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.

翻译：针对具有$1/f^α$功率谱的信号，我们推导了模数转换器（ADC）位深与有效带宽之间的标度关系。量化会引入平坦的噪声基底，其与衰减的信号频谱的交点定义了一个有效截止频率$f_c$。我们证明，每增加一个量化比特，该截止频率会扩展$2^{2/α}$倍；当$α= 2$时，每比特带宽约翻倍。该定律要求量化噪声近似为白噪声，我们证明了满足此条件的最小位深$N_{\min}$与$α$相关。在$α\in \{1.5, 2.0, 2.5\}$的合成$1/f^α$信号上进行验证，使用理论噪声基底$Δ^2/(6f_s)$时预测误差低于3%，而通过量化信号频谱经验估计噪声基底时误差约为14%。我们以真实脑电图（EEG）数据为例说明了其实际意义。