For an overview of the history of channel coding, see Costello and Forney, While concatenated codes showed that the performance-complexity tradeoff problem of channel coding could be solved in principle, they were hardly practical in the technology of the s. The author recalls much eye-rolling when he presented concatenated codes to a Bell Labs research group in , and discussed code lengths up into the thousands. Space applications However, by the s, technology had advanced sufficiently that concatenated codes became standardized by NASA for space applications. Figure 2: NASA standard concatenated coding system.

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This property can be easily shown based on the idea of defining a generator matrix for the concatenated code in terms of the generator matrices of Cout and Cin. Decoding concatenated codes[ edit ] A natural concept for a decoding algorithm for concatenated codes is to first decode the inner code and then the outer code.

For the algorithm to be practical it must be polynomial-time in the final block length. Consider that there is a polynomial-time unique decoding algorithm for the outer code. Now we have to find a polynomial-time decoding algorithm for the inner code. It is understood that polynomial running time here means that running time is polynomial in the final block length.

The main idea is that if the inner block length is selected to be logarithmic in the size of the outer code then the decoding algorithm for the inner code may run in exponential time of the inner block length, and we can thus use an exponential-time but optimal maximum likelihood decoder MLD for the inner code. In other words, it is NO 1 i. As the outer decoding algorithm in step two is assumed to run in polynomial time the complexity of the overall decoding algorithm is polynomial-time as well.

The algorithm also works if the inner codes are different, e. It is still notably used today for satellite communications , such as the DVB-S digital television broadcast standard. For example, within the DVB-S2 standard, a highly efficient LDPC code is combined with an algebraic outer code in order to remove any resilient errors left over from the inner LDPC code due to its inherent error floor. Turbo codes: A parallel concatenation approach[ edit ] The description above is given for what is now called a serially concatenated code.

Turbo codes , as described first in , implemented a parallel concatenation of two convolutional codes, with an interleaver between the two codes and an iterative decoder that passes information forth and back between the codes. However, a key aspect of turbo codes is their iterated decoding approach. Iterated decoding is now also applied to serial concatenations in order to achieve higher coding gains, such as within serially concatenated convolutional codes SCCCs.

An early form of iterated decoding was implemented with two to five iterations in the "Galileo code" of the Galileo space probe.

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## Concatenated error correction code

This property can be easily shown based on the idea of defining a generator matrix for the concatenated code in terms of the generator matrices of C out and C in. While concatenated codes showed that the performance-complexity tradeoff problem of channel coding could be solved in principle, they were hardly practical in the technology of the s. Thus, there are at least D positions in which the sequence of N symbols of the codewords C out m 1 and C out m 2 differ. B1 Data Adaptive Entropy Coder. Although a simple concatenation scheme was implemented already for the Mariner Mars orbiter mission, [5] concatenated codes were starting to be regularly used for deep space communication with the Voyager programwhich launched two space probes in Consider that there is a polynomial-time unique decoding algorithm for the outer code. In other words, it is N O 1 i. The main idea is that if the inner block length is selected to be logarithmic in the size of the outer code then the decoding algorithm for the inner code may run in exponential time of the inner block length, and we can thus use an exponential-time but optimal maximum likelihood decoder MLD for the inner code.

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## FORNEY CONCATENATED CODES PDF

Transactions on Information Theory. A forndy concept for a decoding algorithm for concatenated codes is to first decode the inner code and then the outer code. Concatenated error correction code — Wikipedia The basic concatenated coding scheme considered by Forney is shown in Figure 1. Typically, the inner code is not a block code but a soft-decision convolutional Viterbi-decoded code with a short constraint length.