Question

An amplitude modulated signal is given by $V(t) = 10\left[ {1 + 0.3\cos (2.2 \times {{10}^4}t)} \right]\sin (5.5 \times {10^5}t)$, here t is in seconds. The side band frequencies (in kHz) are, [Given π=22/7]
A. 1785 and 1715
B. 892.5 and 857.5
C. 89.25 and 85.75
D. 178.5 and 717.5

Answer 1

Hint: In this question, we need to determine the side band frequencies in kilo-hertz of an amplitude modulated signal which is defined as $V(t) = 10\left[ {1 + 0.3\cos (2.2 \times {{10}^4}t)} \right]\sin (5.5 \times {10^5}t)$. For this, we will use the compare the given equation with the standard equation for the modulated signal.

Complete step by step answer:
The given amplitude modulated signal is defined by the equation $V(t) = 10\left[ {1 + 0.3\cos (2.2 \times {{10}^4}t)} \right]\sin (5.5 \times {10^5}t) - - - - (i)$.
The standard equation for the modulated signal is given as
$V(t) = {A_c}\left[ {1 + \dfrac{{{A_m}}}{{{A_c}}}\cos {\omega _m}t} \right]\sin {\omega _c}t - - - - (ii)$ where, ${A_c}$ is the amplitude of the carrier wave, ${A_m}$ is the amplitude of the modulated wave, ${\omega _c}$ is the angular frequency of the carrier wave and ${\omega _m}$ is the angular frequency of the modulated wave.
Now, on comparing the given modulated signal (equation (i)) with the standard equation of the modulated signal (equation (ii)), we get
The angular frequency of the modulated wave is ${\omega _m} = 2.2 \times {10^4}$ and the angular frequency of the carrier wave is ${\omega _c} = 5.5 \times {10^5}$.
Now, the angular frequency is also written as $\omega = 2\pi f$.
So,
$
{\omega _m} = 2\pi {f_m} \\
2.2 \times {10^4} = 2\pi {f_m} \\
{f_m} = \dfrac{{2.2 \times {{10}^4} \times 7}}{{2 \times 22}} \\
= 3.5 \times {10^3} \\
= 3.5{\text{ kHz}} \\
 $
Similarly,
$
{\omega _c} = 2\pi {f_c} \\
5.5 \times {10^5} = 2\pi {f_c} \\
{f_c} = \dfrac{{5.5 \times {{10}^5} \times 7}}{{2 \times 22}} \\
 = 87.57 \times {10^3} \\
 = 87.57{\text{ kHz}} \\
 $
Now, the side band frequencies are of two types namely, lower side band frequencies and the upper side band frequencies.
The difference in the modulated frequency and the carrier frequency results in the lower side band frequency. Mathematically, $LSB = {f_c} - {f_m}$. So, substituting the values of the modulated frequency and carrier frequency, we get
$
LSB = {f_c} - {f_m} \\
 = 87.57 - 3.5 \\
 = 84.07{\text{ kHz}} \\
 $
Hence, lower side band frequency is $84.07$ KHz.
The summation of the modulated frequency and the carrier frequency results in the upper side band frequency. Mathematically, $USB = {f_c} + {f_m}$. So, substituting the values of the modulated frequency and carrier frequency, we get
$
USB = {f_c} + {f_m} \\
 = 87.57 + 3.5 \\
 = 91.07{\text{ kHz}} \\
 $
Hence, the upper side band frequency is 91.07 KHz.
The nearest values that are matching with the calculated frequencies are 89.25 KHz and 85.75 KHz.

So, the correct answer is “Option C”.

Note:
Students should be careful while comparing the equations as many times, the positions of the cosine and the sine functions are replaced in the question. Moreover, while substituting the values in the final equation to determine the side band frequencies, the units of the modulated frequency and the carrier frequency must be the same to avoid any error.