 First, . To get it, all we have to do is use the integral equation (5.6). The next four lines show the integral computation to get that ri: (5.17) (5.18) (5.19) Here, Sm1 is just the coefficient of (t) along Q1(t). (In Figure 5.22, it's the X- axis part of the 4-PSK signal.) What about the ni? Using some statistical arguments which I won't present, N1 turns out to be a Gaussian random variable with 0 mean and variance 02 How nice. Next, let's take a look at r2 Again, we can figure this one out by using the inte- gral equation (5.6). The next line shows you what happens when you do it: Chapter Five 12=Sm2+N2 (5.20) This time, Sm2 is the coefficient of Sm(t) along Q2(t). . (In Figure 5.22, it's the y axis component.) Here, N2 is a Gaussian random variable with 0 mean and variance o'n Not only that, but after some statistical wrangling (about 12 lines long), you can show that N2 and N1 are statistically independent. That is, knowing N1 tells you nothing about N2, and knowing N2 tells you zippo about N1. On to r3 Pulling equation (5.6) out of our pocket and popping it down on paper leads us to (5.21) (5.22) (5.23) (5.24) (5.25) Now because Q1(t) and Q3(t) are parts of an orthonormal basis, then by defini- tion (or by looking back up at Equation (5.4)), we see that their integral is 0; and that also happens for Q2(t) and Q3(t). This realization leads us to: 3=Sm1.0+Sm2.0+n3`(5.26) (5.27) Now, N3 is (using a few lines of statistical analysis) a Gaussian random variable, independent of N1 and independent of N2. Similarly, we can compute r4, V5, and SO on, and we'll find r4 = N4 (5.28) r5=N5 (5.29) And SO on. Here the N4, N5, yada yada yada, are all Gaussian random variables independent of all the other Ni terms. The receiver front end looks at this and says: "You know, I really only want to give the rest of the receiver information that's useful for detection. Certainly I'll give it r'1 Getting It from Here to There: Modulators and Demodulators and r2 , because these contain an information term. But there really is no 