Question

A sinusoidal carrier wave of amplitude Ac and angular frequency ωc is modulated in accordance with a sinusoidal information signal of amplitude Am and angular frequency ωm. Show that the amplitude modulated signal contains three frequencies centered around ωc. Draw the frequency spectrum of the resulting modulated signal.

Answer 1

When is small (say less than radians per second, or 10 Hz), the signal is heard as a ``beating sine wave'' with  beats per second. The beat rate is twice the modulation frequency because both the positive and negative peaks of the modulating sinusoid cause an ``amplitude swell'' in  . (One period of modulation seconds--is shown in...... The sign inversion during the negative peaks is not normally audible. Subtracting this fromleads to the identity
Setting and gives us an alternate form for our ``ring-modulator output signal''These two sinusoidal components at the sum and difference frequencies of the modulator and carrier are called side bands of the carrier wave at frequency  (since typically ).

Equation expresses  as a ``beating sinusoid'', while Eq.expresses it as two unmodulated sinusoids at frequencies. Which case do we hear?

It turns out we hear as two separate 

Answer 2

The angular frequency of message signal or informational signal is ωm.

The amplitude of message signal or informational signal is Am.

The angular frequency of carrier signal is ωc.

The amplitude of carrier signal is Ac.

So, the frequency spectrum of the resulting modulated signal is given below where the amplitude is plotted in Y axis and the frequency is plotted on X axis.

*USB stands for Upper Side Band of the modulated signal.

*LSB stands for Lower Side Band of the modulated signal.