s bigger, the frequency of s(t) gets bigger, and as x(t) gets smaller the frequency of s(t) gets smaller. Chapter Twelve Figure 12.11 (a) Information signal x(t) (b) Transmitted FM signal s(t) Let's see if we can characterize the sent signal s(t) in the frequency domain-that is, let's see if we can evaluate the Fourier transform of s(t), called S(f). . To help us out, let's start by rewriting s(t): s(1)=A.cos(a)(+0(1)) = (12.29) S = Re {Ace (12.30) (12.31) Analog Communications x(t) = cosWt highest lowest frequency frequency Figure 12.12 (a) Information signal x(t) (b) Sent signal s(t) in FM (t)= = Re{g(t)e (12.32) where g(t) = A eje(+) . Taking the Fourier transform of s(t) and applying properties of the Fourier transform, we end up with s(f) )=12G(f-fc)+1/2G(-f-fc) (12.33) Chapter Twelve where G(f) is the Fourier transform of and e(t)= The relationship between G(f) and x(t) is SO complex that there is no simple mathematical equation to relate the value G(f) to the value X(f)-that means there is no simple equation to relate S(f) to X(f). In the simple case when x(t) = cos (Wt) and W is small, we can derive an equation relating X(f) to S(f), but this equation is messy, involving Bessel functions, and I just want to offer an introduction to analog communications here. Look in the reference list to see a book that covers the joys of Bessel functions. Example 12.3 Draw the output of an FM modulator when the input corresponds to Figure E12.7. Solution: The output corresponds to equation (12.24), which shows that the output corresponds to a sinusoid with constant amplitude. Equation (12.28) tells us that the frequency of the FM output changes linearly with X (t). Putting this information together leads to the output plot of Figure E12.8. Figure E12.7 Input to FM modulator s(t)=Accos(Wct+(t)) Figure E12.8 Output of FM modulator Analog Communications 12.3.2 The Demodulator in FM We now know about how the FM modulators work and what they do. At the receiver side, you want to build an FM demodulator that gets the received