12.5, the 4x(t) is drawn as a dotted line, and forms the envelope for the signal S (t). A - 4x(t) is also drawn as a dotted line. The signal s(t) is the solid "jiggly" line between the two dotted lines. (b) The output signal corresponds to (n==[x(r-l)+x(r-1)] (E12.3) For the X(f) of Figure E12.4, this S(f) corresponds to Figure E12.6 Figure E12.5 4x(t) The DSB-SC output for (a) -4x(t) -f - 100 -fc + 100 fc - 100 fc+100 Figure E12.6 Output of modulator 12.3 Frequency Modulation (FM) To some, FM is a dial on the radio where you can hear love songs, classic rock, or greatest hits. To engineers, FM is shorthand for frequency modulation. Analog Communications 12.3.1 The Modulator in FM The idea in frequency modulation is to map the information x(t) into the fre- quency of the transmitted signal s(t). Mathematically, what is done is this: Given an information bearing signal x(t), you send out over the channel (12.24) where (12.25) Looking at this equation, you really can't tell that the information x(t) is placed in the frequency of s(t). So let's do a little math that will show that x(t) has indeed been placed in the frequency of s(t). The frequency of s(t), at any moment in time t, is given (12.26) (12.27) (12.28) This tells us that at time t, the frequency of the sent signal s(t) is which indicates that x(t) is determining the frequency of s(t). Let's use pictures to see what is going on in FM. Let's say we have the informa- tion-bearing signal x(t) as shown in Figure 12.11(a). Using this, we can determine some important information about s(t): 1. At times when x(t) = -1, the frequency of s(t) is = 2. At times when x(t) = +1, the frequency of s(t) is w(t)w (t)w = Using this information, we can get the plot of s(t) shown in Figure 12.11(b). Here, we see the variation in the frequency of s(t) as a direct result of changes to x(t). For another example, take a look at Figures 12.12(a) and (b). There, we see the input x(t) = cos (Wt) (where W is a very small number). We also see the output in Figure 12.12(b)-as. x(t) get