Modulation and PCM

Baseband, and the case for a carrier

Send a signal in its own natural range of frequencies and you are transmitting at baseband. A telephone landline does exactly this, speech from about 300 Hz to 3.4 kHz travelling down a private pair of wires, and on a private wire baseband is fine. Radio has no private wires. Every transmitter shares the same air, and if each station broadcast baseband audio they would all occupy the same few kilohertz and land on top of one another.

An audio-frequency radio signal has a wavelength that would require an impractically large resonant aerial, and different broadcasts would be difficult to separate if transmitted at baseband. Modulation uses the information signal to vary a high-frequency carrier. Amplitude modulation varies the carrier amplitude; frequency modulation varies its instantaneous frequency. Pulse-code modulation represents regularly sampled values as binary numbers.

The numbers make the aerial point concrete. An efficient resonant aerial is a substantial fraction of a wavelength long, and a 3 kHz radio wave is 100 km long, so a resonant aerial for it would be impractically large; electrically short aerials can be built, but they are inefficient. A carrier's much shorter wavelength suits a real aerial, and each station is allotted its own carrier frequency, so a tuned circuit at the receiver can select one station and reject the rest.

Amplitude modulation

Amplitude modulation keeps the carrier frequency f_c$f_{c}$f_c fixed and makes its amplitude follow the programme, loud passages swelling the envelope. The surprise is what this does to the spectrum. Modulate a carrier with a single audio tone of frequency f_M$f_{M}$f_M and you transmit exactly three frequencies, the carrier itself plus two sidebands at f_c + f_M$f_{c} + f_{M}$f_c + f_M and f_c - f_M$f_{c} - f_{M}$f_c - f_M. The audio is not sent at its own frequency; it appears as the pair of new frequencies flanking the carrier.

Describe the drawing

A frequency axis carrying three vertical lines: a tall carrier at nine hundred and nine kilohertz with a shorter sideband four and a half kilohertz below it at nine hundred and four point five, and another the same distance above at nine hundred and thirteen point five. A double-headed arrow beneath spans the two sidebands and is labelled bandwidth equals twice the modulating frequency, nine kilohertz.

Real programmes are a band of tones up to some maximum audio frequency f_M$f_{M}$f_M, so each sideband spreads into a band, and the transmission occupies everything from f_c - f_M$f_{c} - f_{M}$f_c - f_M to f_c + f_M$f_{c} + f_{M}$f_c + f_M. The occupied width is the bandwidth:

Frequency modulation

Frequency modulation holds the amplitude constant and swings the carrier's frequency instead, above and below f_c$f_{c}$f_c in step with the programme. Louder audio swings it further; the maximum swing is the frequency deviation Δf$\Delta f$Δf. An FM signal's sidebands in fact trail off indefinitely, so the occupied bandwidth is estimated by Carson's rule, which allows the deviation plus the highest modulating frequency on each side of the carrier:

Noise, whether from motors, lightning or distant sparks, adds to a wave's amplitude. An AM receiver reads the amplitude, so it faithfully reproduces the noise. An FM receiver reads only the frequency, and it clips the amplitude flat before detection, removing most amplitude noise. Noise that disturbs the wave's apparent frequency or phase survives in smaller measure, and the advantage holds only while the signal stays comfortably above the receiver's working threshold.

Describe the drawing

Five traces one above the other, all against the same stretch of time. At the top is one cycle of the modulating tone. Below it the carrier amplitude-modulated by that tone, its dashed envelope swelling to A times one plus m at the tone's peak and shrinking to A times one minus m at the trough. Below that the same wave with a short burst of amplitude noise added, the coral envelope lifted clear of the dashed one where the burst lands. The fourth trace is the same tone sent by frequency modulation: the height never changes and the cycles are visibly crowded together where the tone peaks and stretched apart where it dips. The last trace is that FM signal with the same burst added and the receiver's limiter applied, its peaks flattened where the burst was, and the dashed verticals show that the zero crossings sit at exactly the instants they did before. AM writes the programme where noise lives, in the amplitude; FM writes it in the frequency, which a limiter can clean.

FM also carries audio to 15 kHz rather than 4.5, and its constant amplitude lets the transmitter run steadily at full efficiency. AM's advantage is economy, a twentieth of the spectrum per station and receivers simple enough to build with a diode.

Sending numbers instead: sampling and PCM

A digital signal resists noise and can be regenerated without loss. To give an analogue signal those advantages, digitise it. Step one is sampling, measuring the signal's value at regular instants. Sample too rarely and wiggles between samples are missed entirely. The rule has a sharp edge: the sampling rate must be greater than twice the highest frequency present in the signal. Sampled slower than that, a high frequency masquerades as a low one and the corruption cannot be undone afterwards.

Describe the drawing

A sine wave crossing eight dashed horizontal levels labelled with the three-bit codes 000 to 111. Cyan dots mark twelve equally spaced samples taken on the wave, each joined by a short stem to the amber dot on the nearest level, and the sequence of three-bit codes those levels give is printed underneath as the transmitted bit stream.

Step two is quantisation. Each sample is rounded to the nearest of 2ⁿ$2^{n}$2^n fixed levels, and the level is named by an n-bit binary code. The rounding error is the quantisation error, shrunk by adding bits, since each extra bit doubles the number of levels and halves the step between them. The stream of codes, sent as pulses, is pulse-code modulation, PCM. From that point the signal has the advantages of a digital signal, regeneration included, so noise stops accumulating along the route.