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1. Errors • Transmission errors are a way of life. • In the digital world an error means that a bit value is flipped. • An error can be isolated to a single bit. • Errors on some media come in bursts • Harder to detect and correct than isolated errors. Errors

2. Dealing with Errors • Error detecting codes • provide enough redundant information to enable the receiver to deduce that an error occurred • Error correcting codes • provide enough redundant information to enable the receiver to deduce that an error occurred AND how to fix it • So a message consists of m data bits and r redundant or check bits. Errors

3. Hamming Distance • Hamming distance: • the number of bit positions in which two codewords differ • Simple to calculate find the XOR • If two codewords are a Hamming distance d apart, it will require d single-bit errors to convert one into the other. • The Hamming distance of a code is the minimum Hamming distance between any two codewords. Errors

4. Hamming Distance 2 Code 000 011 101 110 Note that not all of the 8 different bit patterns are included in the code Any single error will convert a valid codeword into an invalid codeword. If we know the valid codewords we can detect the error Errors

5. How Error Detection Works 2e Valid codeword Valid codeword Invalid Code Words Errors

6. Parity • A simple single error detecting code could be constructed by counting bits. • Any codeword with an even number of bits is consider valid (you could also make it the other way around). • The minimum distance of this code is 2, so it is capable of detecting single errors. • This code can be created by adding a parity bit: • chose the parity bit so that the number of ones in the codeword is even (or odd). Errors

7. Parity in Action Want to send: 10 Data Link Sends: 110 Receive: 111 (ERROR) Errors

8. Protecting Blocks • The probability of detecting a burst error on a block using a single parity bit is 50%. • This can be improved by viewing the block as a n by k bit matrix. • A parity bit is then computed for each column. • The check bits are placed in a k-bit row and affixed to the matrix as the last row. • Bursts of length n can be detected. Errors

9. Detecting Burst Errors Data 1001000 1100001 1101101 1101101 1101001 1101110 1100111 0100000 1100011 1101111 1100100 1100101 10010000 11000011 11011011 11011011 11010010 11011101 11001111 01000001 11000110 11011110 11001001 11001010 11001001 VRC (Vertical Redundancy Check) n LRC (Longitudinal Redundancy Check) Errors

10. What About Error Correction? • How do we get error correction? • Must increase the minimum distance of the code • The key to error correction is that it must be possible to detect and locate the error. • The minimum distance must be at least 2e+1 Errors

11. Error Correction The +1 ensures the circles will not overlap Valid codeword Valid codeword Invalid codewords Errors

12. Hamming Codes • Hamming codes are n-bit codes that can correct single errors. • The basic idea is to split the codeword into two portions • information or message bits (m) • parity bits (k) • The result are codewords that consist of m+k bits Errors

13. Choosing m and k • Selecting m is easy, you are usually told what it is. • How do you pick k? • The parity bits are used to generate a k-bit word that identifies where in the codeword the error, if any, occurred. • Consequently, k must satisfy the following: Errors

14. Constructing the Codeword • The codeword consists of m+k bits. • The location of each of the m+k bits is assigned a decimal value, 1 is assigned to the MSB, and m+k to the LSB. • Parity bits go in positions 1, 2, 4, …, 2k-1 m3 p0 p1 m0 p2 m1 m2 p3 m4 ... mm+k 1 2 3 4 5 6 7 8 9 ... m+k Errors

15. Parity Checks • The parity checks must be specified so that when an error occurs, the position number will take on the the value assigned to to location of the error Errors

16. Putting It Together Errors

17. Example Send the message 0010 using a hamming code Step 1: Find k. Here k=3 Step 2: Determine where things go Step 3: Figure out the parity bits p1 will cover 1,3,5,7,9,11,… p2 will cover 2,3,6,7,10,11,… p3 will cover 4,5,6,7,12,13,14,15,... Errors

18. Correcting Burst Errors • Hamming codes can be used to correct burst errors • A sequence of s consecutive codewords are arranged as a matrix, one codeword per row. • Transmit data one column (s bits) at a time. • The matrix is reconstructed by the receiver one column at a time. • If a burst error of size s occurs, only a single column will be affected. Errors

19. Correcting Burst Errors Character ASCII Check Bits H a m m I n g c o d e 1001000 1100001 1101101 1101101 1101001 1101110 1100111 0100000 1100011 1101111 1100100 1100101 00110010000 10111001001 11101010101 11101010101 01101011001 01101010110 01111001111 10011000000 11111000011 10101011111 11111001100 00111000101 s Errors

20. Correcting vs. Detecting • Most often error detection followed by retransmission is more efficient. • Consider a channel with an error rate is 10-6 per bit (one error per million bits sent) • Block size 1000 == 10 check bits ( k == 10 ) • For parity one check bit will suffice • Overhead for sending 1MB • Hamming == 10,000 bits • Parity == 2001 bits (since 1 block will be retransmitted) Errors

21. Checksums • Both sides agree on a checksum function • Sender • Computes checksum while sending message • Attaches result to the end of the message • Receiver • Computes checksum while reading message • Compares result to checksum at end of message Errors

22. Error Detection Errors

23. Simple Example • Checksum function • Sum of bytes in message modulo 256 • Sender • 72 101 108 108 111 87 111 114 108 100 • Checksum  1020 % 256  252 • 72 101 108 108 111 87 111 114 108 100 252 • Receiver • 72 101 118 108 111 87 111 114 108 100 252 • Checksum  1030 % 256  6 • Error, discard message Errors

24. Cyclic Redundancy Check • CRC detects all of the following errors: • All single bit errors • All double bit errors if the divisor is at least three terms • Any odd number of errors, if the divisor contains a factor (x+1) • Any error in which the length of the error (an error burst) is less than the length of the FCS • Most errors with larger bursts Errors

25. Basic Idea • Treat the entire message as a binary number • To calculate the checksum • Divide message by another fixed number • Use the remainder as the checksum • CRC treats bit strings as representations of polynomials with coefficients of 0 and 1. • 110001 == x5+ x4+ x0 Errors

26. The Generator Polynomial • Both the sender and the receiver must agree upon a generator polynomial, G(x). • Both the high and low order bits of the generator must be 1. • The length of the generator is one bit longer than the FCS. • Finally the frame must be longer than the generator. • This is what we use mathematicians for Errors

27. Standard Polynomials • CRC-12 (x12+x11+x3+x2+x1+1) • used when the character length is 6 • CRC-16 (x16+x15+x2+1) • CRC-CCITT (x16+x12+x5+1) • used for 8 bit characters • catches all single and double errors • all errors of an odd length • all bursts of 16-bits or less, 99.997% of 17-bits, and 99.998% of 18-bits and longer. Errors

28. The Algorithm • To compute the checksum • Append n 0s to the end of the message, where n is the number of bits in the checksum • The resulting value is divided by the generator polynomial • Each division step is carried out in the conventional manner, except that we use polynomial arithmetic Errors

29. Polynomial Arithmetic • Subtraction and addition as usual but no borrows or carries • Both operations are identical to XOR 1 1 0 0 -1 -0 -1 -0 -- -- -- -- 0 1 1 0 1 1 0 0 +1 +0 +1 +0 -- -- -- -- 0 1 1 0 Errors

30. Polynomial Arithmetic • Addition and subtraction, are a single operation, that is its own inverse • By collapsing addition and subtraction, the arithmetic discards any notion of magnitude • Beyond the power of the highest bit • 1010 is clearly greater than 10 • 1010 is no longer greater than 1001 • 1010 = 1001 + 0011 • 1010 = 1001 - 0011 Errors

31. Polynomial Multiplication 1101 x 1011 ---- 1101 11010 000000 1101000 ------- 1111111 Errors

32. Polynomial Division 1101 1011 1111111 1011 1001 1011 1011 1011 0 Errors

33. The Algorithm (continued) • The division produces a quotient which is discarded. • The remainder replaces the 0s appended to the frame (subtracted from the frame modulo 2). • The resulting frame is now evenly divisible by the generator polynomial. • The receiver performs the same division, a non-zero remainder indicates that an error occurred. Errors

34. CRC Example (transmit) Frame contents: 111011 Polynomial: 11101 (x4+ x3+x2 + x0) Frame with 0s: 1110110000 100001 11101 1110110000 11101 ----- 10000 11101 ----- 1101 Frame to send: 1110111101 Errors

35. CRC Example (receive) Frame contents: 1110111101 Polynomial: 11101 (x4+ x3+x2 + x0) 100001 11101 1110111101 11101 ----- 11101 11101 ----- 0 Errors

36. Fast Polynomial Division Errors

37. Optimize • For CRC • We do not need the quotient • If the divisor is W bits long • The remainder will be at most W-1 bits long • Only need 1 register Errors

38. CRC Simple Version • Consider the polynomial 10111 with a CRC of size W=4 • To perform the division perform the following: • Load the register with zero bits. • Augment the message by appending W zero bits to the end of it. • While (more message bits) • Shift the register left by one bit, reading the next bit of the augmented message into register bit position 0. • If (a 1 bit popped out of the register during step 3) • Register = Register XOR Poly. • The register contains the remainder. Source: http://www.repairfaq.org/filipg/LINK/F_crc_v33.html Errors