cations 12.2.1 AM Modulators-in Time The basic idea in AM modulators is to take the information signal x(t) and map it into the amplitude of the sent signal s(t). Specifically, what happens is this: 1. Make sure that the amplitudes of x(t) are in the range [-1, 1]. If they exceed this range, update x(t) by multiplying by a scalar SO that it fits into the range [-1, 1]. In what follows, I will assume x (t) has amplitudes in the range [-1, 1]. 2. Send S (1)=A,(1+mx(t))cos = (wt) (12.2) where m is a value in the range [0,1] and is called the modulation index. Let's say the input coming in is the x(t) shown in Figure 12.3(a); that is, the incoming x(t) is a square wave. To create the output s(t), let's assume that m = 1. In this case, we can figure out the output as follows: Figure 12.3 (a) Input to AM modulator (b) Output of AM modulator (m = 1) Chapter Twelve 1. At times when x(t) = -1, we have an output S (t) 4 (1+mx(t)) cos (wet) (12.3) S (t) A (1+(-1)) cos (wt) (12.4) (t)=0 (12.5) 2. At times when x(t) = +1, we have an output S (t) = cos (wt) (12.6) S (t)=A,(1+1) cos (wt) (12.7) :(t)=2A.cos(w)t (12.8) These two results tell us that for the x(t) of Figure 12.3(a), we have the output shown in Figure 12.3(b). You can see from Figure 12.3(b) that the shape (dotted lines) of Figure 12.3(b) is the same as the x(t) in Figure 12.3(a). The x(t) is said to create the "envelope" of the sent signal s(t). Let's look at another example. In Figure 12.4(a), we see the input waveform x(t) = cos(Wt), where W is a very small value (close to 0). That means that s(t) has the form (using m = 1 again) 3(t)=A,(1+cosWt) cos (wt) (12.9) The plot of this is shown in Figure 12.4(b). Again, you can see that the x(t) = cos(Wt) shape forms the envelope of s(t). In general, to plot S (t) given x(t), you first plotx x(t). Then, plot mx(t). Next, plot 1 + mx(t) Continuing on, plot A (1 + mx(t))-most times, it's easy to go right from x(t) to A (1 + mx(t)). Finally, draw in dotted lines at A (1 + mx(t)) and its negative -A (1 + mx(t)). Between the