In this section we will present a few methods that we will
use to not only help describe the convolutional codes, but also help analyze them. The first method
we will use to describe the code is generator matrix or polynomials. Below we show an encoder with
rate 1/2 and a constraint length of K 3. The corresponding generator polynomials are given as
follows [11]:
(5.51)
(5.52)
The block diagram of the R 1/2, K 3 encoder is shown in Fig. 5.22.
g2(x) 1 x2
g1(x) 1 x x2
242 CHAPTER FIVE
Tb Tb
+
+
Input Bit
Sequence
Output Bit
Sequence
Rb Rc = 2Rb
X
Y1
Y2
FIGURE 5.22 Rate 1/2, K 3, convolutional encoder.
Data is presented to the encoder 1 bit at a time at a rate of Rb Bps. Since the constraint length 3,
we can see that 3 bits are used to create the output bit sequence. The modulo 2 addition operations are
controlled by the generator polynomial. These polynomials have special properties (i.e., possess no
catastrophic states) and have been computer searched. Some of the details will be discussed later.
This is a rate 1/2 encoder, which means for every input bit presented to the encoder, 2 output bits
are generated. The output encoded bit rate, Rc, is twice the input bit rate, Rb.
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