Nyquist Rate Calculator (Sampling Theorem)
Find the minimum sample rate to capture a signal without aliasing, or the max frequency a given rate can represent.
Nyquist-Shannon sampling theorem.
The Nyquist-Shannon sampling theorem is the rule that makes digital audio, digital cameras, and every analog-to-digital converter work correctly. It was proved formally by Claude Shannon in 1949, building on Harry Nyquist’s 1928 work at Bell Labs.
The theorem:
f_s ≥ 2 × f_max
To reconstruct a continuous signal perfectly from its samples, the sampling rate (f_s) must be at least twice the highest frequency in the signal (f_max). The threshold 2 × f_max is called the Nyquist rate; the upper limit f_s / 2 of what your samples can represent is the Nyquist frequency.
What happens when you sample too slowly: High-frequency components fold back into the lower part of the spectrum as bogus low frequencies. This is aliasing. Once it happens you cannot undo it; the original frequency information is gone. You have probably seen visual aliasing on screen as moire patterns in fine grid textures, or as the wagon-wheel effect in old movies where car rims appear to spin backwards.
Why CD audio sits at 44,100 Hz: Human hearing tops out around 20 kHz. The Nyquist rate is 40 kHz. Sony and Philips picked 44,100 Hz when designing the CD format in 1980 because it leaves about 10% of headroom for the analog anti-aliasing filter to roll off, and 44,100 has the convenient factorization 2² × 3² × 5² × 7² that made it easy to derive from existing video timing standards. DVD audio at 48 kHz gives a bit more headroom; high-resolution audio at 96 kHz or 192 kHz gives much more (though whether it is audible is a different argument).
Common sample rates and what they capture:
| Rate (Hz) | Nyquist freq (Hz) | Used for |
|---|---|---|
| 8,000 | 4,000 | Telephone speech (human voice 300–3,400 Hz) |
| 16,000 | 8,000 | Wideband VoIP, hearing aids |
| 22,050 | 11,025 | Low-quality web audio |
| 44,100 | 22,050 | CD audio, MP3 standard |
| 48,000 | 24,000 | Video soundtracks, DVD, broadcast |
| 88,200 / 96,000 | 44k / 48k | Studio recording with processing headroom |
| 192,000 | 96,000 | High-resolution archival |
| 2.5 GHz | 1.25 GHz | Modern oscilloscopes |
Worked example, audio recording: You are recording a podcast. Human speech tops out at about 8 kHz of useful content. The Nyquist rate is 16 kHz. Standard practice is to record at 44.1 kHz or 48 kHz anyway, because the extra headroom lets you apply analog and digital filters without artifacts and stay compatible with delivery formats.
Worked example, undersampling: A sensor’s signal has frequencies up to 500 Hz. You sample at 800 Hz. Your Nyquist rate is 1,000 Hz, so 800 Hz is too low. The 500 Hz component will alias to 800 − 500 = 300 Hz in your data and you will never be able to tell the difference between the real signal and the artifact. Either raise the sample rate to at least 1,000 Hz, or insert an analog low-pass filter that kills everything above 400 Hz before you sample.
Where the theorem applies and where it does not: The theorem assumes a strictly band-limited signal (zero energy above f_max) and ideal sampling. Real signals always have some leakage above their nominal cutoff, which is why every ADC pairs with an anti-aliasing filter. The theorem also applies to spatial sampling, which is why your camera sensor’s pixel pitch sets a fundamental limit on the detail it can capture (and why Bayer-pattern cameras need optical low-pass filters or pixel-shift tricks to avoid color moire).
Bandpass sampling exception: If your signal occupies a narrow band that does not start at DC (for example, an FM radio channel at 100 MHz with 200 kHz bandwidth), you can sometimes sample at far less than 2 × 100 MHz and still reconstruct it perfectly. This is called bandpass sampling or undersampling and the rules are more nuanced than the basic theorem.