Chapter 7.3, Problem 8P

A periodic modulated (AM) radio signal has the form $y=(A+B\sin 2\pi ft)\sin 2\pi f_e\left(t-\frac{x}{v}\right)$.

The factor $\sin 2\pi f_c(t-x/v)$ is called the carrier wave; it has a very high frequency (called radio frequency; $f_c$ is of the order of ${10}^6$ cycles per second). The amplitude of the carrier wave is $(A+B\sin 2\pi ft).$ This amplitude varies with time-hence the term "amplitude modulation"-with the much smaller frequency of the sound being transmitted (called audio frequency; $f$ is of the order of ${10}^2$ cycles per second). In order to see the general appearance of such a wave, use the following simple but unrealistic data to sketch a graph of $y$ as a function of $t$ for $x=0$ over two periods of the amplitude function: $A=3,B=1,f=1,f_c=20.$ Using trigonometric formulas, show that $y$ can be written as a sum of three waves of frequencies $f_c$ $f_c+f,$ and $f_c-f;$ the first of these is the carrier wave and the other two are called side bands.