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## Errors

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**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**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**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**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**How Error Detection Works**2e Valid codeword Valid codeword Invalid Code Words Errors**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**Parity in Action**Want to send: 10 Data Link Sends: 110 Receive: 111 (ERROR) Errors**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**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**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**Error Correction**The +1 ensures the circles will not overlap Valid codeword Valid codeword Invalid codewords Errors**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**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**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**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**Putting It Together**Errors**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**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**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**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**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**Error Detection**Errors**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**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**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**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**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**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**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**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**Polynomial Multiplication**1101 x 1011 ---- 1101 11010 000000 1101000 ------- 1111111 Errors**Polynomial Division**1101 1011 1111111 1011 1001 1011 1011 1011 0 Errors**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**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**CRC Example (receive)**Frame contents: 1110111101 Polynomial: 11101 (x4+ x3+x2 + x0) 100001 11101 1110111101 11101 ----- 11101 11101 ----- 0 Errors**Fast Polynomial Division**Errors**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**Faster Polynomial Division**Errors**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