Power and efficiency in amplitude modulation

P_t = P_c(1 + m²/2).

Worked answer.

For standard AM (double sideband with full carrier) modulated by a single tone, the total power is
P_t = P_c (1 + m²/2),
where P_c is the unmodulated carrier power and m is the modulation index.

Step 1: Total power. P_t = 100 × (1 + 0.8²/2) = 100 × (1 + 0.32) = 132 W.

Step 2: Sideband power. Total sideband power = P_c × m²/2 = 100 × 0.32 = 32 W. This splits equally between the upper and lower sidebands, so each sideband carries 16 W.

Step 3: Efficiency. Efficiency is the fraction of power in the sidebands, because only they carry the information:
η = P_sidebands / P_t = 32 / 132 ≈ 24.2%.

Equivalently, η = m² / (2 + m²) = 0.64 / 2.64 ≈ 0.242.

Check. Carrier + sidebands = 100 + 16 + 16 = 132 W, which matches Step 1. The carrier power is unchanged by modulation; modulation adds the sideband power on top.

Interpretation. Even at 100% modulation (m = 1), η = 1/3, so at best 33.3% of the transmitted power carries information. This is why DSB-SC (suppressed carrier) and SSB are more power-efficient, although they need more complex receivers. Standard AM survives in broadcasting because envelope detectors in receivers are very simple and cheap.

Common mistake. Forgetting the factor of 2 and writing P_t = P_c(1 + m²) = 164 W.

Likely follow-up. "What happens if m exceeds 1?" The signal is over-modulated: the envelope crosses zero, an envelope detector produces distortion, and extra sideband frequencies spread into adjacent channels.