Question

In amplitude modulation, equation of messenger wave is ${{x}_{1}}={{A}_{0}}\sin {{\omega }_{m}}t$ and that of carrier wave is ${{x}_{2}}={{A}_{c}}\cos {{\omega }_{c}}t$. The equation of amplitude modulated wave is:
$A)x={{A}_{c}}\cos {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{2}\left[ \sin ({{\omega }_{m}}+{{\omega }_{c}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t \right]$
$B)x={{A}_{c}}\cos {{\omega }_{c}}t-\dfrac{{{A}_{0}}}{2}\left[ \sin ({{\omega }_{m}}+{{\omega }_{c}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t \right]$
$C)x={{A}_{c}}\cos {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{4}\left[ \sin ({{\omega }_{m}}+{{\omega }_{c}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t \right]$
$B)x={{A}_{c}}\sin {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{4}\left[ \sin ({{\omega }_{m}}+{{\omega }_{c}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t \right]$

Answer 1

Hint: In amplitude modulation, the amplitude of the carrier signal is varied in proportion to the amplitude of the message signal. It is a modulation technique, generally used in electronic communication. Amplitude modulation affects only the amplitude of the carrier wave and not the frequency and the phase of the carrier wave. The general expression for amplitude modulation is written down and necessary simplifications and rearrangements using trigonometric relations are done to match the correct option in the provided answers.

Complete step-by-step solution:
The general expression for a carrier signal is represented as
${{x}_{c}}(t)={{A}_{c}}\sin {{\omega }_{c}}t$
where
${{x}_{c}}(t)$ is the carrier signal and is expressed as a function of time $t$
${{A}_{c}}$ is the amplitude of the carrier signal
${{\omega }_{c}}$ is the angular frequency of the carrier signal
Similarly, the general expression for a message signal is given by
${{x}_{m}}(t)={{A}_{m}}\sin {{\omega }_{m}}t$
where
${{x}_{m}}(t)$ is the message signal and is expressed as a function of time $t$
${{A}_{m}}$ is the amplitude of the message signal
${{\omega }_{m}}$ is the angular frequency of the message signal
Note that both carrier signals as well as message signals are sine waves and are expressed accordingly.
They can also be expressed as cosine waves.
The general expression for an amplitude modulated signal is represented as
$x(t)=\left[ {{A}_{c}}+{{x}_{m}}(t) \right]\sin {{\omega }_{c}}t$
where
$x(t)$ is the amplitude modulated signal and is expressed as a function of time $t$
${{A}_{c}}$ is the amplitude of the carrier signal
${{x}_{m}}(t)$ is the message signal
${{\omega }_{c}}$ is the angular frequency of the carrier signal
Let this be equation 1.
It can be easily understood that the amplitude modulated signal is in the form of a carrier signal, but with an extra term in its amplitude, which is nothing but the message signal. This suggests that modulation is done to the amplitude of the carrier signal.
From the question, it is provided that the carrier signal is given by
${{x}_{2}}={{A}_{c}}\cos {{\omega }_{c}}t$
where
${{x}_{2}}$ is the carrier signal
${{A}_{c}}$ is the amplitude of the carrier signal
${{\omega }_{c}}$ is the angular frequency of the carrier signal
Also, the message signal is given as
${{x}_{1}}={{A}_{0}}\sin {{\omega }_{m}}t$
where
${{x}_{1}}$ is the message signal
${{A}_{0}}$ is the amplitude of the message signal
${{\omega }_{m}}$ is the angular frequency of the message signal
Let these two signals superimpose in such a manner that amplitude modulation is carried out between them. Using equation 1, the expression for amplitude modulated wave formed here is given by
$x(t)=x=\left[ {{A}_{c}}+{{x}_{1}}(t) \right]\cos {{\omega }_{c}}t=\left[ {{A}_{c}}+{{A}_{0}}\sin {{\omega }_{m}}t \right]\cos {{\omega }_{c}}t$
where
$x(t)$ or $x$ is the amplitude modulated signal
Let this be equation 2.
Let us simplify equation 2 further by expanding the terms as follows.
$x(t)=\left[ {{A}_{c}}+{{A}_{0}}\sin {{\omega }_{m}}t \right]\cos {{\omega }_{c}}t={{A}_{c}}\cos {{\omega }_{c}}t+{{A}_{0}}\sin {{\omega }_{m}}t\cos {{\omega }_{c}}t$
Multiplying and dividing by 2, the second term of the sum, we have
$x(t)={{A}_{c}}\cos {{\omega }_{c}}t+{{A}_{0}}\sin {{\omega }_{m}}t\cos {{\omega }_{c}}t={{A}_{c}}\cos {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{2}\left[ 2\sin {{\omega }_{m}}t\cos {{\omega }_{c}}t \right]$
Let this be equation 3.
We know from the trigonometric formula that
$2\sin {{\omega }_{m}}t\cos {{\omega }_{c}}t=\sin ({{\omega }_{c}}+{{\omega }_{m}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t$
Substituting this value in equation 3, we have
$x(t)={{A}_{c}}\cos {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{2}\left[ 2\sin {{\omega }_{m}}t\cos {{\omega }_{c}}t \right]={{A}_{c}}\cos {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{2}\left[ \sin ({{\omega }_{c}}+{{\omega }_{m}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t \right]$
Therefore, the amplitude modulated signal is given by
$x(t)=x={{A}_{c}}\cos {{\omega }_{c}}t+\dfrac{{{A}_{0}}}{2}\left[ \sin ({{\omega }_{c}}+{{\omega }_{m}})t+\sin ({{\omega }_{c}}-{{\omega }_{m}})t \right]$
Hence, option $A$ is correct.

Additional Information:
Amplitude modulation is a modulation technique, in which the amplitude of the carrier signal is changed or varied in proportion to the amplitude of the message signal. This modulation technique is generally used in electronic communication to add audio signals to radio signals. Here, carrier signals are the radio signals and message signals are the audio signals. This technique of modulation is widely used in AM broadcasting as well as computer modems.

Note: Students should take utmost care while writing the general expression for amplitude modulated signal. It is to be noted that the amplitude-modulated signal should be in the same form as the carrier wave. If the carrier signal is given in sine form, amplitude modulated signal should be written in sine form. If the carrier signal is provided in cosine form, amplitude modulated signal should be written in cosine form. This can be easily understood from equation 1 and equation 2. This is done in order to facilitate easy calculation.