AM — Concept and Math

The Amplitude Modulation Idea

Amplitude Modulation (AM) is the most direct answer to the problem the previous lesson set up: the message signal directly controls the amplitude — the peak height — of the carrier wave.

Think of AM as making a carrier wave "breathe" in sync with the message. When the message is strong, the carrier swings high. When the message is quiet, the carrier barely moves. The shape of this breathing — the envelope — is an exact replica of the message signal.

AM was the first modulation scheme used commercially. Starting in the 1920s, AM broadcasting transformed society by bringing news, music, and culture into homes across continents. Today, AM remains in use for medium-wave broadcasting and aviation communications (VHF AM).

The AM Signal Equation

For a single-tone message signal x(t) = A_m cos(2πf_mt), the standard AM signal is:

Notice the structure: the AM signal is a carrier cosine whose amplitude coefficient is not a constant A_c, but a time-varying quantity A_c[1 + mcos(2πf_mt)]. This time-varying amplitude is the envelope, and it faithfully tracks the message.

The Modulation Index

The modulation index m is the ratio of message amplitude to carrier amplitude:

Physically, m determines how deeply the carrier amplitude swings:

• When m = 0: pure carrier, no message encoded.
• When m = 0.5: carrier amplitude varies from 0.5A_c to 1.5A_c — 50% modulation.
• When m = 1: carrier amplitude varies from 0 to 2A_c — 100% modulation, maximum depth.
• When m > 1: envelope goes negative — over-modulation, causing distortion.

The AM Spectrum: Three Lines

To understand the AM signal in the frequency domain, expand the product using the cosine multiplication identity: cos(A)·cos(B) = ½[cos(A−B) + cos(A+B)].

This is the AM spectrum: one carrier line flanked by two sidebands. The message information is entirely in the sidebands — the carrier contains no information, only power. This spectral structure directly informs the bandwidth calculation.

Bandwidth

The AM signal occupies frequencies from f_c − f_m,max to f_c + f_m,max. The total bandwidth is:

BW = 2 × f_m,max

For voice communications where f_m,max = 3.4 kHz, BW = 6.8 kHz. Compare that 6.8 kHz to broadcast FM, which occupies about 200 kHz — so AM fits roughly 30× as many voice channels into the same spectrum. (Standard AM broadcast channels are spaced 10 kHz apart, leaving room for guard bands.) This spectral efficiency is one reason the crowded medium-wave band (535–1705 kHz) can support hundreds of simultaneous broadcasters.

Power Efficiency: AM's Achilles Heel

AM's big disadvantage is power efficiency. The total power of an AM signal is:

At m = 1, the total power is 1.5P_c: P_c in the carrier and P_c/2 in the sidebands. The sideband fraction is (P_c/2) / (3P_c/2) = 1/3. In other words, only one-third of transmitted power carries information even at maximum modulation.

This is a fundamental inefficiency. At lower modulation depths, the situation is worse: at m = 0.5, sidebands carry only 1/9 of total power. This is why AM transmitters need to be very high power to achieve the same range as SSB or FM transmitters of much lower power.

AM Variants: Fixing the Efficiency Problem

Engineers developed several AM variants to improve power and bandwidth efficiency:

Getting It Back Off Again

Despite that power inefficiency, standard AM survived for one reason: it is extraordinarily easy to demodulate. Because the envelope is the message, a diode, a capacitor and a resistor are enough to recover the audio — no oscillator, no phase reference, not even a power supply. M3-L3 builds that envelope detector, works out the single time constant it depends on, and shows what has to change once the carrier is suppressed.