An analog-to-digital converter rounds continuous values to discrete levels. The rounding introduces quantization noise — a flat noise floor whose power depends on the step size between levels and therefore on the number of bits. More bits means smaller steps, means lower noise floor. This is textbook signal processing. What is less obvious is the relationship between the noise floor and the effective bandwidth of the digitized signal, and how that relationship depends on the spectral shape of the signal being measured.
Kalcher and Dubcek (arXiv 2602.23252, February 2026) derive a scaling law: for signals with 1/f^alpha power spectra, each additional bit of quantization extends the effective bandwidth cutoff frequency by a factor of 2^(2/alpha).
The geometry is simple. A 1/f^alpha signal has power that decreases with frequency — steeply for large alpha, gently for small alpha. Quantization introduces a flat noise floor that does not depend on frequency. At low frequencies, the signal dominates. At high frequencies, the noise floor dominates. The effective bandwidth is the frequency where the declining signal power crosses the flat noise floor. Below this cutoff, the signal is recoverable. Above it, the signal is buried in quantization noise.
Adding one bit halves the quantization step size, reducing the noise floor by 6 decibels. This lower floor intersects the declining signal spectrum at a higher frequency. How much higher depends on the spectral slope. A steep spectrum (large alpha) crosses any given noise level over a narrow frequency range — lowering the floor by 6 dB extends the cutoff modestly. A gentle spectrum (small alpha) crosses the noise level over a wide range — the same 6 dB extends the cutoff substantially. The scaling exponent 2/alpha captures this: for alpha = 2 (Brownian motion, many physical signals), each bit approximately doubles the bandwidth. For alpha = 1 (pink noise), each bit quadruples it.
The law requires that quantization noise be approximately white — spectrally flat, uncorrelated between samples. This condition fails at very low bit depths, where the quantization error becomes structured. The authors show that the minimum bit depth for the white-noise approximation to hold depends on alpha itself — steeper spectra need fewer bits because the signal-to-noise ratio at low frequencies is higher, making the quantization error look more random.
Validation on synthetic 1/f^alpha signals for alpha in {1.5, 2.0, 2.5} gives prediction errors below 3% using the theoretical noise floor formula. Application to real EEG data — which has approximate 1/f^2 power spectrum — confirms the practical relevance. An EEG system with 16-bit digitization has an effective bandwidth set by this scaling law, and upgrading to 24 bits extends the usable bandwidth by the predicted factor.
The result quantifies something that instrumentation engineers have long known intuitively — more bits buy more bandwidth — but the functional form was not previously stated. The scaling is not linear (each bit does not add the same bandwidth) but multiplicative (each bit multiplies the bandwidth by a fixed factor). The factor depends only on the spectral slope of the signal, not on the sampling rate, the signal amplitude, or the specific ADC architecture. The measurement resolution and the measurement bandwidth are linked by the statistics of the signal being measured.