Crc Burst Error
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reliable link. This is done by including redundant information in each transmitted frame. Depending on the nature of the link and the data one crc calculation example can either: include just enough redundancy to make it possible to detect cyclic redundancy check in computer networks errors and then arrange for the retransmission of damaged frames, or include enough redundancy to enable the receiver cyclic redundancy check ppt to correct any errors produced during transmission. Most current networks take the former approach. One widely used parity bit based error detection scheme is the cyclic redundancy check or crc error detection CRC. The CRC is based on some fairly impressive looking mathematics. It is helpful as you deal with its mathematical description that you recall that it is ultimately just a way to use parity bits. The presentation of the CRC is based on two simple but not quite "everyday" bits of mathematics: polynomial division arithmetic over the field of integers
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mod 2. Arithmetic over the field of integers mod 2 is simply arithmetic on single bit binary numbers with all carries (overflows) ignored. So 1 + 1 = 0 and so does 1 - 1. In fact, addition and subtraction are equivalent in this form of arithmetic. Polynomial division isn't too bad either. There is an algorithm for performing polynomial division that looks a lot like the standard algorithm for integer division. More interestingly from the point of view of understanding the CRC, the definition of division (i.e. the definition of the quotient and remainder) are parallel. When one says "dividing a by b produces quotient q with remainder r" where all the quantities involved are positive integers one really means that a = q b + r and that 0 <=r < b When one says "dividing a by b produces quotient q with remainder r" where all the quantities are polynomials, one really means the same thing as when working with integers except that the meaning of "less than" is a bit different. Fo
Redundancy Check) Data is sent with a checksum. When arrives, checksum is recalculated. Should match the one that was sent. Bitstring represents polynomial. e.g. 110001 represents: 1 . x5 + 1 . x4 + 0 . x3 + 0 . x2 + 0 . x1 + 1 . x0 = x5 + x4 + x0 The order
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of a polynomial is the power of the highest non-zero coefficient. This is polynomial of order 5. Special case: crc-16 We don't allow bitstring = all zeros. Easy to use framing or stuffing to make framed-and-stuffed transmission never all-zero, while still allowing payload within it to be all-zero. hash crc cambridge functions CRC Origin in research of W. Wesley Peterson: W.W. Peterson and D.T. Brown, "Cyclic codes for error detection", Proceedings of the IRE, Volume 49, pages 228-235, Jan 1961. W.W. Peterson, Error Correcting Codes, MIT Press 1961. Modulo 2 arithmetic We are going to define a particular http://www.cs.jhu.edu/~scheideler/courses/600.344_S02/CRC.html field (or here), in fact the smallest field there is, with only 2 members. We define addition and subtraction as modulo 2 with no carries or borrows. This means addition = subtraction = XOR. Here's the rules for addition: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 Multiplication: 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1 Subtraction: if 1+1=0, then 0-1=1, hence: 0 - 0 = 0 0 - 1 = 1 http://www.computing.dcu.ie/~humphrys/Notes/Networks/data.polynomial.html 1 - 0 = 1 1 - 1 = 0 Long division is as normal, except the subtraction is modulo 2. Example No carry or borrow: 011 + (or minus) 110 --- 101 Consider the polynomials: x + 1 + x2 + x ------------- x2 + 2x + 1 = x2 + 1 We're saying the polynomial arithmetic is modulo 2 as well, so that: 2 xk = 0 for all k. Digital Communications course by Richard Tervo Intro to polynomial codes CGI script for polynomial codes CRC Error Detection Algorithms What does this mean? Just consider this as a set of rules which, if followed, yield certain results. When the checksum is re-calculated by the receiver, we should get the same results. All sorts of rule sets could be used to detect error. It is useful here that the rules define a well-behaved field. Consider the polynomials with x as isomorphic to binary arithmetic with no carry. It is just easier to work with abstract x so we don't make the mistake of starting to add, say. 3 x3 to get x4 + x3 if we were thinking of x=2. We work in abstract x and keep "the coefficients of each power nicely isolated" (in mod 2, when we add two of same power, we get zero, not another power). multiplication Multiply 110010 by 1000 Multiply (x5 + x4 + x) by x3 = x8 + x7 + x4 = 110010000 i.e. Just add 3 zeros In general, to multiply by xk, add k zeros. division x2 + 1 = (x+1)(x+1) (since 2x=0) Do long division: Divide (x+1) into x2 + 1 Divide 11 into 101 Subtraction mod 2 Get 11, remainder 0 1
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