PCM, Quantisation and Companding

Sampling discretises time and, above Nyquist, loses nothing. Quantisation discretises value, and it always loses something. The two steps are often conflated, and only one of them is irreversible by nature.

The rounding error is bounded by half a step and, for a complex signal, is effectively random — so it is modelled as noise with power Δ²/12. Each extra bit halves the step, halves the error, and gains 6 dB.

The noise model breaks for simple signals. A pure tone produces an error correlated with the signal, which sounds like distortion rather than hiss — which is why dither is deliberately added in high-quality converters.

With uniform steps the error is the same size regardless of signal level, so the SNR falls with the signal. Speech spends most of its time far below its peak, which makes this precisely the wrong arrangement for telephony.

Compand = compress + expand. Compress the signal before quantising and expand it afterwards, which is equivalent to a non-uniform ladder: fine steps near zero, coarse at the extremes.

• The SNR becomes roughly constant across the signal range, which is what perception rewards.
• Eight companded bits sound like twelve uniform ones — four bits gained for free.
• A-law in Europe and most of the world; µ-law in North America and Japan. Same idea, slightly different curves.
• Which is why a telephone channel has been 64 kbit/s — 8000 samples × 8 bits — since the 1960s.

A digital repeater only has to decide which symbol was sent, then generate a fresh clean one — so noise does not accumulate along a route. An analogue repeater amplifies its input noise at every hop.

Regeneration, not fidelity at the source, is why long-distance telephony went digital. A thousand-hop digital route is as clean as a one-hop route; a thousand-hop analogue route is unusable.

Delta modulation sends one bit per sample — up or down — exploiting the similarity of consecutive samples. It fails in two ways: slope overload when the signal changes faster than the step allows, and hunting around a constant value.

Adaptive delta modulation varies the step size to address both. Its descendant, sigma-delta, is inside nearly every audio converter sold — trading bit depth for sample rate and then filtering digitally.