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If the number of errors within a code word exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more uniform distribution of errors. [ 21 ]
Compute forward probabilities Compute backward probabilities β {\displaystyle \beta } Compute smoothed probabilities based on other information (i.e. noise variance for AWGN , bit crossover probability for binary symmetric channel )
The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes, turbo codes, and fountain codes.
Parity check is the special case where n = k + 1.From a set of k values {}, a checksum is computed and appended to the k source values: + = =. The set of k + 1 values {} + is now consistent with regard to the checksum.
A loop is a sequence of statements which is specified once but which may be carried out several times in succession. The code "inside" the loop (the body of the loop, shown below as xxx) is obeyed a specified number of times, or once for each of a collection of items, or until some condition is met, or indefinitely. When one of those items is ...
The analysis of errors computed using the global positioning system is important for understanding how GPS works, and for knowing what magnitude errors should be expected. The Global Positioning System makes corrections for receiver clock errors and other effects but there are still residual errors which are not corrected.
The backward algorithm complements the forward algorithm by taking into account the future history if one wanted to improve the estimate for past times. This is referred to as smoothing and the forward/backward algorithm computes (|:) for < <. Thus, the full forward/backward algorithm takes into account all evidence.
The general algorithm involves message passing and is substantially similar to the belief propagation algorithm (which is the generalization of the forward-backward algorithm). With an algorithm called iterative Viterbi decoding, one can find the subsequence of an observation that matches best (on average) to a given hidden Markov model.