Nyquist Theorem
The Nyquist-Shannon sampling theorem states that a continuous signal can be perfectly reconstructed from its samples if the sampling rate is at least twice the highest frequency present: f_s ≥ 2f_max. This minimum rate 2f_max is called the Nyquist rate. Sampling below this causes aliasing — high frequencies masquerade as lower frequencies. For example, CD audio samples at 44.1 kHz to capture frequencies up to 22.05 kHz (beyond human hearing at ~20 kHz).
The Nyquist Theorem: Perfect Reconstruction from Samples
Harry Nyquist and Claude Shannon proved that no information is lost when sampling a band-limited signal at or above twice its maximum frequency. This seems counterintuitive — how can discrete points fully capture a continuous curve? The key is band-limiting: if the signal contains no frequencies above f_max, then its behavior between samples is completely determined by the samples themselves. Reconstruction uses sinc interpolation: each sample generates a sinc function (sin(πx)/(πx)), and summing all these sinc functions perfectly recreates the original continuous signal. This mathematical guarantee is the foundation of all digital audio, video, and communications.
Key Formulas
Aliasing: What Happens When You Undersample
When the sampling rate is too low (f_s < 2f_max), frequencies above f_s/2 fold back and appear as lower frequencies in the digital representation. A 15 kHz tone sampled at 20 kHz appears as a 5 kHz tone (|15 - 20| = 5 kHz). This aliased signal is indistinguishable from a genuine 5 kHz signal, so the damage is irreversible. The visual equivalent is the wagon wheel effect in movies: wheels appear to spin backward because the frame rate is too low for the rotation speed. Anti-aliasing filters (analog low-pass) must remove all frequencies above f_s/2 before sampling.
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Open CalculatorNyquist Rate vs. Nyquist Frequency
The Nyquist rate is the minimum sampling rate needed: 2f_max. The Nyquist frequency is half the actual sampling rate: f_s/2. These are related but different concepts. For CD audio: the highest audio frequency is 20 kHz, so the Nyquist rate is 40 kHz. The actual sampling rate is 44.1 kHz (10% above the minimum for practical filter design), giving a Nyquist frequency of 22.05 kHz. Frequencies between 20 kHz and 22.05 kHz form a transition band where the anti-aliasing filter rolls off — this margin makes the filter design practical.
Practical Sampling: Oversampling and Anti-Aliasing
Real systems always sample above the Nyquist rate for practical reasons. Anti-aliasing filters cannot achieve perfect brick-wall cutoffs — they need a transition band. Sampling at 2.5× to 4× the maximum frequency provides this margin. Some systems use deliberate oversampling: modern audio DACs sample at 192 kHz or higher, then digitally filter down. This relaxes the analog anti-aliasing filter requirements while achieving excellent noise performance through noise shaping. Delta-sigma ADCs take this further, sampling at millions of times the signal bandwidth with 1-bit resolution.
Nyquist Beyond Audio: Images, Radar, and Communications
The Nyquist theorem applies to any sampled signal. In digital photography, the pixel pitch determines spatial sampling rate — too few pixels for the lens resolution causes moiré patterns (spatial aliasing). Camera sensors use optical low-pass filters to prevent this. In radar, the pulse repetition frequency must be at least twice the maximum Doppler shift to avoid velocity aliasing. In digital communications, the symbol rate must satisfy Nyquist criteria to avoid inter-symbol interference. The theorem's universality makes it one of the most important results in information theory.
Related Topics in signal processing techniques
Understanding nyquist theorem connects to several related concepts: nyquist sampling theorem, sampling rate theorem, sampling nyquist, and shannon sampling theorem. Each builds on the mathematical foundations covered in this guide.
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