Can you hear Für Elise in quantization noise?

01. The question

A conventional PCM quantizer maps many input amplitudes to the same digital value. We normally listen to that value and dismiss the difference as quantization noise. What happens if we listen to the difference instead?

We use one fixed passage of Für Elise so that any changes are due to the bit depth and not the audio material. The experiment is deterministic: no dither, no additive random noise, and no clipping at the input of the quantizer.

02. Signal model

Let x[n] denote the normalized audio samples. The B-bit, symmetric mid-rise quantizer has L = 2^B output levels and step size

For every input sample, calculate the integer index and corresponding quantization level:

The quantization noise is defined by the difference—not by an independently generated white-noise process:

With no overload, the samplewise error satisfies |e_B[n]| ≤ Δ_B/2. Crucially, the input-dependent structure of e_B depends on the signal and the quantizer. A white, independent-noise approximation is not an identity.

Equation (4) has an especially useful form. Within the unsaturated input interval \(-U\le x[n]<U\), the difference is exactly a centered modulo measurement:

In particular, \(e_B(x+k\Delta_B)=e_B(x)\) for integers k as long as both arguments remain inside the quantizer's unsaturated interval. Quantization noise is therefore not injective sample by sample; signal-class assumptions are essential for recovery from its samples.

For B = 2, the four possible outputs are precisely \(\{-0.75,\,-0.25,\,+0.25,\,+0.75\}\). The arithmetic below and the level histogram later on both use these four levels; there is no implicit one-bit approximation.

03. Listening experiment

Start with the original. Then compare each quantized signal with its quantization noise. The buttons below switch audio at the same time position, making A/B comparisons possible while listening.

Listening level matters. At 8 bits, the raw noise has much lower RMS than the original music. RMS-matched playback amplifies it solely for audibility; it does not change the quantizer, error samples, or SQNR. Uncheck the setting to hear all three representations at their original amplitudes. Start at a comfortable headphone volume.

04. Waveforms & levels

Choose a bit depth and zoom into a few milliseconds of the recording. The blue staircase is computed sample by sample using (2)–(3); the red curve is the actual difference in (4).

Why the 2-bit case is subtle. Although the quantizer has four valid output values, approximately 99.48% of this excerpt's samples use the middle pair, ±0.25. That is a property of this audio amplitude distribution, not a software shortcut to a one-bit quantizer.

05. Spectral analysis

Listening suggests that low-bit quantization noise can retain recognizable musical structure. A spectral comparison provides a complementary, quantitative view. Below, power spectral densities are estimated directly from the original input and from each quantization-noise sequence.

The quantization-noise spectrum depends on both the input and bit depth. Spectral peaks alone do not establish statistical independence, whiteness, or exact reconstruction; these require separate mathematical tests or assumptions.

06. Code & data

07. Interpretation

There is a simple algebraic reason to question the word noise: e_B is not produced independently of x. In our low-bit experiment, the quantization noise contains audible musical structure. By (5), those noise samples are also a centered modulo representation. The conventional digital representation retains Q_B(x), whereas listening to e_B reveals another signal-dependent representation of the same input.

The broader research question is therefore more interesting than perceptual fidelity alone: under which structural assumptions can the information discarded by conventional quantization become a sufficient measurement? This leads naturally to the sampling and recovery viewpoint developed in Unlimited Sensing.