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Sec. 1–11 Coding 23
Convolutional Codes
A convolutional encoder is illustrated in Fig. 1–5. Here k bits (one input frame) are shifted in
each time, and, concurrently, n bits (one output frame) are shifted out, where n 7 k. Thus,
every k-bit input frame produces an n-bit output frame. Redundancy is provided in the output,
because n 7 k. Also, there is memory in the coder, because the output frame depends on the
3
previous K input frames, where K 7 1. The code rate is R = k/n, which is
4
in this illustration.
The constraint length, K, is the number of input frames that are held in the kK-bit shift
register. † Depending on the particular convolutional code that is to be generated, data from the
kK stages of the shift register are added (modulo 2) and used to set the bits in the n-stage
output register.
For example, consider the convolutional coder shown in Fig. 1–6. Here,
k = 1, n = 2, k = 3, and a commutator with two inputs performs the function of a
two-stage output shift register. The convolutional code is generated by inputting a bit of
data and then giving the commutator a complete revolution. The process is repeated for
successive input bits to produce the convolutionally encoded output. In this example, each
k = 1 input bit produces n = 2 output bits, so the code rate is R = k/n = 1 2
. The code
tree of Fig. 1–7 gives the encoded sequences for the convolutional encoder example of
Fig. 1–6. To use the code tree, one moves up if the input is a binary 0 and down if the input
is a binary 1. The corresponding encoded bits are shown in parentheses. For example, if
Convolutional encoder
k-bit frame
Constraint length = K input frames
Shift register
(kK bits)
Uncoded input data
•••••••••••••
Logic
n bit frame
Shift register
(n bits)
n bits
Coded output data
Figure 1–5 Convolutional encoding ( k = 3, n = 4, K = 5, and R = 3 4 ) .
† Several different definitions of constraint length are used in the literature [Blahut, 1983; Clark and Cain,
1981; Proakis, 1995].