hes of the trellis shown in Figure 8.6. Chapter Eight Solution: First, the 8-ASK constellation is separated into parts, separating the constellation into points further and further apart, as shown in Figure E8.1. Next, we assign each point in the 8-ASK constellation to a branch in the trellis according to the two rules spelled out by Ungerboeck: all parallel transi- tions separated by a maximum distance, and all transitions diverging or merging get the next maximum distance. Applying this rule leads to the first two columns of Figure 8.9, where this time the O's, 1's, 7's refer to the points marked in Figure E8.1. The PSK constellation drawn in its orthonormal basis 01(t) Figure E8.1 Separating the 8-PSK points 8.3 The Receiver End of Things At the receiver side, Ungerboeck had an unusual idea: he would combine the demodu- lator and channel decoder together and they would operate as a single device, as shown in Figure 8.10. He would call his new unit the TCM decoder. We'll explain how it works using an example, shown in Figure 8.11. Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness Channel Demodulator decoder Combine to get T C M Decoder r(t) = s(t) + n(t) Figure 8.10 Ungerboeck's idea at the receiver side 8.3.1 The Input First, we'll figure out what is coming into the decoder (its input). Figure 8.11 (a) shows the communication system under consideration. The input to the channel coder is m = (00 00 00 00), which makes its output u = (000 000 000 000). These bits enter into the modulator. Looking back at the modulator in the previous section (we studied it in Section 8.2, and detailed it in Figure 8.9), we know that for each input of 000, the modulator outputs the signal corresponding to "0", which corresponds to - More importantly to our case in hand, for the input u = (000 000 000 000), the entire output corresponds to (as seen in Figure 8.11): (8.1) s(t) = s'(t) s2(t) (8.2) where si(t) = Acos(wt)n(t-(i-1)T). Graphically, we have s(t) corresponding to the plot shown in Figure 8.11(