Amplitude Modulation

DSB-SC from the modulation property, AM with a carrier, the modulation index, envelope and coherent detection, and frequency-division multiplexing

2 November 1920 — election night in Pittsburgh

On the evening of the Harding–Cox presidential election, Westinghouse puts a transmitter on the air from a shack on the roof of a nine-storey building at its East Pittsburgh works, using the temporary call sign 8ZZ. A publicity man, Leo Rosenberg, reads the election returns into a microphone, and listeners with the new "wireless sets" hear the results as they arrive. The station becomes KDKA, considered by many historians the first commercially licensed radio station (KDKA (AM)). Election returns had gone out by radio before, but in Morse code, to the few who could read it. KDKA was not alone that night either: at least three other stations also transmitted the returns by voice (KDKA (AM)). What was new was the idea of broadcasting — one transmitter, an audience of anyone with a receiver. Underneath it is the modulation property of Lesson 4-2: multiply a message by a carrier, and its spectrum moves to wherever you want it.

Why not simply transmit the audio? An antenna radiates efficiently only when it is comparable in size to the wavelength, and audio wavelengths are kilometres long. And if every station transmitted its audio directly, all of them would occupy the same few kilohertz. Modulation solves both problems: it moves each message up to its own carrier frequency, where antennas are practical and stations can sit side by side.

Derivation

Double-sideband, suppressed-carrier modulation

Multiply a message

m(t)

, bandlimited to

W

, by a carrier. By the modulation property (Lesson 4-2),

x(t) = m(t)\cos\omega_c t \;\leftrightarrow\; X(j\omega) = \tfrac12 M\big(j(\omega - \omega_c)\big) + \tfrac12 M\big(j(\omega + \omega_c)\big).

The message's spectrum is copied to

\pm\omega_c

: an upper sideband above the carrier frequency and a lower sideband, its mirror image, below it, occupying

2W

in all. To recover

m

, multiply by the same carrier again,

x(t)\cos\omega_c t = m(t)\cos^2\omega_c t = \tfrac12 m(t) + \tfrac12 m(t)\cos 2\omega_c t,

and low-pass filter away the copy at

2\omega_c

. This coherent (synchronous) demodulation needs a local carrier with the right frequency and phase: a phase error

\theta

scales the output by

\cos\theta

, and at 90° the message vanishes.

💡AM with a carrier

Broadcast AM adds the carrier itself:

x(t) = A\big[1 + \mu\,m(t)\big]\cos\omega_c t, \qquad |m(t)| \le 1,

with modulation index

\mu

. For a tone, the spectrum is the carrier plus two sidebands of relative amplitude

\mu/2

— the lab finds lines at 19, 20 and 21 kHz with amplitudes 0.25, 1 and 0.25 for

\mu = 0.5

. The carrier carries no information, and it takes most of the power: the sidebands get only

\mu^2/(2 + \mu^2)

of it, 11.1 % at

\mu = 0.5

and at most a third at

\mu = 1

💡Envelope detection

The carrier buys a very simple receiver. As long as

\mu \le 1

, the envelope

A[1 + \mu m(t)]

never goes negative, so it traces the message exactly; a diode and a capacitor — or, digitally, the magnitude of the analytic signal

|x + j\mathcal H\{x\}|

— recover it without any local carrier. Push

\mu

above 1 and the envelope folds over where

1 + \mu m

would go negative: overmodulation, and distortion that no receiver can undo. The lab recovers the message to within 0.000 at

\mu = 0.5

and 0.9, but with an error of 0.617 at

\mu = 1.5

💡Frequency-division multiplexing

Each modulated message occupies

2W

around its own carrier. Space the carriers at least

2W

apart and many messages share one channel: a receiver selects one with a band-pass filter, then demodulates it. In the lab, two stations on 10 and 15 kHz share a wire; station 2 comes back with an error of 0.0004 and no trace of station 1. Broadcast radio, early telephone trunk lines and cable television all worked this way.

The medium is the message.

– Marshall McLuhan, Understanding Media (1964), quoted on Wikiquote

🕰A patent for the receivers

Westinghouse's push into radio after the First World War was as much about patents as about stations: to strengthen its position in receivers, it bought control of the "heterodyne" patent originally issued to Reginald Fessenden (KDKA (AM)). The heterodyne idea — mixing a signal with a local oscillator to move it to a convenient frequency — is the modulation property used inside the receiver.

⚠Half the band is redundant

For a real message, the lower sideband is the mirror image of the upper one: both carry the same information. Transmitting one of them only — single-sideband modulation, invented by John Carson of AT&T — halves the bandwidth and saves the carrier's power, at the price of a more demanding receiver (Edwin Howard Armstrong).

💡Where you will use this

Two DSB-SC carriers in quadrature,

\cos\omega_c t

and

\sin\omega_c t

, carry two independent messages in the same band — the basis of quadrature amplitude modulation in modems and Wi-Fi. FM (Lesson 10-2) moves the message into the phase; spread spectrum (Lesson 10-3) hides it under a code.

📚Further reading

Oppenheim and Willsky, Signals and Systems, sections 8.1–8.4 — sinusoidal amplitude modulation, demodulation and frequency-division multiplexing.
KDKA (AM) on Wikipedia — the election-night broadcast in detail.

↓But there is a problem

Static — the crackle of lightning, motors and switches — adds straight onto an AM signal's amplitude, and no filter can separate it from the message sharing that amplitude. In 1922 a respected AT&T mathematician proved that changing the carrier's frequency instead would not help. A decade later, working in secret in a Columbia University basement, Edwin Armstrong proved him wrong.