Error Detection And Correction

Communication channels have a range of characteristics.  Some channels, like optical fiber in telecommunications networks, have  tiny error rates so that transmission errors are a rare occurrence. But other channels,  especially wireless links and aging local loops, have error rates that are orders  of magnitude larger. For these links, transmission errors are the norm. They cannot be avoided at a reasonable expense or cost in terms of performance. The  conclusion is that transmission errors are here to stay. We have to learn how to  deal with them.

      Network designers have developed two basic strategies for dealing with errors. Both add redundant information to the data that is sent. One strategy uses error-correcting codes to deduce what the transmitted data must have been. The other uses error-detecting codes to deduce that an error has occurred (but not which error). We will examine Hamming code for error detection and CRC (Cyclic Redundancy Check) for error correction separately.

      In Hamming codes,  the bits of the codeword are numbered  consecutively, starting with bit 1 at the left end, bit 2 to its immediate right, and so  on. The bits that are powers of 2 (1, 2, 4, 8, 16, etc.) are check bits. The rest (3,  5, 6, 7, 9, etc.) are filled up with the m data bits. This pattern is shown for an  (11,7) Hamming code with 7 data bits and 4 check bits in the Figure below. Each check bit  forces the modulo 2 sum, or parity, of some collection of bits, including itself, to  be even (or odd). A bit may be included in several check bit computations. To  see which check bits the data bit in position k contributes to, rewrite k as a sum of  powers of 2. For example, 11 = 1 + 2 + 8 and 29 = 1 + 4 + 8 + 16. A bit is  checked by just those check bits occurring in its expansion (e.g., bit 11 is checked  by bits 1, 2, and 8). In the example, the check bits are computed for even parity  sums for a message that is the ASCII letter "A".

                

This construction gives a code with a Hamming distance of 3, which means that it can correct single errors (or detect double errors). The reason for the very careful numbering of message and check bits becomes apparent in the decoding process. When a codeword arrives, the receiver redoes the check bit computations including the values of the received check bits. We call these the check results. If the check bits are correct then, for even parity sums, each check result should be zero. In this case the codeword is accepted as valid. If the check results are not all zero, however, an error has been detected. The set of check results forms the error syndrome that is used to pinpoint and correct the error. In the figure above, a single-bit error occurred on the channel so the check results are 0, 1, 0, and 1 for k = 8, 4, 2, and 1, respectively. This gives a syndrome of 0101 or 4 + 1=5. By the design of the scheme, this means that the fifth bit is in error. Flipping the incorrect bit (which might be a check bit or a data bit) and discarding the check bits gives the correct message of an ASCII "A".

      The CRC(Cyclic Redundancy Check), also known as a polynomial code. Polynomial codes are based upon treating bit strings as representations of polynomials with coefficients of 0 and 1 only. A k-bit frame is regarded as the coefficient list for a polynomial with k terms, ranging from x^(k−1) to x^0. Polynomial arithmetic is done modulo 2, according to the rules of algebraic field theory. It does not have carries for addition or borrows for subtraction. Both addition and subtraction are identical to exclusive OR.

      When the polynomial code method is employed, the sender and receiver must agree upon a generator polynomial, G(x), in advance. Both the high- and low- order bits of the generator must be 1. To compute the CRC for some frame with m bits corresponding to the polynomial M(x), the frame must be longer than the generator polynomial. The idea is to append a CRC to the end of the frame in such a way that the polynomial represented by the checksummed frame is divisible by G(x). When the receiver gets the checksummed frame, it tries dividing it by G(x). If there is a remainder, there has been a transmission error.
      The algorithm for computing the CRC is as follows:

        1. Let r be the degree of G(x). Append r zero bits to the low-order end of the frame so it now contains m + r bits.

        2. Divide the bit string corresponding to G(x) into the bit string corresponding to x rM(x), using modulo 2 division.

        3. Subtract the remainder from the bit string using modulo 2 subtraction. The result is thepolynomial T(x) to be transmitted.

The figure below illustrates the calculation for a frame 1101011111 using the generator G(x) = x^4 + x + 1.

                  

      It should be clear that T(x) is divisible (modulo 2) by G(x). In any division problem, if you diminish the dividend by the remainder, what is left over is divisible by the divisor. Imagine that a transmission error occurs, so that instead of the bit string for T(x) arriving, T(x) + E(x) arrives. Each 1 bit in E(x) corresponds to a bit that has been inverted. If there are k 1 bits in E(x), k single-bit errors have occurred. A single burst error is characterized by an initial 1, a mixture of 0s and 1s, and a final 1, with all other bits being 0.