§ 1.2  Module 1 — When Voice Became Data

Sound into Numbers: Sampling, PCM, and G.711

If a function contains no frequencies higher than W, it is completely determined by giving its ordinates at a series of points spaced 1/2W seconds apart.— Claude Shannon, “Communication in the Presence of Noise,” 1949

1.2.1The sampling theorem, and what aliasing punishes

An analog microphone signal is a continuously varying voltage. To digitize it, a sampler measures that voltage at strictly regular intervals — the sample rate f_s, in samples per second. The counterintuitive claim, the one Reeves’s contemporaries doubted, is that nothing is lost: Shannon’s sampling theorem says a signal containing no energy above some bandwidth B is exactly recoverable from its samples, provided f_s > 2B. Twice the highest frequency is not a rule of thumb; it is a mathematical boundary, and the frequency f_s/2 is called the Nyquist frequency in honor of the 1928 result.

The boundary is enforced by a specific failure mode. A sampler cannot tell a tone at frequency f from a tone at f + f_s, or from f_s − f: their sample values are identical. Any input energy above the Nyquist frequency is therefore folded back into the band below it, masquerading as a lower frequency. This is aliasing, and it is not distortion in the forgiving, analog sense — it is a new, unrelated tone sitting where no tone was played. The alias of a tone at f is found by reducing f modulo f_s, then reflecting anything above f_s/2 down to f_s − f. Sample a 5 kHz whistle at 8,000 Hz and your listener hears 3 kHz; sample 9 kHz and they hear 1 kHz. Figure 1.2 shows the trap: the samples of the 5 kHz input lie perfectly on a 3 kHz sinusoid, and no algorithm downstream can ever know which one was played.

Because folding is irreversible, every practical digitizer places an analog low-pass anti-aliasing filter before the sampler, discarding energy above Nyquist while it is still merely inaudible rather than destructive. This is also where the phone network’s numbers come from. Bell engineers had long band-limited speech channels to roughly 300–3,400 Hz — intelligible, if thin. A 4 kHz channel allocation, doubled per Nyquist, gives 8,000 samples per second, with the 3,400–4,000 Hz gap left as transition room so a realizable analog filter could roll off in time. The sample rate of every telephone call since 1962 is, quite literally, a filter specification with margin.

1.2.2Quantization: the price of each bit

Sampling makes time discrete; quantization makes amplitude discrete, and unlike sampling it genuinely destroys information. A b-bit quantizer rounds each sample to the nearest of 2^b levels spread across the converter’s full scale. The rounding residue behaves, for busy signals, like an added noise floor: uniformly distributed within ±half a step, with power step²/12. Work through the algebra for a full-scale sine wave and a b-bit quantizer, and the signal-to-noise ratio comes out to

SNR = 6.02·b + 1.76 dB     (full-scale sinusoid, uniform quantizer)

Listing 1.1 — The 6 dB-per-bit rule. Each added bit doubles the level count, halves the step, and buys 20·log₁₀(2) ≈ 6.02 dB.

Amplitudes in digital audio are measured against the converter’s ceiling in dBFS — decibels relative to full scale, where 0 dBFS is the largest representable value and everything real is negative. The 6 dB/bit rule then reads directly as dynamic range: 16-bit audio (CD, and the Int16 samples you will meet in Web Audio worklets) offers about 98 dB between full scale and the noise floor; 8-bit linear offers about 50 dB. Fifty decibels sounds adequate until you remember that the rule was derived for a full-scale signal. Quantization noise is a fixed floor: a quiet talker at −40 dBFS on an 8-bit linear channel enjoys not 50 dB of SNR but roughly 10 — audibly gritty, like sand poured over the whisper. Speech routinely swings 40 dB between a raised voice and trailing consonants, before you add the variation between loud and faint telephone lines.

So the 1962 designers faced an ugly trade. Twelve or thirteen linear bits would cover speech comfortably — but T1’s budget was 8 bits per sample, because 24 channels × 8,000 Hz × 8 bits had to fit the 1.544 Mbit/s the copper could carry. The escape was to make the steps unequal.

From the archive1919

1.2.3Companding: G.711’s logarithmic bargain

The insight is perceptual as much as mathematical: hearing judges loudness ratios, not differences, so quantization error matters relative to the signal carrying it. A quantizer with fine steps near zero and coarse steps near full scale spreads a roughly constant signal-to-noise ratio across the whole dynamic range — compressing on encode and expanding on decode, hence companding. G.711, standardized by the CCITT in 1972, defines two logarithmic curves: μ-law (μ = 255), deployed in North America and Japan, and A-law (A = 87.6), deployed in Europe and most elsewhere; international trunks convert on the μ-law side. Both compress each sample into 8 bits: one sign bit, three bits selecting one of eight exponentially sized segments, and four bits placing the sample linearly within its segment — a piecewise-linear, hardware-friendly approximation of a logarithm.

// G.711 mu-law, the 1972 algorithm in JavaScript
const BIAS = 0x84;                       // 132
const SEG_END = [0xFF, 0x1FF, 0x3FF, 0x7FF,
                 0xFFF, 0x1FFF, 0x3FFF, 0x7FFF];

function muLawEncode(sample) {           // Int16 -> byte
  const mask = sample < 0 ? 0x7F : 0xFF;
  let mag = Math.min(Math.abs(sample), 32635) + BIAS;
  let seg = 0;
  while (seg < 8 && mag > SEG_END[seg]) seg++;
  return ((seg << 4) | ((mag >> (seg + 3)) & 0x0F)) ^ mask;
}

Listing 1.2 — μ-law encoding: sign, segment, mantissa, inversion. Eight bits that behave like fourteen.

The bargain pays off exactly where linear 8-bit fails. For small signals, 8-bit μ-law performs like roughly 14-bit linear PCM (A-law like 13-bit), holding SNR near 38 dB across some 40 dB of input level instead of letting it collapse with the signal. The cost is a slightly noisier ceiling at full scale — a trade speech accepts gladly. The resulting stream, 8,000 samples × 8 bits = 64 kbit/s, is the DS0, the indivisible unit around which the entire digital telephone hierarchy — T1, E1, ISDN, SS7 trunking — was sized. And it never left: G.711’s two flavors survive inside WebRTC as the static RTP payload types 0 (PCMU) and 8 (PCMA), and RFC 7874 makes G.711 — alongside Opus — mandatory for every WebRTC endpoint, precisely so the browser can always meet a 1972 telephone gateway on its own terms.

1.2.4The sample-rate zoo, and why WebRTC settled on 48 kHz

Telephony’s 8 kHz was only the first entry in a menagerie of rates, each an artifact of the medium that bore it. 16 kHz — “wideband,” a 7 kHz audio band — arrived with ITU-T G.722 in 1988 and was marketed decades later as HD Voice. 44.1 kHz is a videotape number: Sony and Philips fixed it in the 1980 Compact Disc Red Book because early digital masters were stored on video recorders, and three samples per video line worked out to exactly 44,100 per second on both NTSC and PAL machines. 48 kHz is the professional and broadcast rate — adopted by the AES, used by DAT recorders from 1987 — chosen for clean arithmetic against film and video frame rates and generous anti-alias filter margins. Modern audio hardware descends from that lineage: virtually every laptop and phone codec chip clocks natively at 48 kHz.

WebRTC standardized on the last of these. Opus (RFC 6716, 2012 — Jean-Marc Valin, Koen Vos, and Timothy Terriberry, fusing Skype’s SILK with Xiph.Org’s CELT) is a 48 kHz codec: whatever bandwidth it actually encodes, its RTP timestamps always tick at 48 kHz (RFC 7587), and its full-band mode covers everything human hearing can use. libwebrtc’s voice pipeline — echo cancellation, noise suppression, gain control — likewise runs at a common internal rate of 48 kHz, resampling whatever the microphone delivers through getUserMedia (often 44.1 kHz on consumer hardware) on the way in. When your WebRTC call sounds present and close while the circuit-switched leg patched in from a conference bridge sounds distant and thin, you are hearing sixty years of sampling decisions side by side: sibilants and consonant edges live above 3.4 kHz, and G.711’s anti-alias filter removed them in the name of 1962’s copper budget.

The through-line of this lesson is worth stating plainly: every sample rate is a bandwidth claim, and every bit depth is a noise-floor claim. 8 kHz/8-bit companded says “intelligible speech, telephone budget, 1962.” 48 kHz says “all of human hearing, with room for the filters, 2012.” In the next lesson these samples meet the packet network — and the timing guarantees the circuit gave them for free disappear.

Problems for § 1.2Check your understanding

Build the two halves of 1962 yourself: a sampler-quantizer that lets you measure the 6 dB/bit rule and predict aliases, then the actual G.711 μ-law compander, graded byte-for-byte against the reference tables.

¶Field quirks

What the specifications say, and what the browsers actually do, are two different documents. These are the divergences that cost production teams their weekends.