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A periodic modulated (AM) radio signal has the form $\mathit{y}\mathbf{=}\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{Ï€}\mathit{f}\mathit{t}\mathbf{)}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{Ï€}{\mathit{f}}_{\mathbf{e}}\left(t-\frac{x}{v}\right)$.

The factor$\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{Ï€}{\mathit{f}}_{\mathbf{c}}\mathbf{(}\mathit{t}\mathbf{-}\mathit{x}\mathbf{/}\mathit{v}\mathbf{)}$ is called the carrier wave; it has a very high frequency (called radio frequency;${\mathit{f}}_{\mathbf{c}}$ is of the order of ${\mathbf{10}}^{\mathbf{6}}$cycles per second). The amplitude of the carrier wave is$\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{Ï€}\mathit{f}\mathit{t}\mathbf{)}$ . This amplitude varies with time-hence the term "amplitude modulation"-with the much smaller frequency of the sound being transmitted (called audio frequency;fis of the order of ${\mathbf{10}}^{\mathbf{2}}$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 oftfor x = 0 over two periods of the amplitude function:$\mathit{A}\mathbf{=}\mathbf{3}\mathbf{,}\mathit{B}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{f}\mathbf{=}\mathbf{1}\mathbf{,}{\mathit{f}}_{\mathbf{c}}\mathbf{=}\mathbf{20}$ . Using trigonometric formulas, show thatycan be written as a sum of three waves of frequencies${\mathit{f}}_{\mathbf{c}}{\mathit{f}}_{\mathbf{c}}\mathbf{+}\mathit{f}$ , and${\mathit{f}}_{\mathbf{c}}\mathbf{-}\mathit{f}$ ; the first of these is the carrier wave and the other two are called side bands.

Step by step solution

01

Given information

A periodic modulated (AM) radio signal has the form .

$y=(A+B\sin 2\mathrm{Ï€}ft)\sin 2\mathrm{Ï€}{f}_c\left(t-\frac{x}{v}\right)$

02

Step 2: Formula of trigonometry

The formula of trigonometry is $\mathit{s}\mathit{i}\mathit{n}\mathit{a}\mathit{s}\mathit{i}\mathit{n}\mathit{b}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\left[}\mathit{c}\mathit{o}\mathit{s}\mathbf{\right(}\mathit{a}\mathbf{-}\mathit{b}\mathbf{)}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathit{a}\mathbf{+}\mathit{b}\mathbf{\left)}\mathbf{\right]}$.

03

Write the sum of three waves of frequencies

The sum of three waves of frequencies are$f_c,f_c+f$, and$f_c-f$ .

$$\begin{gathered} y=(A+B\sin 2\mathrm{π}ft)\sin 2\mathrm{π}{f}_c(t\left.-\frac{x}{v}\right) \\ y=A\sin 2\mathrm{π}{f}_c\left(t-\frac{x}{v}\right)+B\sin (2\mathrm{π}ft)\sin 2\mathrm{π}{f}_c\left(t-\frac{x}{v}\right) \\ y=A\sin 2\mathrm{π}{f}_c\left(t-\frac{x}{v}\right)+\frac{B}{2}\left[\cos 2\mathrm{π}\left(t\left(f-f_c\right)+\frac{x}{{\mathrm{λ}}_c}\right)-\cos 2\mathrm{π}\left(t\left(f+f_c\right)-\frac{x}{{\mathrm{λ}}_c}\right)\right] \end{gathered}$$

04

Computer plot over two periods of the amplitude T = 1   is

The plot over two periods is shown below.

Thus, the sum of three waves frequencies are

$y=A\sin 2\mathrm{π}{f}_c\left(t-\frac{x}{v}\right)+\frac{B}{2}\left[\cos 2\mathrm{π}\left(t\left(f-f_c\right)+\frac{x}{{\mathrm{λ}}_c}\right)-\cos 2\mathrm{π}\left(t\left(f+f_c\right)-\frac{x}{{\mathrm{λ}}_c}\right)\right]$

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