e path through the trellis that has a corresponding S = (s , s Superscript(2), S3, s4) closest to r is that shown in Figure 8.14. Now, looking at the branches of this path through the trellis (solid branch indicates M2= = 0, top branch indicates M1 = 0), we immediately figure out that the decoder should output m'=(00 00 00 00 00). Chapter Eight so=(1,0) so=(1,0) so=(1,0) so=(1,0) Figure 8.14 Best path through trellis given r = ((1.1,0.1), (1.1, 0.1), (1.1, 0.1), (1.1, 0.1)) Example 8.2 Assume, as in the section just before this example, that a rate 2/3 convolutional coder is followed by an 8-PSK modulator. Let the trellis diagram describing these operations correspond to Figure 8.8. Assume the input bits are m = (00 00 00 00), which means the output from the convolutional coder is then V = (000 000 000 000), and the output from the modulator is thus the 8-PSK output that can be expressed on an orthonormal basis as S = (s , s Superscript(2), S3, s4) = ((1,0), (1,0), (1,0), (1,0)) (E8.1) Now, assume that the channel is particularly noisy, and somehow the de- modulator receives r=s+n = ((1,0), (1,0), (1,0), (1,0)) + ((0.9, 0), (0.9,0), (0.9, 0), (0.9, 0)) = ((1.9,0), (1.9,0), (1.9, 0), (1.9, 0)) (E8.2) Determine the output of an optimal decoder. Solution: At this point in the chapter, an exhaustive search through the trellis would have to be performed to determine the best path. However, in this case, a little bit of thinking let's us avoid an exhaustive search. As you can see in Figure E8.2, the received value (1.9, 0) is closer to (1, 0) than any other point in the 8-PSK constellation. This tells us that if the path corre- sponding to outputs of (1, 0) exists, this would be the closest path to inputs (1.9, 0). Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness (1, 0) (1.9, 0) closest point in an 8-PSK constellation Figure E8.2 Closest point to (1.9, 0) in our 8-PSK constellation That (1, ,0)'s path does indeed exist-it's the top line through the trellis. So, following this top line, whi