In telecommunication, a convolutional code is a type of error-correcting code in which (a) each m-bit information symbol (each m-bit string) to be encoded is transformed into an n-bit symbol, where m/n is the code rate (n >= m) and (b) the transformation is a function of the last k information symbols, where k 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.
Convolutional Encoding
To convolutionally encode data, start with k memory registers, each holding 1 input bit. Unless otherwise specified, all memory registers start with a value of 0. The encoder has n modulo-2 adders, and n generator polynomials -- one for each adder (see figure below). An input bit m1 is fed into the leftmost register. Using the generator polynomials and the existing values in the remaining registers, the encoder outputs n bits. Now bit shift all register values to the right (m1 moves to m0, m0 moves to m-1) and wait for the next input bit. If there are no remaining input bits, the encoder continues output until all registers have returned to the zero state.
The figure below is a rate 1/3 (m/n) encoder with constraint length (k) of 3. Generator polynomials are G1 = (1,1,1), G2 = (0,1,1), and G3 = (1,0,1). Therefore, output bits are calculated as follows:
n1 = mod2 (m1 + m0 + m-1)
n2 = mod2 (m0 + m-1)
n3 = mod2 (m1 + m-1)
Decoding Convolutional Codes
Several algorithms exist for decoding convolutional codes. For relatively small values of k, 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.
Popular Convolutional Codes
An especially popular Viterbi-decoded convolutional code, used at least since the Voyager program has a constraint length k of 7 and a rate r of 1/2.
- Longer constraint lengths produce more powerful codes, but the complexity of the Viterbi algorithm 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 k of 15 and a rate of 1/6; this code performs about 2 dB better than the simpler k=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 sequential decoding algorithms, of which the 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 codes, 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.