8.6.1 Description

In convolutional codes, a codeword is obtained by passing a binary information sequence through a finite-length shift register. A typical convolutional encoder is depicted in Fig. 8.15. Note that N _in = 1 and N _out = 2. We will consider only codes with N _in = 1. The encoder consists of a sequence of L memory blocks (registers), and binary adders. In our example, L = 3. Observe that the code is systematic, since .

Figure 8.15: A recursive systematic convolutional encoder with N _in = 1 and N _out = 2.

Convolutional codes are usually described by feedforward and feedback polynomials, reflecting the relation between the outputs and the values in the registers. In our example, the feedback polynomial is g _FB( D) = 1 + D ² + D ³ (since we feed back the output of the second and third register), while the feedforward polynomial is g _FF( D) = 1 + D + D ³ (since we feed forward the output of the first and third register). When the feedback polynomial is trivial (i.e., g _FB( D) = 1), we say that the code is non-recursive. In our example from Fig. 8.15, we have a recursive systematic convolutional code.

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