Dither and quantisation: what 16 bits actually buys you

Every converter rounds. A 16-bit sample can only land on one of 65,536 levels, so the difference between the signal and the nearest level is an error that the recording keeps forever. Whether that error sounds like a faint hiss or like a gritty distortion riding on the signal depends on a single decision — whether you dither it.

The calculator works out where the noise sits for a given depth and dither, and then moves the signal down from full scale, because that is the point where the numbers stop being an abstraction and start describing a recording.

The 6 dB rule, and where the 1.76 comes from

A converter with N bits divides the peak-to-peak range into 2^N steps, so one step is q = 2^1−N of the peak amplitude. Rounding to the nearest step produces an error spread evenly across ±q/2, and the power of that error is q²/12 — the variance of a uniform distribution, which is where the 12 comes from and why it never changes.

A full-scale sine has power 1/2, so the ratio is

The 6.02 is the part worth remembering: every bit buys 6 dB. The 1.76 is not a property of the converter at all — it is the crest factor of the sine that was assumed. Measure the same converter with pink noise and the constant moves, while the 6.02 does not.

This is also where the famous 96 dB for CD audio comes from: 6.02 × 16 is 96.3, and the 1.76 has been quietly dropped. The exact figure for a full-scale sine at 16 bits is 98.09 dB. Neither number is wrong; they describe different signals.

What the error actually is

Quantisation error is not random. For a periodic input it is a periodic function of the signal, which means it is harmonic distortion, not noise. A 997 Hz tone at −80 dBFS in a 16-bit file does not sit in a bed of hiss: it is approximated by a staircase, and a staircase is a square-wave-like sequence of harmonics at odd multiples of 997 Hz.

The audible consequence is specific. On a fade, the distortion products move with the music and stay harmonically related to it, so the ear tracks them as part of the sound rather than as a separate noise. Musicians describe it as grain or as a "digital" edge, and the mechanism is correlation: the error knows what the signal is doing.

Dither breaks that correlation by adding a small amount of noise before the rounding. The error is then dominated by the dither, which is independent of the signal, so the distortion turns into a steady hiss. The hiss is measurable; the distortion was audible.

Rectangular and triangular

Rectangular dither one step wide is the minimum that removes the distortion. It works because the added noise is at least as large as the rounding error it is trying to hide. Its weakness is that the resulting noise level still depends on the signal: it modulates with the programme material, and on a quiet sustained note that modulation is itself audible.

Triangular dither, two steps wide, is the sum of two independent rectangular sources. It removes the modulation as well, at the cost of twice the noise power. That is the entire difference between the two, and it is why every serious tool defaults to triangular.

• None — full 98.09 dB at 16 bits, but correlated error and audible distortion on quiet material.
• RPDF, 1 LSB — 3.01 dB of SNR given up; distortion removed, noise level still signal-dependent.
• TPDF, 2 LSB — 4.77 dB given up; distortion and modulation both removed. The correct default.

Noise shaping: moving it rather than removing it

Shaping filters the error before it is added back, pushing it out of the band where the ear is most sensitive and into the top octave. The total noise power goes up, so the wideband SNR gets worse while the audible result gets better — which is why a shaped 16-bit master can sound cleaner than an unshaped one at the same nominal SNR.

The price is paid above roughly 15 kHz, and it is not small: a second-order shaper can buy 10 to 12 dB in the 2–5 kHz region while adding a similar amount above 15 kHz. On material with little high-frequency content that is a good trade. On cymbals and sibilance it is a trade you can hear, which is why the choice is exposed rather than fixed.

The extreme version of the same idea is the 1-bit converter running at 2.8 MHz: the noise is spread over a band 64 times wider than audio, then a decimation filter discards everything outside 20 kHz and takes the in-band noise down with it.

The floor does not follow the fader

Dither noise and quantisation noise are both absolute. They are defined against full scale and they stay there regardless of what the signal does. A passage at −40 dBFS therefore has 40 dB less headroom over the noise than a full-scale one, and the SNR figure that matters for that passage is 40 dB worse than the headline number.

This is the argument that the "96 dB dynamic range" claim hides. A finished mix typically sits around −18 dBFS RMS with peaks near −1 dBFS, so the usable margin above the noise on the quietest passages is smaller than the specification suggests. At 16 bits with triangular dither, a passage 40 dB down has about 53 dB of SNR — fine for most material, marginal for a solo instrument in a quiet room.

Five mistakes worth avoiding

1. Dithering twice. Dither belongs at the final word-length reduction, once. Applying it at every intermediate stage accumulates noise with nothing gained.
2. Dithering before a gain change. If the signal is turned down afterwards, the dither goes down with it and no longer covers the rounding that follows.
3. Using rectangular dither because it is 1.76 dB better. The 1.76 dB is real and the noise modulation it buys is worse.
4. Claiming 16-bit is 96 dB of dynamic range. It is 98.09 dB for a full-scale sine and considerably less for anything that is not at full scale.
5. Shaping noise into a lossy encoder. The encoder's psychoacoustic model expects a flat floor; a shaped one can be coded as signal and cost bits.

Summary

Six decibels per bit, plus 1.76 for the crest factor of the test signal, minus 4.77 if you dither properly. Then subtract the distance your signal sits below full scale, because the floor does not move. That last term is the one that decides whether 16 bits is enough for a particular recording.

Values shown are engineering aids rather than measurements; see the disclaimer and the tool index.