Coding for error detection and correction in .NET Produce Code 128A in .NET Coding for error detection and correction

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Coding for error detection and correction using barcode encoding for .net framework control to generate, create code 128b image in .net framework applications. ISO/IEC 18004:2000 Table 7.3 Viterbi algorithm, 5-interval operation, K = 3, rate-1/2 coder Possible pat hs 2 3 4 01 00 11 01 01 01 01 00 01 00 01 01 10 01 10 00 10 00 10 11 01 11 01 11 10 11 10. interval received bits 1 01 11 11 11 11 00 11 00 11. 5 10 00 11 11 00 10 01 01 10. Previous state a c a c b d b d Current state a a b b c c d d Hamming distance 2 + 1 = 3 3+1=4 2 + 1 = 3 3+1=4 2 + 0 = 2 3+2=5 2+2=4 3 + 0 = 3 Convolutiona Visual Studio .NET barcode 128 l encoders were introduced as a particularly ef cient and effective way of correcting errors in digital transmission. We described the Viterbi algorithm as an ingenious method of reducing the computational and storage requirements in carrying out maximum-likelihood decoding of convolutionally coded digital signals.

How effective is convolutional coding How well do convolutional coders perform in practice How does their performance depend on the constraint-length K, the rate 1/ , and the function generators used By performance we mean here the residual probability of bit error after using convolutional coding to both code and decode messages. To determine the performance of a convolutional coder note that an error occurs when the path selected on reception differs from the correct one, as transmitted. The probability of error is then found by calculating the probability a given (incorrect) path deviates from a speci ed correct path, and averaging over all such paths.

Since no particular path is likely to always be the correct one, we assume all possible paths equally likely to occur this is, in fact, the basic assumption behind the concept of maximum-likelihood decoding, the decoding procedure implemented by the Viterbi algorithm. In particular, we assume for simplicity and without loss of generality that the correct path is the all-zero one. This implies that the message transmitted is a sequence of zeros.

Focusing again on the K = 3, rate-1/2 encoder example we have been using throughout this section, note from Fig. 7.10, the trellis representation of this encoder, that, with this correct sequence of transmitted bits, the path the encoder follows is the upper horizontal one, staying in state a at all times.

An error then occurs if the path chosen at the decoder strays from this horizontal one. The probability of error is then found by determining the possible ways the decoder can deviate from the all-zero path and the probability of doing so. But recall the cyclic nature of entering and re-entering states that we have referred to previously and as readily seen from the trellis representation of Fig.

7.10. We can thus pick any interval to start and determine the various ways of leaving the all-zero path and then returning to it.

In particular, we choose to start with interval 0 for simplicity. We now list the various paths that could be taken in leaving the all-zero transmitted path and then returning to it. Mobile Wireless Communications later, in or barcode 128 for .NET der of increasing Hamming distance with respect to the all-zero path. Note that the only way to leave the all-zero path is to rst move to state b with bits 11 received.

From state b there are two possible ways to move, the shortest-distance one being to state c. (Note that moving to state d from b also incrementally adds a 1 to the metric, but takes us further away from the all-zero state.) From this state the shortest-distance return to state a is the one moving directly back to a.

The Hamming distance of this path is then readily found to be 5. This is then the path with the smallest Hamming distance. This minimum Hamming distance of 5 is termed the minimum free distance dF .

It turns out that the performance of convolutional decoders depends critically on the minimum free distance, i.e., the smallest Hamming distance.

It is left for the reader to show that the next larger value of Hamming distance for this particular convolutional decoder is 6. There are two possible paths with this distance: a-b-c-b-c-a and a-b-d-c-a. It is left for the reader to show as well that there are four paths of distance 7.

Many more paths of higher and higher values of Hamming distance exist as well. In general, for any convolutional decoder, let the function a(d) represent the number of paths with a Hamming distance of d deviating from, and then returning to, the all-0 test path. The error-probability Pe of deviating from the correct path is then upper-bounded by Pe <.

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