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Alternative methods of describing a convolutional code are the tree diagram,the trellis diagram, and the state diagram. Example 6.6 will be used to explorethese alternative methods.Example 6.6: Consider a binary convolutional encoder, rate of 1=3, shown inFig. 6.4. This figure is similar to Fig. 6.3 except that the output ‘‘v 1 ’’ is feddirectly from r 0 . For each message (input) bit, the sequence (v 1 v 2 v 3 ) isgenerated. It follows from Fig. 6.4 that the output sequence is given byv 1 ¼ r 0v 2 ¼ r 0 r 2ð6:34Þv 3 ¼ r 0 r 1 r 2Since the first bit in the output sequence is the message bit, this particularconvolutional code is systematic. As a result, v 2 and v 3 can be viewed as paritycheck bits.The output sequence for an arbitrary input sequence is often determinedby using a code tree. For example, the tree diagram for the above convolutionalencoder is illustrated in Fig. 6.5. Initially, the encoder is set to an all-zerostate. The tree diagram shows that if the first message (input) bit is ‘‘0’’, theoutput sequence is ‘‘000’’ and, if the first input bit is ‘‘1’’, the output sequenceis 111. If the first input bit is now ‘‘1’’ and the second bit is ‘‘0’’, the second setof three output bits will be 001. Continuing through the tree, we will be able toshow that if the third input bit is ‘‘0’’ the output will be 011. If the third inputbit is ‘‘1’’, the output will be 100. Supposing that a particular sequence takesFIGURE 6.4Rate 1/3 convolutional encoder.Copyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.