Build an AM signal in the sandbox and read off its spectrum, its efficiency and its envelope — and predict, before the readouts confirm it, where the two sidebands land, how little of the power carries information, and what happens when you push the modulation index m past 1.
Standard AM multiplies a carrier by [1 + m·cos(2πf_m t)]. Expand that product and the spectrum is three lines: a carrier at f_c and a sideband a message-frequency away on each side, at f_c ± f_m. The carrier carries no information yet holds most of the power, so the efficiency η = m²/(2+m²) is low — even at its useful maximum m = 1, two thirds of the power is wasted. And the envelope A_c(1+m·cos) dips to a minimum A_c(1−m): push m past 1 and that minimum goes negative, the envelope crosses zero, and a simple envelope detector distorts the message. Reading those numbers off a live signal is how AM stops being a formula and becomes something you can compute.
Open the sandbox. It opens on the default AM signal (m = 0.5, f_m = 2 Hz, f_c = 20 Hz), so a Reset gets you back to Step 1. For this lab, drag the Modulation index slider to the value each step names, and leave f_m and f_c where they are. Each step names the one knob to move.
Modulation index m -> set per step (0.5, then 1, then 1.5) f_m -> 2 Hz f_c -> 20 Hz
Carrier amplitude A_c = 1 is fixed (not a knob). Reset -> m 0.5, f_m 2 Hz, f_c 20 Hz
Watch the Sidebands f_c ± f_m, Power efficiency and Envelope min A_c(1−m) readout cards — every number you predict is printed there.
Leave the defaults (m = 0.5, f_m = 2 Hz, f_c = 20 Hz). Expanding the AM product puts a sideband at f_c − f_m and one at f_c + f_m. Predict the upper sideband frequency, then read the Sidebands f_c ± f_m card.
The card reads 18 / 22 Hz — the lower sideband at 20 − 2 = 18 Hz and the upper at 20 + 2 = 22 Hz. The two sidebands straddle the carrier one message-frequency out, so the occupied bandwidth is 2 f_m = 4 Hz.
Keep m = 0.5. The power efficiency is η = m²/(2+m²) — the fraction of the transmitted power that sits in the two sidebands, the only part carrying information. Predict the efficiency as a percentage, then read the Power efficiency card.
The card reads 11.1% — 0.25 / 2.25. At m = 0.5 barely a ninth of the power carries the message; the other 88.9% is spent on the information-free carrier.
Now drag the Modulation index up to m = 1 — as deep as standard AM can go without distorting. Predict the new efficiency η = m²/(2+m²) as a percentage, then read the Power efficiency card.
The card reads 33.3% — 1 / 3. Even at the deepest undistorted modulation, two thirds of the power still sits in the carrier. That waste is exactly why suppressed-carrier schemes (DSB-SC, SSB) exist.
Drag the Modulation index to m = 1.5. The envelope dips to a minimum A_c(1−m) with A_c = 1. Predict that minimum, then read the Envelope min A_c(1−m) card and watch the recovered message distort.
The card reads -0.5 — 1·(1 − 1.5). A negative envelope minimum is overmodulation: the envelope crosses zero, so a simple envelope detector (which outputs |envelope|) folds the negative dips back up and the recovered message is clipped and distorted.
Everything above is waiting in the sandbox. Drag the Modulation index and watch the envelope deepen and the sideband lines grow, raise it past 1 to see the envelope cross zero and the detector output clip, and move f_m to slide the sidebands in and out along the spectrum.
- Placed the sidebands at 18 / 22 Hz — f_c ± f_m for f_c = 20, f_m = 2
- Found only 11.1% efficiency at m = 0.5 — m²/(2+m²)
- Saw even the deepest undistorted m = 1 reach just 33.3%, two thirds wasted in the carrier
- Pushed m = 1.5 to an envelope minimum -0.5 — A_c(1−m) gone negative, which is overmodulation
- Every value you predicted is the demo's own exact arithmetic, not a picture