AM Modulation

How it works

The transmitted signal is the carrier with its amplitude offset by the message:

s(t) = A_c [1 + m · m(t)] cos(2π f_c t)

• A_c — the unmodulated carrier amplitude
• m — the modulation index (0–1), how deeply the carrier is modulated
• m(t) — the message, normalised to [−1, 1]
• f_c — the carrier frequency

The factor [1 + m · m(t)] is the envelope — the dashed outline the carrier peaks trace in the middle panel. At $m = 0$ there's no modulation, just the bare carrier. At $m = 0.5$ the envelope swings with 50% depth; at $m = 1$ it kisses zero at the message troughs, the deepest you can go cleanly. Push past 1 and you overmodulate: the envelope tries to go negative, the carrier inverts, and a simple envelope detector recovers a distorted, clipped mess.

Why the spectrum has exactly three lines

Multiplying two signals in time is convolution in frequency, so the product of a tone $\cos(2\pi f_m t)$ with the carrier shifts the message spectrum up to sit around $f_c$. For a single message tone the result is three spikes: the carrier at $f_c$, and sidebands at $f_c \pm f_m$, each with amplitude $m/2$ relative to the carrier. This is the frequency-shift property in action — the carrier "lifts" a baseband signal to a radio frequency you can actually radiate from an antenna.

It's also why AM is power-hungry: the carrier line carries no information yet hogs most of the power. That insight drives the variants — DSB-SC suppresses the carrier, SSB drops one redundant sideband too — trading transmit efficiency for trickier demodulation.

The knobs

• Mod index (m) — modulation depth; below 1 is clean, above 1 overmodulates and clips.
• Carrier freq — $f_c$, the rate of the fast oscillation being shaped.
• Message freq — $f_m$; raising it pushes the sidebands further from the carrier.
• Speed — playback rate of the animation; purely cosmetic.

Push m past 1 and watch the overmodulation warning fire as the carrier clips; raise the message frequency and watch the sidebands spread apart in the spectrum below.

Further reading

• Haykin, Communication Systems — Chapter 3 builds AM, DSB-SC, and SSB from first principles.
• Proakis, Digital Communications — the modulation/demodulation theory behind the spectra.
• Amplitude modulation (Wikipedia)