Figure E7.1 Convolutional Coder Channel Coding and Decoding: Part 2-Convolutional Coding and Decoding Figure E7.2 The trellis diagram 7.2 Channel Decoding Let's now explore what the channel decoder does. First, we'll define the task of the channel decoder. In Figure 7.6, at the channel coder, a bit sequence m comes in; we'll consider the channel coder of Figure 7.2 with m = (000) coming in. This is mapped to the output, which we'll call u; in the example we're considering, u = (00 00 00). These six bits are sent across the channel by the modulator and returned to six bits by the de- modulator. The bits that come out of the demodulator, which feed the channel decoder, will be referred to as V. These bits V may or may not be equal to u. An error may have occurred in transmission, resulting in V =u+e, where e represents the bit error. For the example we're considering, we'll say we receive V =u+e = (00 00 00) + (00 00 01) = (00 00 01) In this case, one bit error has occurred in transmission at position 6. The goal of the channel decoder, Convolutional Coder then, is to take the of Figure 7.2 Modulator bits V that it has m=(000) (takes in 1 bit at a time u = (00 00 00) and outputs 2 bits) received, and come up with the best guess at m, the original information bits. We'll call the channel decoder's guess at m the vector Convolutional Demodulator m'. We want to find a m' = best guess on m Decoder V = (00 00 01) way to make m' as a bit error occurred here close to m as pos- Figure 7.6 The role of the channel (convolutional) decoder sible. Chapter Seven 7.2.1 Using a Trellis Diagram How does the channel decoder use a "trellis diagram" to help it figure out how to put out an m' that matches m? The trellis diagram of Figure 7.5(b) shows the possible output bits of the channel coder. If you look at the top of the dotted and dashed lines of the trellis diagram, you see those possible outputs. Let's return to our example of Figure 7.6 and consider the transmitted output = (00 00 00). We can use the trellis diagram to verify 