A Scaling Law for Bandwidth Under Quantization

TL;DR

This paper establishes a mathematical relationship between ADC bit depth and effective bandwidth for signals with 1/f^α spectra, showing how each additional bit extends the bandwidth approximately exponentially, with validation on synthetic and real EEG data.

Contribution

It introduces a scaling law linking ADC bit depth to bandwidth for 1/f^α signals, providing theoretical predictions and empirical validation, including practical implications for EEG data.

Findings

Each additional bit extends bandwidth by a factor of approximately 2^{2/α}.

Prediction errors below 3% on synthetic signals using the theoretical noise floor.

Empirical noise floor estimation yields about 14% prediction error.

Abstract

We derive a scaling law relating ADC bit depth to effective bandwidth for signals with $1/f^\alpha$ power spectra. Quantization introduces a flat noise floor whose intersection with the declining signal spectrum defines an effective cutoff frequency $f_c$. We show that each additional bit extends this cutoff by a factor of $2^{2/\alpha}$, approximately doubling bandwidth per bit for $\alpha = 2$. The law requires that quantization noise be approximately white, a condition whose minimum bit depth $N_{\min}$ we show to be $\alpha$-dependent. Validation on synthetic $1/f^\alpha$ signals for $\alpha \in \{1.5, 2.0, 2.5\}$ yields prediction errors below 3\% using the theoretical noise floor $\Delta^2/(6f_s)$, and approximately 14\% when the noise floor is estimated empirically from the quantized signal's spectrum. We illustrate practical implications on real EEG data.