irst, there's S(t), which is called the impulse function (or delta function, or just impulse). This function is very tall and very skinny, and centered around 0. To be a bit more specific regarding what it looks like, take a look at Figure 3.5. There, you'll see a function of height 1/T and of duration T. As T goes to 0, this function becomes very, very skinny and very, very tall. When T goes to 0, the plot in Figure 3.5 corresponds to S(t). So, looking closely at Figure 3.5 as T goes to 0, we can say three things about S(t): (1) S(t) is infinitely tall; (2) S(t) is infinitely skinny; and (3) the area under the S(t) function is 1; that is, mathematically, [8(t) dt ==1 S(t) = this plot as T-> 0 Figure 3.5 Describing d(t) The most common way to represent S(t) using the plot is shown in Figure 3.6. The next function that we'll use frequently is a square wave function, which I'll call (t). This function is of height 1 and duration T. Mathematically, you can write (t) using the simple equation A Review of Some Important Math, Stats, and Systems 0, else (3.11) You can also see what (t) looks like graphically in Figure 3.7. Figure 3.6 The most common way to represent S(t) graphically Figure 3.7 The signal (t) 3.3.2 Another Way to Represent a Signal: The Fourier Transform There are lots of ways to represent a signal s(t). One easy way is to write a mathemati- cal equation such as S (t) = cos (2 ft t.ru (t) (3.12) Chapter Three s(t) = COS (2rfct) (t) Figure 3.8 Plot of s(t) Another way is to draw a picture, which shows you what s(t) looks like and how it changes with time. For example, for the s(t) of equation (3.12), you can see its plot in Figure 3.8. But there's another way to describe signals, discovered by a fellow named Fou- rier. He realized that any time signal s(t) can be created by adding together cosine and sine waveforms with different frequencies. You can describe s(t) by indicating how much of each frequency f you need to put together to make your signal s(t). To figure this out, all you have to do is the si