signal look like it has a 90-degree phase shift than it is to make it look like it has a 180- degree phase shift. So it's easier to make symbol errors at the receiver in 4-PSK than is with BPSK. We'll talk more about this when we look at demodulators, coming soon to a subsection near you. 5.3 Just-in-Time Math, or How to Make a Modulator Signal Look Funny When I was taking my Ph.D. comprehensive oral exam, one of my professors asked how I would get by with the little math I knew. I explained to him the concept, which he later called "just-in-time math." Basically, this is the idea that when my research required that I learn some more math, I'd read the math books then. In keeping with the spirit of "just- in-time math," I'm going to share with you some math tools that we'll need in order to understand demodulators (coming up next). The particular math skill you're going to learn is how to represent a signal as a point in space. Take a look at Figure 5.20. In part (a) of the figure, we've got a point in space. In part (b), you see that we can characterize that point as (1,1) by introducing an x-axis and a y-axis. It turns out that you can do the same thing with signals. Take a look at part (c) of the figure. There you see a square wave. In part (d), you see that we can represent this square wave as a point (1,1), which means that the signal is made up of one part of the x-axis signal and one part of the y-axis signal. All kinds of detail will follow. Figure 5.20 Representing points and signals P=(1,1) s(t)=(1,1) 01(t): 134 Chapter Five 5.3.1 The Idea The idea is this. Let's say you throw me a set of signals {s,(t), S2(t), ..., SM(t)}, and you make sure they have finite energy (and all the signals we look at will be finite energy signals). Then I can always represent these signals like this: (5.2) (5.3) {0,(t),...,QN(t)} are called an orthonormal basis. The number of Q ((t)'s (N) is always less than or equal to the number of s;(t)'s (M) you sent me. The {Q1 1(t),...,ON(t); , have two particularly noteworthy propert