AM modulation depth requirement: For the amplitude-modulated signal s(t) = Ac cos(ωc t) + 2 cos(ωm t) cos(ωc t), what is the minimum value of carrier amplitude Ac for distortionless envelope detection?

Introduction / Context:
Envelope detection requires the modulation index μ ≤ 1 for distortionless demodulation. The modulation index is the ratio of the message amplitude to the carrier amplitude. If μ > 1, overmodulation occurs and the envelope detector distorts the signal.

Given Data / Assumptions:

• AM signal: s(t) = Ac cos(ωc t) + 2 cos(ωm t) cos(ωc t).
• The message term has amplitude 2.
• Carrier amplitude is Ac (unknown).

Concept / Approach:

Standard AM: s(t) = Ac cos(ωc t) [1 + (Am/Ac) cos(ωm t)]. Here, Am = 2. Modulation index μ = Am / Ac. For envelope detector to work: μ ≤ 1 ⇒ Ac ≥ Am.

Step-by-Step Solution:

Verification / Alternative check:

General rule: minimum carrier amplitude equals or exceeds the message amplitude to avoid overmodulation. Here Ac must be ≥ 2.

Why Other Options Are Wrong:

• Ac = 1 or 0.5: gives μ = 2 or 4 ⇒ overmodulation, distortion.
• Ac = 0: no carrier; signal cannot be demodulated by envelope detector.

Common Pitfalls:

• Forgetting to compare message amplitude to carrier amplitude.
• Misinterpreting the cosine product term without factoring it into AM standard form.

Final Answer:

2