ies, both of which have to do with integrating. And here they are. If you integrate two different Q (t) 's together you get 0, and if you integrate a Qj (t) with itself you get a 1. That is, in a nice concise equation form: [ 9,(1)9,(1)di=0,i = j (5.4) (5.5) Also in Equations (5.2) to (5.3), the coefficients {s,,i=1,2,...,M,j = N} are just numbers. These numbers, like S12 for example, tell you how much Q2(t) there is in S1 (t). Now, Equations (5.2) to (5.3) tell me there is a new way I can express signal S1 (t) . In the past, I either had to write out a mathematical equation (for example, S1 (t) = Vt ) or I had to draw you a picture. Now, I can write S1 = and you know that to make S1 (t) -all you do is the math of Equation (5.2). This is all very nice, but there are two very important things I haven't yet told you. How do we figure out the coefficients, {Sij i=1,2,... M, j = 1, 2, N}, and how do we figure out the signals The coefficents, from S11 to SMN, , are a piece of cake. All you have to do is this integration: (5.6) Getting It from Here to There: Modulators and Demodulators The orthonormal set {Q((t),...,ON(t)} is a little bit more work, but it's still quite easy. You simply follow the mathematical algorithm that I'm going to give you next, called the Gram-Schmidt orthogonalization procedure (big words, simple algorithm): (1) To get Q(t), just compute where E, = [s,(1)s,(1)da. (2) To Q2(t), compute where - and (3) To get P3(t), just compute where and Eq3= = (4) Keep going, up to QM(t), and if you get Qk(t)= = 0 along the way, just throw that one out, because you don't need it. Well, now you know this rather neat trick. If someone gives you a set of signals, you can figure out the set of signals (0,(t),...,ON(t)}, and you can figure out a set of coefficients (values) {Sipi, = 2, M, j = 1, 2, ... N}, and you can write their set of signals in terms of your Q;(t)'s and Sij's. Moreover, you can represent their signal as a vector, taking their S1 (t) and writing it as S1 = (S11,....,SIN). Chapter Five Example 5.3