Question

State the meaning of amplitude modulation. Show that an amplitude modulated wave can be regarded as ‘made up’ of three sinusoidal waves, one having amplitude ${E_C}$ and other two having amplitudes of $\dfrac{{{E_M}}}{2}$ each.

Answer 1

Hint: In electronics and communication physics, modulation is a process of changing various parameters of a primary signal caller carrier signal with another signal called modulation signal and these parameters such as their amplitude or frequency. The general signal waves are mathematically written in sine and cosine trigonometric forms.

Complete step by step answer:
When the modulation of a carrier wave or carrier signal is made by varying the amplitude of the modulation wave, such modulation is known as amplitude modulation. Mostly, amplitude modulation is used to send messages through radio carrier waves.

Let us suppose wave having amplitude of ${E_C}$ is represented by C (t) which can be written as $C(t) = \sin {\omega _c}t$ and wave having amplitude of $\dfrac{{{E_M}}}{2}$ is represented by M (t) which can be written as,
$M(t) = \cos ({\omega _m}t + \phi )$
Where $\phi $ is the phase difference between these two waves.
Now, the complete modulated wave can be written as
$y(t) = [A + M \times M(t)]C(t)$
Putting the values of each parameters we get,
$y(t) = [A + M \times M(t)]C(t)$
$\Rightarrow y(t) = [A + M\cos ({\omega _m}t + \phi )]\sin {\omega _c}t$
$\Rightarrow y(t) = A\sin {\omega _c}t + M\sin {\omega _c}t \times \cos ({\omega _m}t + \phi )$

Multiply and divide by two we get,
$y(t) = A\sin {\omega _c}t + \dfrac{M}{2}(2\sin {\omega _c}t \times \cos ({\omega _m}t + \phi ))$
Using trigonometric formula, $\sin (A + B) + \sin (A - B) = 2\sin A\cos B$ we get,
$y(t) = A\sin {\omega _c}t + \dfrac{M}{2}[(\sin ({\omega _c} + {\omega _m})t + \phi ) + (\sin ({\omega _c} - {\omega _m})t - \phi ))]$
Where, $A = {E_C}$ and $M = {E_M}$ hence,
The equation $y(t) = A\sin {\omega _c}t + \dfrac{M}{2}[(\sin ({\omega _c} + {\omega _m})t + \phi ) + (\sin ({\omega _c} - {\omega _m})t - \phi ))]$
Represents a modulation wave made of three sinusoidal waves having amplitudes of ${E_C}$ and other two having amplitudes of $\dfrac{{{E_M}}}{2}$ each.

Note: It should be remembered that, not only amplitude modulation exist, we can also change the frequency of the carrier waves in order to modulate them and such modulation where frequency is changed for modulation are known as Frequency modulation, the most common example of frequency modulation is Signals of music and radio sounds. The amplitude modulation and frequency modulation are generally written as AM and FM.