ETM Learn · 08 / 08
Enlarge quantization steps and learn why a little noise can make error more predictable.
Digital audio separates when samples are taken from which amplitude values are available. Bit depth sets the number of values in a fixed-point format. Fewer bits over the same range mean larger steps, not fewer samples per second.
Quantization chooses an allowed value using a rounding rule. Imagine a ruler with widely spaced marks: a number between them must move to a mark. The difference is quantization error. Repeating signal-related error can create distortion. Dither adds noise before rounding to change that behaviour; it costs noise rather than removing it. Start with large visible steps, then increase bit depth.
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Mapping amplitude to a value from a finite set. The spacing between adjacent values is quantization step Δ.
Noise with a triangular probability distribution. This example adds two-step peak-to-peak noise before quantization without subtracting it from output.
Redistributing quantization noise across frequencies using filtered error feedback. Less noise in one region comes with more elsewhere.
Choose 8 bits, a −40 dBFS sine and no magnification. Compare rounding alone, TPDF and shaping, then increase bit depth.
Predict → change one control → compare → explain the result.
Watch both signal and error spectrum. Magnify attenuates before quantization and restores gain afterwards: it changes signal level relative to steps, not just listening volume. Nonzero tail samples may be noise alone.
Do nonzero dithered tail samples prove the original signal is preserved?
You must examine the relationship to the source; nonzero samples alone are insufficient.For an ideal uniform N-bit quantizer, full-scale sine and uncorrelated uniform error over ±Δ/2, . This is not universal measured dynamic range or every dithered signal’s SNR. Ideal non-subtractive two-step gives total error variance Δ²/4, versus Δ²/12 in the idealized undithered noise model.
feeds quantization error through a filter to redistribute its spectrum. Lower noise in one band costs more elsewhere; stability and overload matter. Per-bin FFT noise depends on length and window; it is not integrated noise power or SNR.
Dither is relevant when requantizing to fixed point, including 24-bit when technically appropriate. Apply it at that delivery’s final quantization. Further processing and quantization may require another dither stage. Independent noise powers add; there is no universal fixed penalty per export. Tail motion does not prove signal preservation: test its relationship to the source, for example through coherent analysis or averaging.
SNR ≈ 6.02N + 1.76 dB
Multiply bit depth N by 6.02 and add 1.76. At 16 bits this gives about 98.08 dB. This assumes a uniform quantizer, full-scale sine and uniform uncorrelated error, not every signal or dithered system.
(Δ²/4) / (Δ²/12) = 3
In the lesson's ideal models, total error variance with non-subtractive TPDF is three times the undithered quantization-noise model. A power ratio of 3 is about 4.77 dB. This adds noise in exchange for reduced signal dependence under the stated assumptions.
Substitute N = 16. The rounded constants slightly change the final decimal.
6.02 × 16 + 1.76 = 98.08 dB (about 98.09 with more precise constants). This is the specified ideal model, not the measured dynamic range of every 16-bit device or every dithered output.
Variance is power: use 10 log₁₀ of the variance ratio, not 20 log₁₀.
10 log₁₀[(Δ²/4)/(Δ²/12)] = 10 log₁₀3 = 4.7712 dB. This compares two specified ideal models. It is not a fixed penalty for adding dither to arbitrary existing noise or for every repeated export. With overload or different shaping, the assumptions change.
Further reading: Lipshitz, Wannamaker & Vanderkooy: Quantization and Dither