comes in creates two bits, each of duration T/2. Finally, the modulator, a QPSK modulator, takes every two bits and maps them to a symbol as shown in Figure 8.3. Figure 8.3 shows the output of the system in the time and frequency domain. Looking at this output in the frequency domain, we see that the signal output has a bandwidth (null-to-null bandwidth) of BW = 2/T. This bandwidth is the same band- width as that of the original (no channel coding system) of Figure 8.2(a). Basically, what Ungerboeck realized (and what Figures 8.2 and 8.3 confirm) is that if you introduce a channel coder, and follow it by a modulator that takes in more bits, then you can use channel coding without increasing the bandwidth of the trans- mitted signal. That turned out to be a very good idea. Convolutional Source coder Channel Modulator Coder s(t)= Acos(w)t33t/2) Acos(w(t+r/2) Acos(w(t+1/2) s(t) = Acos(Wct + 3/2)(t) + Acos (Wct +1/2)r(t-T) + Acos(Wct + t/2)r(t-2T) f-1/T fc fc+1/T BW = 2/T Figure 8.3 Ungerboeck's idea for a new transmitter Trellis-Coded Modulation (TCM): The Wisdom of Modulator and Coder Togetherness 8.2 Improving on the Idea There were still a number of details that had to be resolved before this idea was em- braced by the engineering community. In Figure 8.4(a) you see the outputs of a BPSK modulator, which are and Acos (wt + 180°) nt - iT). An error will only occur if the noise of the channel is big enough to make a 180-degree phase shift occur. In Figure 8.4(b) you see a drawing of the outputs of the QPSK modulator, which shows that the signals sent by the modulator are Acos(wt) nt - iT), - Acos (wt + 180°) (t - iT) and Acos(at+270°) (t -iT). In this case, an error occurs if the noise is big enough to make it look like a 90-degree phase shift has occurred. By adding a QPSK modulator in place of a BPSK modulator, the noise in the system will introduce more errors-possibly SO many more that the system of Figure 8.3 will be lousy when compared to the systems of Figure 8.2(a) and (b). Modulator Output in signal space 