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Transmission Errors Error Detection and Correction

Transmission Errors Error Detection and Correction. Transmission Errors Outline. Error Detection V.S. Error Correction Hamming Distances and Codes Parity Checksum Polynomial Codes Cyclic Redundancy Checking (CRC) Properties for Detecting Errors with Generating Polynomials. Error Control.

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Transmission Errors Error Detection and Correction

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  1. Transmission ErrorsError Detection and Correction

  2. Transmission Errors Outline • Error Detection V.S. Error Correction • Hamming Distances and Codes • Parity • Checksum • Polynomial Codes • Cyclic Redundancy Checking (CRC) • Properties for Detecting Errors with Generating Polynomials Advanced Computer Networks Transmission Errors

  3. Error Control Error control mechanisms are techniques (e.g., Checksum, CRC) that enable reliable delivery of data over unreliable communication channels

  4. Error Detection and Correction • Error detection • Error detection refers to a class of techniques for detectinggarbled messages • Adding some Extra bits to detect Occurance of error • Not enough to detect the position of errors • Error correction • Adding enough redundant bits to deduce what the correct bits are • Error Correction are too expensive and hard Advanced Computer Networks Transmission Errors

  5. Error Detection and Correction • Error Detection +ARQ • Automatic Repeat reQuest (ARQ) • When only Error Detection is possible • Error is detected • At data link layer, information is automatically resent • Types of ARQ protocols • Stop-and-wait ARQ • Go-Back-N ARQ • Selective Repeat ARQ • Sliding Window Advanced Computer Networks Transmission Errors

  6. Error Detection and Correction • ARQ • A.K.A Automatic Repeat Query • Using Acknowledgement (Ack) (Ack is the messages sent by receiver indicating the correction of reception) • Using Timeout (A specified time period of time allowed to elapse before an acknowledgment is to be received) • Sender automaticallyre-transmits the frame/packet until the reception of acknowledgement (or exceeds a predefined number of re-transmissions) Advanced Computer Networks Transmission Errors

  7. Error Detection and Correction Single error V.S. Burst error • Errors • Single-bit error • Only one bit in the given data is in error • Easy to correct • Burst error • Two or more (continuous) bits from a data are in error. The length of burst is measured from the first corrupted bit to the last one • Very common • Hard to correct Advanced Computer Networks Transmission Errors

  8. Error Detection and Correction Single error V.S. Burst error • Example: • Error rate=0.001 • Frame size=1000 bits • Message length= 100 frame • Burst error length=100 bits • Single bit error • All frames might be corrupted averagely • Burst error • Only one frame is corrupted averagely Advanced Computer Networks Transmission Errors

  9. Hamming Codes • Code Word A legal data word consisting of mdata bits and rredundant bits. • Code System The set of all code words • Code Word Distance The number of bit positions in which two code words differ • Code System Distance The Smallest (minimum possible) distance between every pair of words of a code system. Advanced Computer Networks Transmission Errors

  10. Hamming Codes • Error Detection Mechanism • Comparison of received message with code system • If it Matches a code word There is no error (not always) Else Error is detected • Some errors change a legal code to another legal code (not detected) • Correction to the nearest legal code, if possible Advanced Computer Networks Transmission Errors

  11. o o o o x x x o x o x o x x o o o o o o x x o o o o x x x o o o o o o x x o Hamming Distance Error detection is possible only if the error changes a legal code word to an illegal A code with good distance properties (b) (a) A code with poor distance properties x = codewordso = non-codewords

  12. Hamming Codes • To detect dsingle-bit errors, a (d+1) code distance is needed • In a code system with distance d, (d-1)-bits errors can be detected at most. • Example: 00000000 00001111 11110000 11111111 Code distance=4 00001111 single bit error 10001111 (illegal) 00001111 two bit errors 11001111 (illegal) 00001111 three bit errors 11101111 (illegal) 00001111 four bit errors 11111111 (legal) Advanced Computer Networks Transmission Errors

  13. Hamming Codes • To correct d single-bit errors, a (2d+1) code distance is needed. • In a code system with distance d, (d-1)/2-bits errors can be corrected at most. • Example: 0000000000 0000011111 1111100000 1111111111 Code Distance= 5 Can Detect & Correct 1, 2 errors Can Detect but Not correct 3,4 errors Can't Detect 5 errors. Advanced Computer Networks Transmission Errors

  14. Error Detection • If data is m bit and total number of bits is m+r, then to correct single bit errors r is defined as 2r m + r + 1 (why?) Example: For m =7(ASCII) , the smallest r value that can satisfy above equation is 4 24 7 + 4 + 1 Advanced Computer Networks Transmission Errors

  15. Hamming Code Positions of Redundant bits in Hamming Code Example for 7 bits data r1 = bits [3, 5, 7, 9, 11] r2 = bits [3, 6, 7, 10, 11] r4 = bits [5, 6, 7] r8 = bits [9, 10, 11] Advanced Computer Networks Transmission Errors

  16. Hamming CodeCalculation of Redundant Bits Advanced Computer Networks Transmission Errors

  17. Hamming CodeCalculation of Redundant Bits Advanced Computer Networks Transmission Errors

  18. Hamming CodeError Detection & Correction R1 R2 Advanced Computer Networks Transmission Errors

  19. Hamming CodeError Detection & Correction R3 R4 Advanced Computer Networks Transmission Errors

  20. Hamming CodeError Detection & Correction The binary number R4R3R2R1 represents the position of the happened error in sequence (m+r) bit. Advanced Computer Networks Transmission Errors

  21. Single Parity Check Code Horizontal Parity Check 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 00 0 0 1 1 0 0 1 0 0 1 011 1 0 Parity bits for each Row Even Parity Check for each row Parity bit=XOR of all data bits Detection of all Odd bit errors Detection of Even bit errors is impossible Error detection rate=50% Advanced Computer Networks Transmission Errors

  22. Two-dimensional Parity Check Code Horizontal Vertical Parity Check 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 check bits for each Row Check bit for each Column Advanced Computer Networks Transmission Errors

  23. Multiple Errors 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 Two errors One error 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 Four errors Three errors Advanced Computer Networks Transmission Errors

  24. Checksum • Data is divided into K sections (words), each of n bits • All K sections are added (discarding any overflow) • Checksum= Two's complement of the total • Receiver adds all the words including the checksum • If the result is a word full of zeros, there is no error • Otherwise, error is detected. Advanced Computer Networks Transmission Errors

  25. Checksum • Example: • Data=ABC • A=65 (0041), B=66(0042), C=67(0043) • Checksum= 65+66+67=198 • Or 0041+0042+0043=00C6 • Checksum= 003A • Transmitting data= 0041, 0042, 0043, 003A Advanced Computer Networks Transmission Errors

  26. Polynomial Codes • A polynomial can be represented by a binary word • Each bit from right to left is mapped onto a power term. • The rightmost bit represents the “0” power term. The bit next to it the “1” power term, etc. • If the bit is of value zero, the power term is deleted from the expression. Advanced Computer Networks Transmission Errors

  27. CRC Codeword n bit codeword n-k check bits k information bits Advanced Computer Networks Transmission Errors

  28. Polynomial Codes • A polynomial of degree k-1can be represented by ak-bit bit string • At each position i • Bit 0=Coefficient 0x • Bit 1=Coefficient x(i-1) • i(x) = ik-1xk-1 + ik-2xk-2+ … + i1x + i0 Example: i(x) = x6+ x4 + x3 1 0 1 1 0 0 0 Advanced Computer Networks Transmission Errors

  29. Polynomial Notation • Let the codeword be n bits with k bits information bits and n-k redundant bits (check bits). • We need a generator polynomial g(x)of degree n-k of the form g(x) = 1xn-k + gn-k-1xn-k-1 + gn-k-2xn-k-2 +… + g1x + 1 Note – the first and last coefficient are always 1. Advanced Computer Networks Transmission Errors

  30. Polynomial ArithmeticModule 2 Arithemetic Addition: Multiplication: = q(x)quotient x3 + x2 + x Division: x3 + x+ 1 ) x6 + x5 x6 + x4 + x3 dividend divisor x5 + x4 + x3 x5 + x3 + x2 x4 + x2 x4 + x2 + x x = r(x)remainder Advanced Computer Networks Transmission Errors

  31. CRC Algorithm CRC Steps: 1) Multiply i(x) by xn-k Putting zeros in (n-k) low order positions 2) Dividing i(x).xn-k by g(x) 3) Adding remainder r(x) to i(x).xn-k i(x). xn-k+ r(x) = g(x) q(x) quotient remainder i(x). xn-k= g(x) q(x) + r(x) b(x) = i(x). xn-k+ r(x) transmitted codeword Advanced Computer Networks Transmission Errors

  32. CRC Algorithm • b(x)/g(x)=0 • b(x) is dividable by g(x) • Let y(x) denotes the received Data at receiver • Let e(x) denotes the Error mixed with sent data • y(x)=b(x)+e(x) (message with error) • y(x)/g(x)= b(x)/g(x)+e(x)/g(x) • First term b(x)/g(x) is divisible by definition • e(x)/g(x) determines the error. Advanced Computer Networks Transmission Errors

  33. CRC Algorithm • In a cyclic code, • If e(x)/g(x) ≠ 0, one or more bits is corrupted. • If e(x)/g(x) = 0, • No bit is corrupted • Some bits are corrupted, but the generator failed to detect them • Choosing a generator polynomial by which a larger number of errors are detected is of great importance. Advanced Computer Networks Transmission Errors

  34. CRC Algorithm Generator polynomial catch all single-bit errors, if • It has more than one term • Coefficient of x0 is 1 • Example (x+1) Note: No polynomial xi is divisible by (x + 1). Therefore, single-bit errors can be caught by the generator having factor (x + 1). (why?) Advanced Computer Networks Transmission Errors

  35. CRC Example Information: (1,1,0,0) i(x) = x3 + x2 Generator polynomial: g(x) = x3 + x + 1 Encoding: i(x).x3 = x6 + x5 x3 + x2 + x 1110 x3 + x+ 1) x6 + x5 1011 ) 1100000 x6 + x4 + x3 1011 x5 + x4 + x3 1110 1011 x5 + x3 + x2 1010 x4 + x2 1011 x4 + x2 + x x 010 • Transmitted codeword: • b(x) =i(x).x3 + r(x)= x6+ x5 + x • b= (1,1,0,0,0,1,0) Advanced Computer Networks Transmission Errors

  36. CRC Example Information: 1101011011 Generator: 10011 (x4+x+1) Transmitted Data: 11010110111110 Advanced Computer Networks Transmission Errors

  37. Detecting Errors 1. Single bit errors: e(x) = xi0in-1 • If a single-bit error happens at ith position of the sent message, error e(x) = xiis added to the message b(x) • Received message y(x)=b(x)+ xi • b(x) is dividable by g(x) • y(x)/g(x)=e(x)= xi • Xi shows the position of the error position Advanced Computer Networks Transmission Errors

  38. Detecting Errors Single–bit error • CRC Coding • Information=1001 • g(x)=1011 • r(x)=110 • b(x)=1001110 Advanced Computer Networks Transmission Errors

  39. Detecting Errors Single–bit error • No Error • y(x)=1001110 • g(x)=1011 • Y(x)/g(x)=0 • e(x)=000 Advanced Computer Networks Transmission Errors

  40. Detecting Errors Single–bit error • Single error at position 3 • y(x)=1000110 • g(x)=1011 • y(x)/g(x)=011 • e(x)=x3 Advanced Computer Networks Transmission Errors

  41. Detecting Errors 2. Double bit errors:e(x)= xi+xj0 i < j  n-1 • What kind of generator polynomial is capable of detecting isolated double-bit errors? Why? Advanced Computer Networks Transmission Errors

  42. Detecting Errors 3. Odd number of bit errors: • If an odd number of errors happen in the message, polynomial of the error has odd number of terms. • For example: Error e(x)=x5+x2+1 • A generator that contains a factor of (x+1)can detect all odd-numbered errors (why?). Advanced Computer Networks Transmission Errors

  43. Detecting Errors 4. Burst errors: • Let r be the number of redundant bits • Let l be the length of the burst error • If burst error happens between bits m and n, where m<n • e(x)=xn+…+xm(n-m+1=l) • All burst errors with length • L≤r is always detected • L=r+1 is detected with probability 1–(1/2)r–1 • L>r+1 is detected with probability 1–(1/2)r Advanced Computer Networks Transmission Errors

  44. Standard Generating Polynomials • A good polynomial generator needs to have the following characteristics: • 1. It should have at least two terms. • 2. The coefficient of the term x0 should be 1. • 3. It should not dividext + 1 • 4. It should have the factor (x + 1). Advanced Computer Networks Transmission Errors

  45. Standard Generating Polynomials CRC-8: 100000111 CRC-10: 11000110011 CRC-16: 11000000000000101 CRC-CCITT: 10001000000100001 CRC-32: 100000100110000010001110110110111 Advanced Computer Networks Transmission Errors

  46. Transmission Errors Summary • Error Detection versus Error Correction • Hamming Distances and Codes • Parity • Internet Checksum • Polynomial Codes • Cyclic Redundancy Checking (CRC) • Properties for Detecting Errors with Generating Polynomials Advanced Computer Networks Transmission Errors

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