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A convolutional encoder is a finite state machine. An encoder with n binary cells will have 2 n states. Imagine that the encoder (shown on Img.1, above) has '1' in the left memory cell (m 0), and '0' in the right one (m −1). (m 1 is not really a memory cell because it represents a current value). We will designate such a state as "10".
The commonly used rule of thumb of a truncation depth of five times the memory (constraint length K-1) of a convolutional code is accurate only for rate 1/2 codes. For an arbitrary rate, an accurate rule of thumb is 2.5( K - 1)/(1− r ) where r is the code rate.
The name trellis derives from the fact that a state diagram of the technique closely resembles a trellis lattice. The scheme is basically a convolutional code of rates ( r , r +1). Ungerboeck's unique contribution is to apply the parity check for each symbol , instead of the older technique of applying it to the bit stream then modulating the bits.
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Convolutional code trellis diagram. A trellis is a graph whose nodes are ordered into vertical slices (time) with every node at almost every time connected to at least one node at an earlier and at least one node at a later time. The earliest and latest times in the trellis have only one node (hence the "almost" in the preceding sentence).
The encoder is usually a simple circuit which has state memory and some feedback logic, normally XOR gates. The decoder can be implemented in software or firmware. The Viterbi algorithm is the optimum algorithm used to decode convolutional codes.
The Viterbi algorithm is named after Andrew Viterbi, who proposed it in 1967 as a decoding algorithm for convolutional codes over noisy digital communication links. [2] It has, however, a history of multiple invention, with at least seven independent discoveries, including those by Viterbi, Needleman and Wunsch, and Wagner and Fischer. [3]
The example encoder is composed of a 16-state outer convolutional code and a 2-state inner convolutional code linked by an interleaver. The natural code rate of the configuration shown is 1/4, however, the inner and/or outer codes may be punctured to achieve higher code rates as needed.