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us to a particular node in the tree, the branching rule allows us to follow theupper branch if the next input bit is ‘‘0’’ and the lower branch if the input bit is‘‘1.’’ Consequently, a particular path through the tree can be traced for aspecific input sequence. It can be observed that the tree generated by thisconvolutional encoder (Fig. 6.4) shows that the structure repeats itself after thethird stage. The tree diagram is thus shown up to the third stage, as in Fig. 6.5.This behavior is consistent with the fact that the constraint length is 3(L c ¼ 3): interpreted as the 3-bit output sequence at each stage is determinedby an input bit and 2 bits contained in the first two stages (r 0 ; r 1 ) of the shiftregister.It should be noted that the bit in the last stage (r 2 ) of the register isshifted out and does not affect the output: just as demonstrated in Table 6.2 forExample 6.5. In essence, it could be said that the 3-output bit for each input bitis determined by the input bit and four possible states labelled a, b, c and d inFig. 6.5 and denoted respectively by 00, 01, 10, and 11. With this labelling, itcan be observed in Fig. 6.5 that, at the third stage,There are two nodes each with the label a, b, c or d,All branches originating from two nodes and having the same labelgenerate identical output sequence. This implies that two nodeshaving the same label can be merged.By merging two nodes having the same label in the code tree diagram of Fig.6.5, another diagram emerges, as shown in Fig. 6.6. This diagram is called thetrellis diagram. The dotted lines denote the output generated by the lowerbranch of the code tree with input bit ‘‘1’’, while the solid lines denote theoutput generated by the upper branch of the code tree with input bit ‘‘0.’’The completely repetitive structure of the trellis diagram in Fig. 6.6suggests that a further reduction is possible in the representation of the code tothe state diagram. A state diagram, shown in Fig. 6.7, is another way ofrepresenting states (a, b, c, and d) transitioning from one state to another.Arrows represent the transitions from state to state. The states of the statediagram are labelled according to the states of the trellis diagram. The threebits shown next to each transitory line represent the output bits.From the preceding discussions, we are in a position to draw the codetree, trellis diagram, and state diagram for the fixed 1=2-rate convolutionalcoder of Fig. 6.3. The message sequence 10111 is used as input to the encoderin Fig. 6.3. The code tree of Fig. 6.8 is drawn. The tree-drawing procedure isthe same as described previously for the encoder in Fig. 6.3 by moving up atthe first branching level, down at the second and third, and up again at thefourth level to produce the outputs appended to the traversed branches. Afterthe first three branches, the structure becomes repetitive: a behavior that isCopyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.