Amplitude Modulation

Expanding A_c[1 + m·cos(ω_m t)]·cos(ω_c t) with the product-to-sum identity gives three components: the carrier at f_c, and sidebands at f_c ± f_m.

So AM occupies twice the message bandwidth. A 5 kHz audio signal needs 10 kHz of spectrum, which is why AM broadcast channels are spaced at 9 or 10 kHz and why AM sounds the way it does.

m is the ratio of message amplitude to carrier amplitude. Larger m puts more power into the sidebands — the only part carrying information — so broadcasters compress their audio to keep it high.

Past m = 1 the envelope tries to go negative and an envelope detector rectifies it, folding the peaks back. The distortion is irrecoverable, and the resulting splatter into adjacent channels makes it a licensing violation rather than merely a quality problem.

Total power is P_c(1 + m²/2). Even at m = 1 the carrier takes two thirds and carries nothing. At realistic speech levels it is closer to 90%. A 50 kW AM transmitter radiates 33 kW of pure carrier.

The two sidebands are mirror images, so one is genuinely redundant. SSB gets the same message through in half the bandwidth at a fraction of the power, which is why amateur and marine HF use it.

Because the carrier is present, the message is recovered by a diode, a capacitor and a resistor — no local oscillator, no phase synchronisation. That is why a crystal set works with no power supply at all.

The design traded transmitter efficiency for receiver simplicity, which was exactly right when there was one transmitter and a million receivers. It is an economic argument, not a technical one.

Most natural and man-made interference — lightning, brush motors, ignition systems — is amplitude noise. AM puts its message in exactly that quantity, so the receiver cannot separate them. That is the crackle, and it is structural.