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In [[telecommunication]], a '''convolutional code''' is a type of [[error-correcting code]] in which (a) each <i>m</i>-[[bit]] [[information]] symbol (each <i>m</i>-[[bit string]]) to be encoded is transformed into an <i>n</i>-bit symbol, where <i>m/n</i> is the code <i>rate</i> (<i>n</i> >= <i>m</i>) and (b) the transformation is a function of the last <i>k</i> information symbols, where <i>k</i> is the constraint length of the code.
===Where Convolutional Codes are used===
Convolutional codes are often used to improve the performance of [[radio]] and [[satellite]] links.
===Decoding Convolusional Codes===
Several [[algorithm]]s exist for decoding convolutional codes. For relatively small values of <i>k</i>, the [[Viterbi algorithm]] is universally used as it provides [[maximum likelihood]] performance and is highly parallelizable. Viterbi decoders are thus easy to implement in [[VLSI]] hardware and in software on CPUs with [[SIMD]] instruction sets. An especially popular Viterbi-decoded convolutional code, used at least since the [[Voyager program]] has a constraint length <i>k</i> of 7 and a rate <i>r</i> of 1/2. Longer constraint lengths produce more powerful codes, but the [[complexity]] of the Viterbi algorithm [[exponential growth|increases exponentially]] with constraint lengths, limiting these more powerful codes to deep space missions where the extra performance is easily worth the increased decoder complexity. [[Mars Pathfinder]], [[Mars Exploration Rover]] and the [[Cassini probe]] to Saturn use a <i>k</i> of 15 and a rate of 1/6; this code performs about 2 dB better than the simpler <i>k</i>=7 code at a cost of 256x in decoding complexity (compared to Voyager mission codes).
===Design Issues===
Longer constraint length codes are more practically decoded with any of several <b>sequential</b> decoding algorithms, of which the [[Shannon-Fano_coding|Fano]] algorithm is the best known. Unlike Viterbi decoding, sequential decoding is not maximum likelihood but its complexity increases only slightly with constraint length, allowing the use of strong, long-constraint-length codes. Such codes were used in the [[Pioneer program]] of the early 1970s to Jupiter and Saturn, but gave way to shorter, Viterbi-decoded codes, usually concatenated with large [[Reed-Solomon error correction]] codes that steepen the overall bit-error-rate curve and produce extremely low residual undetected error rates.
===Turbo Codes: replacing Convolutional Codes===
Simple Viterbi-decoded convolutional codes are now giving way to [[turbo code]]s, a new class of iterated short convolutional codes that closely approach the theoretical limits imposed by [[Shannon's theorem]] with much less decoding complexity than the Viterbi algorithm on the long convolutional codes that would be required for the same performance.
Turbo codes have not yet been concatenated with solid (low complexity) [[Reed-Solomon error correction]] codes. However, in the interest of planetary exploration this may someday be done.
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