Question

An amplitude modulated signal consists of a message signal of frequency 1 KHz and peak voltage of 5 V, modulating a carrier frequency of 1 MHz and peak voltage of 15 V. The correct description of this signal is
A.) $5[1+3sin(2\pi10^6 t)]sin(2\pi 10^3 t)$
B.) $15[1+{\dfrac{1}{3}}sin(2\pi10^3 t)]sin(2\pi 10^6 t)$
C.) $[5+15sin(2\pi10^3 t)]sin(2\pi 10^6 t)$
D.) $[15+5sin(2\pi10^6 t)]sin(2\pi 10^3 t)$

Answer 1

Hint: The peak voltage and frequency of both carrier and message signal wave has been given. So, we will first find out the wave equation of the signal. Then by using the equation for amplitude modulated waves, we can reach the answer.

Formula used:
Equation of a signal wave with frequency $f$ and peak voltage $A$ over time $t$ is given by
$s(t)=Asin(2\pi ft)$

Complete step by step solution:
We have been given that the frequency and peak voltage of the message signal is 1 KHz and 5 V, therefore, the equation of message signal $m(t)={A}_{m}sin(2\pi {f}_{m}t)$, where ${A}_{m}$ is the peak voltage of the message signal and ${f}_{m}$ is the frequency of the message signal.

Thus, $m(t)=5sin(2\pi \times 10^3 t)$ ………. (i)

Similarly, the peak voltage and frequency of the carrier wave is given as 15 V and 1 MHz. Therefore, the equation of the carrier signal will be $c(t)={A}_{c}sin(2\pi {f}_{c} t)= 15sin(2\pi \times 10^6 t)$

So, we know that the equation of the amplitude modulated wave is given by
$s(t)=[{A}_{c}+{A}_{m}sin(2\pi {f}_{m}t)]sin(2\pi {f}_{c}t) = 15$
$\implies s(t)=[15+5sin(2\pi \times 10^3 t)]sin(2\pi \times 10^6 t)$
$\implies s(t)=15[1+\dfrac{1}{3}sin(2\pi \times 10^3 t)]sin(2\pi \times 10^6 t)$

Hence, option b is the correct answer.

Additional information:
Modulation index $\mu$ is defined as the ratio of the ratio of the amplitude of modulated signal to the carrier signal and is given by $\mu=\dfrac{{A}_{m}}{{A}_{c}}$

Note: We should be careful while writing the equation for the modulated wave, as we may write the amplitude of the carrier wave in place of the message wave and vice versa. Some questions may also have the cosine value in the equation that can be found out by having a phase difference of $90^{\circ}$.