QUANTIZATION NOISE IS NOT NOISE

WHAT YOU ARE LOOKING AT

The usual story is that a quantizer adds white noise of power Δ²/12. That story is a convenience. Feed a quantizer a pure sine and the error is not random at all: it repeats exactly once per signal period, so it can only contain harmonics of the signal. Writing that Fourier series out gives

e(t) = Σ_p A_p sin(pωt),    A_p = δ_p12^n-1 + Σ_m (2/mπ) J_p(2mπ2^n-1),   p odd

where J_p is a Bessel function of the first kind. The red stems are that formula, evaluated live. The grey stems are a direct FFT of a quantized sine. They land on top of each other, which is the point: the formula is not an approximation.

THINGS TO TRY

• Start at 1 bit. Two levels, so the output is a square wave and the harmonics fall off as 1/p — enormous, nothing like a noise floor. The white noise line sits far below the real harmonics.
• Walk the bits up. Each individual harmonic drops about 9 dB per bit — not the 6 dB you might expect, because 6 dB per bit is the total noise power, and the number of harmonics doubles at the same time, adding 3 dB back. They also get denser and more irregular, because the Bessel functions oscillate. By 8 bits the spikes are so many and so small that calling them “a noise floor” finally becomes a fair description — which is exactly when the Δ²/12 model starts being useful.
• Drag terms m down to 1. One Bessel term already gets the shape roughly right; the sum converges slowly (the terms fall off like m^-3/2) so a few hundred terms are needed before the small harmonics settle. Watch A₃ stop wobbling.
• Back the amplitude off to about 0.5. The measured harmonics move but the Bessel curve does not: the formula is derived for a full scale sine. This is why a real converter's distortion changes with input level.
• At any setting, compare SQNR with 6.02B+1.76. They agree within a fraction of a dB even though the noise is manifestly not white — the total power is right, only its distribution is wrong.

WHY IT MATTERS

Harmonics fold. An odd harmonic above f_s/2 lands somewhere inside your band, and no amount of oversampling removes it, because it is not noise you can average away — it is a tone. That is why real converters dither: adding a little genuine noise before the quantizer breaks the harmonics up into something that actually is white, trading a slightly higher noise floor for the absence of spurs.

The static version of this figure is Figure 18 and the new quantization noise figure in the ADC chapter. Bessel values here come from a small series implementation in this page; the lecture figures use scipy.special.jv and agree to about 0.1 %.