Redundancy bits calculation
- 46. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.16 Example of redundancy bit calculation
- 47. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.17 Error detection using Hamming code
- 48. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 10.18 Burst error correction example
- 49. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Automatic-repeat request (ARQ): ARQ procedures require the transmitter to resend the portions of the exchange in which error have been detected. Generally, ARQ procedures include the following actions by the receiver or the sender:
- 50. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Receiver: Discard those frames in which errors are detected. For frames in which no error was detected, the receiver returns a positive acknowledgment to the sender. For the frame in which errors have been detected, the receiver returns negative acknowledgement to the sender.
- 51. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Sender: Retransmit the frames in which the receiver has identified errors. After a pre-established time, the sender retransmits a frame that has not been acknowledged.
- 52. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Three Common ARQ Techniques are: Stop-and-Wait Go-back-n Selective-repeat
- 53. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques • Stop-and-Wait The sender sends a frame and waits for acknowledgment from the receiver. This technique is slow Suited for half-duplex connection.
- 54. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques •Stop-and-Wait
- 55. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Go-back-n: The sender sends frames in a sequence and receives acknowledgements from the receiver. On detecting an error, the receiver discards the corrupted frame, and ignores any further frames. The receiver notifies the sender of the number of frame it expects to receive.
- 56. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Go-back-n: On receipt of information, the sender begins re-sending the data sequence starting from that frame. This technique is faster than stop-and-wait technique.
- 57. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 ARQ Techniques Selective-repeat: Used on duplex connections. The sender only repeats those frames for which negative acknowledgment are received from the receiver, or no acknowledgment is received. The appearance of a repeated frame out of sequence may provide the receiver with additional complications.
- 58. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Forward error correction (FEC): FEC techniques employ special codes that allow the receiver to detect and correct a limited number of errors without referring to the transmitter. Possible for the receiver to detect and correct errors without reference to the sender. This convenience is bought at the expense of adding more bits.
- 59. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction For example If we only have two massages to send. we represent one (A) by the bits 10101010, And the other (B) by the bits 01010101. If the receiver knows that the message is A or B and no other, and it is provided with the ability to determine the logical distance between each incoming massage and the two known messages, this strategy will allow the receiver to correct for up to three bits in error. The prove is as follows.
- 60. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Proof: Suppose A is in error by 1 bit, so that A’ = 00101010 The logical distance between the received pattern and A is 1 And logical distance between the received pattern and B is 7; Thus A’ is likely to be A.
- 61. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Suppose A is in error by 2 bit, so that A’ = 01101010 The logical distance between the received pattern and A is 2 And logical distance between the received pattern and B is 6; Thus A’ is likely to be A.
- 62. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Suppose A is in error by 3 bit, so that A’ = 01001010 The logical distance between the received pattern and A is 3 And logical distance between the received pattern and B is 5; Thus A’ is likely to be A.
- 63. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Suppose A is in error by 4 bit, so that A’ = 01011010 The logical distance between the received pattern and A is 4 And logical distance between the received pattern and B is 4; Thus A’ is likely to be A or B.
- 64. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Continuing the sequence to higher levels of error makes A’ more likely to be B than A. For this particular case, the limit of correction is 3- bits in error.
- 65. McGraw-Hill ©The McGraw-Hill Companies, Inc., 2004 Forward Error Correction Codes used to provide FEC (Forward Error Correction) are more sophisticated than our example. They can be divided into two types. Linear Block Codes Convolutional Codes