Build an FM signal in the sandbox and read off its modulation index, its Carson bandwidth and its noise advantage over AM — and predict, before the readouts confirm it, how wide wideband FM spreads, how much cleaner it recovers a message than AM, and what you give up when you narrow it.
FM writes the message into the carrier's frequency: s(t) = A_c·cos(2πf_c t + β·sin(2πf_m t)), with modulation index β = Δf/f_m. Two numbers follow from β. Carson's rule says the signal occupies BW = 2(Δf + f_m) = 2(β+1)f_m — so wideband FM (large β) is spectrum-hungry. In return, above a ~10 dB input-SNR threshold, FM recovers the message with an output-SNR advantage over AM of G = 3β²(β+1), which climbs fast with β. That is the FM bargain: trade bandwidth for noise immunity. Reading those numbers off a live signal is how the AM-vs-FM choice stops being a slogan and becomes something you can compute.
Open the sandbox. It opens on the default FM signal (Δf = 15 Hz, f_m = 3 Hz, f_c = 24 Hz, σ = 0.10), which already sits at β = 5 — the broadcast-FM index — so a Reset gets you back to Step 1. Each step names the one knob to move; leave the rest where they are.
Peak deviation df -> 15 Hz (Steps 1-3), then 3 Hz (Step 4) f_m -> 3 Hz f_c -> 24 Hz sigma -> 0.10
Reset -> df 15 Hz, f_m 3 Hz, f_c 24 Hz, sigma 0.10 (beta = 5, above threshold)
Watch the Modulation index β, Carson bandwidth, FM SNR gain and Threshold readout cards — every number you predict is printed there.
Leave the defaults (Δf = 15 Hz, f_m = 3 Hz, f_c = 24 Hz). The FM index is β = Δf/f_m. Predict β, then read the Modulation index β card.
The card reads β = 5.00 — 15 / 3 — the wideband regime (β ≥ 1). That is exactly broadcast FM's index: a real FM station uses Δf = 75 kHz and f_m = 15 kHz, and 75 / 15 = 5 too. Because β is a pure ratio, the sandbox's few-hertz signal and a real broadcast share the identical modulation index.
The Carson bandwidth card shows BW = 2(Δf + f_m). At the sandbox's few-hertz scale it reads 36 Hz (= 2(15 + 3)) — confirm it. Now feed that same formula the real broadcast numbers, Δf = 75 kHz and f_m = 15 kHz (the same β = 5). Predict the broadcast Carson bandwidth.
180 kHz — 2(75 + 15). That is why an FM station is allotted a 200 kHz channel: Carson's rule says essentially all the FM power fits inside 180 kHz, with margin. Push the deviation higher and the bandwidth would spill past the channel.
Keep β = 5, with the noise low enough that the Threshold card reads ABOVE (the default σ = 0.10 gives about 17 dB input SNR). The wideband-FM output-SNR advantage over AM of equal transmitted power is G = 3β²(β+1). Predict that gain in decibels, then read the FM SNR gain card.
The card reads 26.5 dB — G = 3·25·6 = 450×, and 10·log₁₀(450) = 26.5 dB. Wideband FM recovers the message with about 26 dB more signal-to-noise than AM would from the same transmitted power. That advantage is the whole reason broadcast FM sounds clean.
Drag Peak deviation Δf down to 3 Hz, so β = Δf/f_m = 1 (narrowband FM). Watch β and the Carson bandwidth shrink. Predict the new FM SNR gain in decibels, then read the FM SNR gain card.
The card reads 7.8 dB — G = 3·1·2 = 6×. Cutting β from 5 to 1 collapses the gain from 26.5 dB to 7.8 dB, and the Carson bandwidth from 36 Hz to 12 Hz (180 kHz to 60 kHz at broadcast scale). That is the FM trade in one move: more deviation buys noise immunity but costs spectrum — and the gain only exists above the ~10 dB input-SNR threshold; below it FM breaks into clicks with no advantage at all.
Everything above is waiting in the sandbox. Drag Peak deviation Δf to watch β, the Carson bandwidth and the SNR gain move together, raise the Noise level σ until the Threshold card flips to BELOW and the FM advantage vanishes, and compare the recovered message against AM at every setting.
- Read the FM modulation index β = 5.00 — Δf/f_m for Δf = 15 Hz, f_m = 3 Hz — the same index as broadcast FM (75 / 15)
- Predicted the broadcast Carson bandwidth 180 kHz — 2(Δf + f_m) — which fits inside the 200 kHz FM channel
- Found the wideband-FM noise advantage 26.5 dB at β = 5 — 3β²(β+1) = 450× over AM, above the ~10 dB threshold
- Narrowed to β = 1 and watched the gain collapse to 7.8 dB — the deviation-immunity-bandwidth trade in one move
- Every value you predicted is the demo's own exact arithmetic, not a picture