ning modulation symbols to the modulator output labeled 4 incoming bits using the trellis (see Figure 8.8, marked (b)). Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness Next we'll move to the branches going from node 0 to node 2. There are two branches here, SO following rule (1), we want to assign to these branches two points as far apart as possible-giving the branches either the points 1 and 5, or 2 and 6, or 3 and 7 (Figure 8.7). We seem to have a choice here. But now rule (2) comes into play, and it tells us which of these to choose. Rule (2) says that all branches leaving the same node must be given the next maximum distance-in our case, since "0 and 4" are already leaving the node 0, we'll want to choose "2 and 6" to be the other points leaving the node 0 (since these points are farther from 0 and 4 than our other choices of "3 and 7" or "1 and 5"). This is shown in Figure 8.8 (marked (c)). From here, we can see that the input bits (0 1 1) are mapped to the modulator output labeled 2, and the input bits (111) are mapped to the modulator output labeled 6 (see Figure 8.8, marked (d)). We can continue in this way, moving through our entire trellis, and we quickly end up with the trellis shown in Figure 8.8. This indicates that the modulator maps input bits to output symbols as you can see in Figure 8.9. And with that, Ungerboeck smiled, for he had just devised a smart way to build modulators that were matched to channel coders. Output symbol number Output symbol Output signal = (AVT/2, O) so(t)=Acos(Wct)r(t-iT) s,(t) = s3(t) = Acos(Wct+3/4)r(t-iT) S2=(0,AVT/2) s2(t) S4(t)=Acos(Wct+r)(t-iT) ss(t) =Acos(Wct+5/4)(t-iT) s.(t) Acos(Wct+7/4)r(t-iT) se=(0,-A/V2) s6(t) =Acos(Wct+3/2)r(t-iT) Figure 8.9 8-PSK modulator input and output created using trellis of Figure 8.8 Example 8.1 For this example, we'll use the convolutional coder of Figure 8.5. But this time, an 8-ASK modulator will be used, in place of an 8-PSK modulator. Match the outputs of the 8-ASK modulator to the different branc