Question

A sinusoidal carrier wave with amplitude \( A_c \) and frequency \( f_c \) is amplitude modulated with a message signal \( m(t) \) having frequency \( 0<f_m \ll f_c \) to generate the modulated wave \( s(t) \) given by \[ s(t) = A_c[1 + m(t)]\cos(2\pi f_c t) \] The message signal that can be retrieved completely using envelope detection is _________

• A. ( m(t) = 0.5 \cos(2\pi f_m t) \)
• B. ( m(t) = 1.5 \sin(2\pi f_m t) \)
• C. ( m(t) = 2 \sin(4\pi f_m t) \)
• D. ( m(t) = 2 \cos(4\pi f_m t) \)

Hint:

Envelope detection is used to retrieve the low-frequency modulating signal from an amplitude-modulated carrier.

Answer 1

The Correct Option is A

Step 1: Understanding envelope detection.
Envelope detection is used to recover the message signal from a modulated carrier wave. The key idea is that the message signal \( m(t) \) is the low-frequency variation in the amplitude of the carrier wave. The modulated wave is typically of the form \( s(t) = A_c[1 + m(t)]\cos(2\pi f_c t) \), where the message signal modulates the amplitude of the carrier.
Step 2: Analyzing the options.
- (A) Correct, \( m(t) = 0.5 \cos(2\pi f_m t) \) is the correct form for the message signal that can be recovered using envelope detection.
- (B) Incorrect, this is not the correct form for the message signal in envelope detection.
- (C) Incorrect, the frequency is too high for the message signal in envelope detection.
- (D) Incorrect, this is a higher frequency than the message signal. Step 3: Conclusion.
Thus, the correct answer is (A) \( m(t) = 0.5 \cos(2\pi f_m t) \).