S (t) = S1 (t) =Acos( iT) -AV(T/2) W(T/2) 2AV(T/2) Modulator Output in signal space so(t) = S1 (t) = Acos(wit+1t/2)r(t iT) S2(t) = iT) Q (t) 53(t) = Acos(ust+3r/2)r(t - iT) AVT/2 2 AVT/2 Q1(t) AVT/2 Figure 8.4 Comparing BPSK to QPSK Chapter Eight Ungerboeck resolved this problem by coming up with "mapping by set partition- ing," shown in Figure 8.5. There you see a source coder, a new convolutional coder which takes two bits and maps into three bits, and a modulator using an 8-PSK constel- lation. Let's examine this figure. Source 8-PSK coder m1 m2 modulator mapping by set partitioning Convolutional Coder 2 bits in 3 bits out Figure 8.5 Ungerboeck's new transmitter First, the channel coder takes in two bits (m, m2) at each time and outputs three bits (u1, 1' U2,U3) at each time. To better understand its workings, we'll draw a trellis diagram for the channel coder as shown in Figure 8.6. First, the nodes (dots): there are four nodes, one for each possible (bit at position 1, bit at position 2) pair. Then the branches: the solid branch indicates that incoming bit M2 is 0; the dotted branch indicates that the incoming bit m2 is 1. If the branch is drawn on top (i.e., if it is the top branch of a pair) then that means the incoming bit M1 is 0; if it corresponds to the bottom branch, then that means the incoming bit M1 is 1. For a given state (node) and a given input (branch), the three-bit output that comes out of the channel coder is drawn above the branch. time O time 1 time 2 Now let me explain the modulator and how Ungerboeck said it would work. His idea was this: First, draw pretty pictures of the modulator outputs. Look at Figure 8.7, where we see on top the eight possible outputs of an 8- PSK constellation, drawn on the orthonormal basis (look back to Chapter 5 for a refresher if you'd like). Figure 8.6 Trellis diagram describing channel coder Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness Q2(t) = /(2/T)sin(0),t)r((t-it) 1 (t) = v(2/T)cos((i),t)r(t-it) Figure 8.7 Splitting the 8