A Scaling Law for Bandwidth Under Quantization

We derive a scaling law relating ADC bit depth to effective bandwidth for signals with $1\mathrm{/}{f}^{\alpha }$ 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 $2^{2\mathrm{/}\alpha }$, 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 $1\mathrm{/}{f}^{\alpha }$ signals for $\alpha \in \{1.5,2.0,2.5\}$ yields prediction errors below 3% using the theoretical noise floor ${\mathrm{\Delta }}^2\mathrm{/}(6f_s)$, and approximately 14% when the noise floor is estimated empirically from the quantized signal’s spectrum. We illustrate practical implications on real EEG data.