B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

830 ERROR CORRECTING CODES
Fi g ure 14.7
(a) State and
(b) state transition
diagram of
the encoder in
Fig. 14.5.
State
labels
a:
h:
0
0
0
b: Dill
c:
OQ]
c:
d:
0
1/1 I
(a)
(b)
Fi g ure 14.8
Trellis diagram
for the encoder
in Fig. 14.5.
a: @Q]
0/00 0/00
Branch labels: input bits / output bits
0/00 0/00 0/00
0/00
a: [QQJ
h: @I]
h: [ill
c:
[iQ]
c: OQ]
d:
IT!]
the encoder is in state a and the input is 0, the output is 00 (solid line), and the encoder remains
in state a. Note that the encoder cannot go directly from state a to states c or d. From any given
state, the encoder can go to only two states directly by inputting a single data bit. This is an
extremely important observation, which will be used later. The encoder goes from state a to
state b (when the input is 1), or to state a (when the input is 0), and so on. The encoder cannot
go from a to c in one step. It must go from a to b to c, or from a to b to d to c, and so on. We
can also verify these facts from the code tree. Figure 14.7b contains the complete information
of the code tree.
Trellis Diagram
Another useful way of representing the code tree is the trellis diagram (Fig. 14.8). The diagram
starts from scratch (all Os in the shift register, i.e_, state a) and makes transitions corresponding
to each input data digit. These transition branches are labeled just as we labeled the state
transition diagram. Thus, when the first input digit is 0, the encoder output is 00, and the trellis
branch is labeled 0/00. This is readily seen from Fig. 14.7b. We continue this way with the
second input digit. After the firsttwo input digits, the encoder is in one of the four states a, b, c, or
d, as shown in Fig. 14.8. If the encoder is in state a (previous two data digits 00), it goes to state
b if the next input bit is 1 or remains in state a if the next input bit is 0. In so doing, the encoder
output is 11 (a to b) or 00 (a to a). Note that the structure of the trellis diagram is completely
repetitive, as expected, and can be readily drawn by using the state diagram in Fig. 14_7b.