2(t) Figure 8.12 The TCM decoder front end 8.3.3 The Rest of the TCM Decoder Ungerboeck noted that he had in his TCM decoder something he could write as one big vector: r=(r',r2,r',44) (8.9) He could write this as r=s+n (8.10) where S = (s1, s Superscript(2), s3, s4) (and si = (si,s2") represents the sent signal from the modula- tor and n = (n , n°, n°, n 4 (and represents the channel noise. He wanted to go from this long vector r to what he called m', a decision on the incoming bits m. The idea is this. You have r = (r1, r2, r3, r4). You'd like to decide on the four sent symbols S = (s 1, s Superscript(2), S3, s4). If you knew S, the signals sent from the modulator, then you could, by looking at the table of Figure 8.9, figure out m, the sent bits. So, given r, you want to figure out what I'll call s', your best guess on the transmit- ted S. To do this, you look at the trellis diagram, which you'll find redrawn in Figure 8.13. You find the one path through the trellis that creates an S as close as possible to the sent r. By this, I mean you find the path through the trellis that minimizes the value (8.11) Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness RECEIVE = time 1 time 2 time 3 time 4 Figure 8.13 The trellis diagram of Figure 8.8 - representing channel coding and modulation where S2 refers to the ith output symbol of the selected path in the trellis and r - si - I refers to the distance between ri = (ri,r2i) and S Superscript(2) = The one path through the trellis closest to r indicates the sent S, and from this you can then determine the m. Let's look at an example. In the example we've been considering throughout Section 8.3, the sent S corresponds to s=(si,s,s,s4) (8.12) where Let's assume A = 1 V and T = 2 sec, which leads us to S Superscript(a) = (siis) = We'll also assume that the noise n is a noise which leads to =s+n=(s+n,snsns4 = (8.13) where +n'=(1,0)+(0.1,0.1) =(1.1,0.1) (8.14) We can easily show, by an exhaustive search of the trellis diagram of Figure 8.13, that th