AM Carrier
y = sin(20x − 10t) · (1 + 0.8 · sin(x − t))
Open in the app The dials and keys named below are the app's.
What it draws
A fast wave is multiplied by a slow one. The carrier is sin(20x − 10t). Its wavenumber is 20, so crests are 2π/20 ≈ 0.314 apart. Its angular frequency is 10, so each point repeats every 0.628 seconds at Speed 1. Together the two carry a crest right at 10/20 = 0.5 units per second.
The envelope is 1 + 0.8·sin(x − t), one long wave 2π ≈ 6.28 across that runs right at 1 unit per second. It swings between 0.2 and 1.8, so the loud parts stand nine times as tall as the quiet ones. The 0.8 is the modulation index, and below 1 the line never falls silent.
The sidebands
Multiplying two sines is the same as adding two more, and that is the whole of amplitude modulation. Expanding the product gives sin(20x − 10t) + 0.4·cos(19x − 9t) − 0.4·cos(21x − 11t): the carrier untouched, plus one wave just below its wavenumber and one just above. Those are the sidebands, and they are where a radio signal's information sits. The carrier itself never changes. Their spacing from it, 1 in wavenumber and 1 in frequency, is the envelope's own.
Two speeds
Watch a single crest and it slides through the bulge that carries it. Crests travel at 0.5 and the bulge at 1, so a crest is born at the back of a loud patch, crosses it and dies at the front. The first speed is the phase velocity ω/k, the second the group velocity dω/dk, and the three components give it directly: (11 − 9)/(21 − 19) = 1. A medium in which those two differ is dispersive, and this formula is the plainest picture of what that means.
History
Amplitude modulation carried the first voices over radio. Reginald Fessenden sent speech about a mile at Cobb Island in December 1900, and on Christmas Eve 1906 made the first AM entertainment broadcast.
Try
- Front in the deck: side on, where the envelope reads as the outline of the ink.
- Overmodulate by changing 0.8 to 1.2: the envelope dips below zero, the carrier flips phase where it does, and the outline crosses itself instead of pinching.
- Hold the carrier still with sin(20x)·(1 + 0.8·sin(x − t)): the crests stop dead and only the bulge travels, which separates the two motions.
- Halve the envelope's wavelength, sin(20x − 10t)·(1 + 0.8·sin(2x − 2t)): twice as many bulges, and sidebands at 18 and 22.