mple integral (3.13) This is called the Fourier transform of s(t). For the s(t) of equation (3.12), the S(f) is computed to be (3.14) While this looks messy, you can see a plot of part of S(f) (the |S(f)|) in Figure 3.9. So, now you can plot your signal in time, showing how it changes in time, or you can plot your signal in frequency, showing how much of each frequency you need to make up your time signal S (t). Your choice. Example 3.3 Determine the Fourier transform of the signal (t) = =8(t). We'll turn to equation (3.13) for help, where we find: (E3.23) A Review of Some Important Math, Stats, and Systems (E3.24) = e-j2nfo (E3.25) (E3.26) (E3.27) This S(f) is plotted in Figure E3.2. Interestingly, what we see here is that the signal s(t)=8(t) is made up of an equal amount of all the frequencies. Figure E3.2 The Fourier transform of S(t) IS(f) fc-1/T fc fc+1/T Figure 3.9 Plot of |s(f) 3.3.3 Bandwidth Oftentimes, engineers want to know one thing about a signal (t), its bandwidth. The bandwidth, s(t), is a way to tell someone how many frequency components you need to make up your signal s(t). This definition of "bandwidth of a signal" is a rather vague definition. Engineers have tried to come up with a more specific definition, but, for some reason, they just couldn't agree on one way to exactly define the words "band- width of a signal." They decided, just to please everyone, they'd use many different definitions. Here are some of these: Chapter Three IS(f)| absolute BW is 8 IS(f) peak value of S(f) null-to-null BW = 2/T fc-1/T fc+1/T IS(f) peak value = P value = P/2 value = P/2 3dB BW Figure 3.10 For a particular S(f), plot shows (a) absolute bandwidth, (b) null-to-null bandwidth, and (c) 3-dB bandwidth A Review of Some Important Math, Stats, and Systems Absolute bandwidth: the range of frequencies over which S(f) (or |S(f)|) is not zero (Figure 3.10(a)). For example, for the |S(f) of Figure 3.9, redrawn in Figure 3.10(a), this value is infinity. Null-to-null bandwidth: the range of frequencies in S(f) (or |S(f|