s changed to g(t). As a result, the output of the modulator is now (9.5) (9.6) where g(t) may be any shape you'd like, and I is shorthand for Ae16, . So far in all the modulators we've looked at, we've used g(t) = (t). Soon, as you read on, you'll see that will be a reason for using a different g(t). But for now, simply know that you have a new way to build modulators and an easy way to generalize them. Chapter Nine ejwct pulse coder shaper Symbol, li=Ajejo Ajej0ig(t-iT) Figure 9.3 Construction of new modulator with different pulse shaper Example 9.1 Consider a BPSK modulator. Express the output of a BPSK modulator in an equation that looks similar to Eq. (9.6). Solution: For starters, let's consider the output of a BPSK modulator at one time. It looks like (E9.1) where O = 0° or 180° (E9.2) Now, in a more general form (we can generalize the pulse shape), we write this equation according to (E9.3) Applying a little bit of math to the output of a BPSK modulator, we end up with (E9.4) Now, if we consider the output at a whole bunch of times, we have (E9.5) (E9.6) Channel Filtering and Equalizers where Aei180° (E9.7) 9.2 The Channel That Thought It Was a Filter We have just re-expressed the modulators that we saw in Chapter 5-in fact, we've found a way to generalize them somewhat. In Chapter 5 we also saw channels. Those channels added a noise, and the channel was modeled as shown in Figure 9.4(a). We are now going to present a new channel, a more general model. A channel may do more than add a noise-it may act as a filter c(t) and add a noise n (t). In this case, the channel is modeled as shown in Figure 9.4(b). The Channel r(t) = s(t) + n(t) Channel channel filter r(t)=s(t)=c(t)+((n(t) Figure 9.4 (a) Channel model seen so far (b) New channel model Let's say we have our new modulator from Figure 9.3, and the signal s(t) out of this modulator is sent across the channel, which acts as a filter, as shown in Figure 9.4(b). We therefore have the system shown in Figure 9.5. Let's evaluate the channel output r(t): -(t)=s(