Nyquist-Shannon Sampling Theorem & Anti-Aliasing Filters in DSP

The foundation of all modern digital telecommunications and audio engineering rests on the Nyquist-Shannon Sampling Theorem, developed by Harry Nyquist in 1928 and mathematically proven by Claude Shannon in 1949.

The Nyquist Criterion

The theorem states that a continuous analog signal containing frequencies up to \(f_{\text{max}}\) can be completely and perfectly reconstructed without error if it is sampled at a uniform rate \(f_s > 2 f_{\text{max}}\).

If an incoming sound wave contains a frequency component higher than half the sampling rate (\(f > f_s / 2\)), the ADC converter cannot distinguish between the high frequency and a lower frequency. Instead, the energy folds back across the Nyquist boundary, generating an artificial, non-harmonic tone called an alias:

Anti-Aliasing Reconstruction Filters

To prevent aliasing distortion, every professional ADC converter places an analog anti-aliasing low-pass filter immediately before the sampling stage to brickwall-filter all energy above \(f_s / 2\). Modern converters utilize high-order sigma-delta oversampling topologies to ensure these filters maintain linear phase in the audible band.