A Scaling Law for Bandwidth Under Quantization

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.