130x1g2 - CCSDS

TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALEillustrative example, and is not the CCSDS recommended code.) The output bits andtransitions between states can be recorded by the trellis diagram of figure 4-5.Figure 4-4: (3,1/2) Convolutional EncoderFigure 4-5: Trellis Representation of (3,1/2) Convolutional CodeThe diagram starts in the all-‘zero’ state, node a, and makes transitions corresponding to thenext data bit. These transitions are denoted by a solid line (branch) for a ‘0’ and by a dottedline for a ‘1’. Thus node a proceeds to node a or b with outputs bits ‘00’ or ‘11’. A branchweight is the number of ‘1’s in the n code symbols in the branch.It has been shown (see reference [25]) that the Viterbi algorithm implements, in fact,maximum-likelihood decoding. An exhaustive search maximum-likelihood decoder wouldcalculate the likelihood of the received data for code symbol sequences on all paths throughthe trellis. The path with the largest likelihood would then be selected, and the informationbits corresponding to that path would form the decoder output. Unfortunately, the number ofpaths for an L bit information sequence is 2 L ; thus this exhaustive search decoding quicklybecomes impractical as L increases.With Viterbi decoding, it is possible to greatly reduce the effort required for maximumlikelihooddecoding by taking advantage of the special structure of the code trellis. Referringto figure 4-5, it is clear that the trellis assumes a fixed periodic structure after trellis depth Kis reached.CCSDS 130.1-G-2 Page 4-5 November 2012