Error correction code: Difference between revisions

Revision as of 03:31, 21 August 2018

Revision as of 18:56, 27 August 2018

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ECC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, ECC is an integral part of the initial [[Analog-to-digital converter|analog-to-digital conversion]] in the receiver. The [[Viterbi decoder]] implements a [[Error correction code#Types of ECC|soft-decision algorithm]] to demodulate digital data from an analog signal corrupted by noise. Many ECC encoders/decoders can also generate a [[bit-error rate]] (BER) signal which can be used as feedback to fine-tune the analog receiving electronics.

The maximum fractions of errors or of missing bits that can be corrected is determined by the design of the ECC code, so different error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective signal-to-noise ratio. The [[noisy-channel coding theorem]] of Claude Shannon answers the question of how much bandwidth is left for data communication while using the most efficient code that turns the decoding error probability to zero. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives no insight of how to build a capacity achieving code. Fortunately, afterAfter years of research, some advanced ECC systems nowadays come very close to the theoretical maximum.

==How it works==

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The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate<ref>{{citation | author = David Tse, Pramod Viswanath | title = Fundamentals of Wireless Communication | publisher = Cambridge University Press, UK | year=2005}}</ref>. In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e. a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection.

One interesting question is the following: how efficient in terms of information transfer can be a ECC that has a negligible decoding error rate? This question was answered by Claude Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC whose error rate tends to zero:<ref name="shannon paper">C. E. Shannon: ''A mathematical theory of communication.'' Bell System Technical Journal, vol. 27, pp. 379–423 and 623–656, July and October 1948</ref>. His proof, unfortunately, relies on Gaussian random coding, which is not suitable of real-world applications. This upper bound given by Shannon's work set up a long journey in designing ECCs that can go close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement.

The most popular codes ECCs have a trade performance and computational complexity. Usually their parameters give a range of possible code rates, which can be optimized depending of the scenario. Usually, this optimization is done in order to achieve a low decoding error probability without hurting too much the data rate. Another criteria for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication<ref>