ce based on Equation (5.60). There's really just one popular way to do it, shown in Figure 5.30. The value marked A in Figure 5.30 is actually equal to r.s, and the value marked B is actually equal to r.sm. . With just a little looking and a little thinking, I believe it becomes apparent that the demodulator of Figure 5.30 carries out the optimization of Equation (5.60). Let's do some math that shows that indeed A is equal to r.s: (5.61) (5.62) (5.63) Getting It from Here to There: Modulators and Demodulators s1(t) Choose table SM(t) Figure 5.30 The matched filter receiver-Version 2 (5.64) A=r.s (5.65) In an identical manner, we can show that B is r.SM . And that's it. Take this, add some looking, throw in some thinking, and, voila, Figure 5.30 does indeed implement the decision shown in Equation (5.60). 5.5 How Good Is It Anyway (Performance Measures) Of course, anytime you build something, you want to be able to tell people how good it is. "It's great!" you might declare. "A four-star performance!" While these descriptions work well for movie reviewers, they're rarely sufficient for number-oriented engineers. "Exactly how wonderful is it?" those engineers will ask. "Give me a number." You might think to reply "A perfect 10!", but they'll want a more descriptive number. This section is all about providing such a number. 5.5.1 A Performance Measure First, we've got to decide what number we want to give those engineers, and what that number will describe. Let's come up with a way to numerically measure the perfor- mance of a modulator-demodulator pair. The best way is to ask a simple question: What's the most important thing in a modulator-demodulator pair? Well, the most important thing is that the demodulator correctly decide what the modulator sent. So, if you want to tell someone how your modulator-demodulator pair is doing, you'll want to tell them P(e), the probability that the demodulator makes an error when it's deciding what signal was sent. The smaller the P(E), the better your demodulator. Chapter Five 