Following the procedure just described we find that the encoded sequence for an information sequence ( 10011 ) is ( 11, 10, 11, 11, 01 ) which agrees with the first 5 pairs of bits of the actual
Information Theory and Coding 10EC55 Dept. of ECE/SJBIT Page 214 encoded sequence. Since the encoder has a memory = 2 we require two more bits to clear and re-set the encoder. Hence to obtain the complete code sequence corresponding to an information sequence of length kL , the tree graph is to extended by n(m-l) time units and this extended part is called the " Tail of the tree ", and the 2kL right most nodes are called the " Terminal nodes " of the tree. Thus the extended tree diagram for the ( 2, 1, 2 ) encoder, for the information sequence ( 10011 ) is as in Fig 8.19 and the complete encoded sequence is ( 11, 10, 11, 11, 01, 01, 11 ). At this juncture, a very important clue for the student in drawing tree diagrams neatly and correctly, without wasting time appears pertinent. As the length of the input sequence L increases the number of right most nodes increase as 2 L . Hence for a specified sequence length, L , compute 2 L . Mark 2 L equally spaced points at the rightmost portion of your page, leaving space to complete the m tail branches. Join two points at a time to obtain 2 L-l nodes. Repeat the procedure until you get only one node at the left most portion of your page. The procedure is illustrated diagrammatically in Fig 8.20 for L = 3. Once you get the tree structure, now you can fill in the needed information either looking back to the state transition table or working out logically.
Information Theory and Coding 10EC55 Dept. of ECE/SJBIT Page 215 From Fig 8.18, observe that the tree becomes " repetitive ' after the first three branches. Beyond the third branch, the nodes labeled S 0 are identical and so are all the other pairs of nodes that are identically labeled. Since the encoder has a memory m = 2, it follows that when the third information bit enters the encoder, the first message bit is shifted out of the register. Consequently, after the third branch the information sequences ( 000u 3 u 4 --- ) and ( 100u 3 u 4 --- ) generate the same code symbols and the pair of nodes labeled S 0 may be joined together. The same logic holds for the other nodes. Accordingly, we may collapse the tree graph of Fig 8.18 into a new form of Fig 8.21 called a " Trellis ". It is so called because Trellis is a tree like structure with re-merging branches (You will have seen the trusses and trellis used in building construction). The Trellis diagram contain ( L + m + 1 ) time units or levels (or depth) and these are labeled from 0 to ( L + m ) ( 0 to 7 for the case with L = 5 for encoder of Fig 8.15 as shown in Fig8.21. The convention followed in drawing the Trellis is that " a code branch produced by an input '0' is drawn as a solid line while that produced by an input '1' is shown by dashed lines ". The code words produced by the transitions are also indicated on the diagram. Each input sequence corresponds to a specific path through the trellis. The reader can readily verify that the encoder output corresponding to the sequence
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