Question

In amplitude modulation, sinusoidal carrier frequency used is denoted by ${\omega _c}$ and the signal frequency is denoted by ${\omega _m}$. The bandwidth ($\Delta {\omega _m}$) of the signal is such that, $\Delta {\omega _m} < < {\omega _c}$. The frequency not contained in the modulated wave is then,
(A) ${\omega _c} - {\omega _m}$
(B) ${\omega _m}$
(C) ${\omega _c}$
(D) ${\omega _m} + {\omega _c}$

Answer 1

Hint: The amplitude modulation is modulation of a wave by varying its amplitude. In amplitude modulation, the amplitude of the carrier wave is varied in proportion to that of the message signal being transmitted. The modulation wave has namely three frequencies f1, f2 and f3 corresponding to $({\omega _c} + {\omega _m})$, ${\omega _c}$ and $({\omega _c} - {\omega _m})$ in terms of angular frequencies. ${\omega _c}$ is the carrier wave frequency, $({\omega _c} + {\omega _m})$ is the upper side band frequency and $({\omega _c} - {\omega _m})$ lower side band frequency.
If the bandwidth $\left( {\Delta {\omega _m}} \right)$ of the signal is such that $\Delta {\omega _m} < < {\omega _c}$, then the frequency spectrum lies within $({\omega _c} + {\omega _m})$,${\omega _c}$ and $({\omega _c} - {\omega _m})$.

Complete step by step solution:
Given,
The sinusoidal carrier frequency is ${\omega _c}$.
The signal frequency is ${\omega _m}$.
The bandwidth is given as $\Delta {\omega _m} < < {\omega _c}$.
Here, $({\omega _c} - {\omega _m})$ is called the lower side frequency band and $({\omega _c} + {\omega _m})$ is called upper side frequency band. The modulated wave consists of the carrier wave frequency ${\omega _c}$ and addition and subtraction of the signal frequency in the side bands. The frequency spectrum is shown in the figure below.

It can be observed that the frequency of the modulated wave lie between $({\omega _c} + {\omega _m})$ and $({\omega _c} - {\omega _m})$ but ${\omega _m}$ does not lie in between the spectrum. It means the frequency which is not contained in the modulated wave is ${\omega _m}$.

Hence, the correct answer is (B).

Note: The students have to understand the amplitude modulation and the frequency corresponding to in amplitude modulation. The frequency of the carrier waves and the modulated wave lies in between a band called upper sideband and lower side band. One can also obtain the answer from the frequency spectrum graph of the amplitude modulated wave.