Question:
An amplitude modulated wave is represented by $C_m(t) = 30 \sin 300\pi t + 10(\cos 200\pi t - \cos 400\pi t)$. Then the carrier wave frequency, signal frequency and modulation index are respectively
An amplitude modulated wave is represented by $C_m(t) = 30 \sin 300\pi t + 10(\cos 200\pi t - \cos 400\pi t)$. Then the carrier wave frequency, signal frequency and modulation index are respectively
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The carrier frequency is always the average of the two sideband frequencies: $f_c = \frac{f_{\text{USB}} + f_{\text{LSB}}}{2}$.
The signal frequency is half of the difference between the sidebands: $f_m = \frac{f_{\text{USB}} - f_{\text{LSB}}}{2}$.
The signal frequency is half of the difference between the sidebands: $f_m = \frac{f_{\text{USB}} - f_{\text{LSB}}}{2}$.
Updated On: Jul 22, 2026
- 200 Hz , 50 Hz , 1/2
- 150 Hz , 50 Hz , 2/3
- 150 Hz , 30 Hz , 1/3
- 200 Hz , 30 Hz , 1/2
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
This question requires us to analyze the expression of an Amplitude Modulated (AM) wave.
We need to extract the carrier wave frequency ($f_c$), modulating signal frequency ($f_m$), and the modulation index ($\mu$).
Step 2: Key Formula or Approach:
The mathematical expression for a standard single-tone amplitude modulated wave is:
\[ C_m(t) = A_c \sin(\omega_c t) + \frac{\mu A_c}{2} \cos(\omega_c - \omega_m)t - \frac{\mu A_c}{2} \cos(\omega_c + \omega_m)t \] We compare the given equation with this standard format:
\[ C_m(t) = 30 \sin 300\pi t + 10\cos 200\pi t - 10\cos 400\pi t \]
Step 3: Detailed Explanation:
• Comparing the first term (the carrier component):
\[ A_c \sin(\omega_c t) = 30 \sin 300\pi t \] This gives the carrier amplitude $A_c = 30$ and carrier angular frequency $\omega_c = 300\pi$ rad/s.
The carrier frequency $f_c$ is:
\[ 2\pi f_c = 300\pi \implies f_c = 150\text{ Hz} \]
• Comparing the sideband components:
The lower sideband frequency is $\omega_c - \omega_m = 200\pi$ rad/s, and the upper sideband frequency is $\omega_c + \omega_m = 400\pi$ rad/s.
Solving for the modulating angular frequency $\omega_m$:
\[ (\omega_c + \omega_m) - (\omega_c - \omega_m) = 400\pi - 200\pi \] \[ 2\omega_m = 200\pi \implies \omega_m = 100\pi\text{ rad/s} \] The signal frequency $f_m$ is:
\[ 2\pi f_m = 100\pi \implies f_m = 50\text{ Hz} \]
• Comparing the sideband amplitudes to find the modulation index $\mu$:
\[ \frac{\mu A_c}{2} = 10 \] Substituting the value of $A_c = 30$:
\[ \frac{\mu \times 30}{2} = 10 \] \[ 15\mu = 10 \implies \mu = \frac{10}{15} = \frac{2}{3} \]
Step 4: Final Answer:
The carrier wave frequency is 150 Hz, the signal frequency is 50 Hz, and the modulation index is 2/3.
This question requires us to analyze the expression of an Amplitude Modulated (AM) wave.
We need to extract the carrier wave frequency ($f_c$), modulating signal frequency ($f_m$), and the modulation index ($\mu$).
Step 2: Key Formula or Approach:
The mathematical expression for a standard single-tone amplitude modulated wave is:
\[ C_m(t) = A_c \sin(\omega_c t) + \frac{\mu A_c}{2} \cos(\omega_c - \omega_m)t - \frac{\mu A_c}{2} \cos(\omega_c + \omega_m)t \] We compare the given equation with this standard format:
\[ C_m(t) = 30 \sin 300\pi t + 10\cos 200\pi t - 10\cos 400\pi t \]
Step 3: Detailed Explanation:
• Comparing the first term (the carrier component):
\[ A_c \sin(\omega_c t) = 30 \sin 300\pi t \] This gives the carrier amplitude $A_c = 30$ and carrier angular frequency $\omega_c = 300\pi$ rad/s.
The carrier frequency $f_c$ is:
\[ 2\pi f_c = 300\pi \implies f_c = 150\text{ Hz} \]
• Comparing the sideband components:
The lower sideband frequency is $\omega_c - \omega_m = 200\pi$ rad/s, and the upper sideband frequency is $\omega_c + \omega_m = 400\pi$ rad/s.
Solving for the modulating angular frequency $\omega_m$:
\[ (\omega_c + \omega_m) - (\omega_c - \omega_m) = 400\pi - 200\pi \] \[ 2\omega_m = 200\pi \implies \omega_m = 100\pi\text{ rad/s} \] The signal frequency $f_m$ is:
\[ 2\pi f_m = 100\pi \implies f_m = 50\text{ Hz} \]
• Comparing the sideband amplitudes to find the modulation index $\mu$:
\[ \frac{\mu A_c}{2} = 10 \] Substituting the value of $A_c = 30$:
\[ \frac{\mu \times 30}{2} = 10 \] \[ 15\mu = 10 \implies \mu = \frac{10}{15} = \frac{2}{3} \]
Step 4: Final Answer:
The carrier wave frequency is 150 Hz, the signal frequency is 50 Hz, and the modulation index is 2/3.
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