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A Scaling Law for Bandwidth Under Quantization

Maximilian Kalcher, Tena Dubcek

Abstract

We derive a scaling law relating ADC bit depth to effective bandwidth for signals with $1/f^α$ 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 $2^{2/α}$, approximately doubling bandwidth per bit for $α= 2$. The law requires that quantization noise be approximately white, a condition whose minimum bit depth $N_{\min}$ we show to be $α$-dependent. Validation on synthetic $1/f^α$ signals for $α\in \{1.5, 2.0, 2.5\}$ yields prediction errors below 3\% using the theoretical noise floor $Δ^2/(6f_s)$, and approximately 14\% when the noise floor is estimated empirically from the quantized signal's spectrum. We illustrate practical implications on real EEG data.

A Scaling Law for Bandwidth Under Quantization

Abstract

We derive a scaling law relating ADC bit depth to effective bandwidth for signals with power spectra. Quantization introduces a flat noise floor whose intersection with the declining signal spectrum defines an effective cutoff frequency . We show that each additional bit extends this cutoff by a factor of , approximately doubling bandwidth per bit for . The law requires that quantization noise be approximately white, a condition whose minimum bit depth we show to be -dependent. Validation on synthetic signals for yields prediction errors below 3\% using the theoretical noise floor , and approximately 14\% when the noise floor is estimated empirically from the quantized signal's spectrum. We illustrate practical implications on real EEG data.
Paper Structure (18 sections, 7 equations, 3 figures, 2 tables)

This paper contains 18 sections, 7 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Signal and Noise Model
  4. Cutoff Frequency
  5. Derivation of the Scaling Law
  6. Validity Conditions
  7. Experiments
  8. Quantization Noise Spectrum
  9. Scaling Law Validation
  10. Robustness
  11. Sensitivity to $\alpha$
  12. Spectral peaks
  13. Application to Electrophysiology Data
  14. Discussion
  15. The $\alpha$-Dependent Threshold
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Scaling law validation: $f_c(N{+}1)$ predicted from Eq. \ref{['eq:scaling']} vs. measured values. Empirical noise floor estimation introduces spectral artifacts (a), while the theoretical noise floor yields close agreement (b).
  • Figure 2: Robustness analysis. (a) Quantization noise spectral slope versus bit depth for different $\alpha$; the green band marks $|$slope$|<0.1$ (approximately white). (b) Scaling prediction error versus $\alpha$ estimation error.
  • Figure 3: EEG from PhysioNet physionet. (a) PSD showing the flat noise floor at 4 bits. (b) Ratio of quantized to original band power. At 4 bits the Gamma band is distorted; at 6 bits and above all bands are preserved.