Calculate the minimum Nyquist sampling rate for any signal, the Nyquist frequency of a sampling rate, and the alias frequency created by undersampling.
Nyquist Sampling Rate Formula
The Nyquist rate boundary is twice the highest frequency of a band-limited signal. For reliable reconstruction, sample faster than this boundary; a component exactly at half the sampling rate can lose amplitude and phase:
f_N = 2 * f_max
The same relationship works in reverse. A system that samples at a fixed rate has a Nyquist boundary at half that rate; keep the input band below it. This limit is the Nyquist frequency:
f_Nyq = f_s / 2
If a frequency above the Nyquist frequency reaches the sampler, it does not disappear. It folds back into the measurable band and shows up at a false lower frequency called an alias:
f_alias = |f - n * f_s|
Variables:
- f_N is the Nyquist rate boundary, twice the highest input frequency, in samples per second
- f_max is the highest frequency component in the signal, in Hz
- f_s is the actual sampling rate of the system, in Hz (1 Hz equals 1 sample per second)
- f_Nyq is the Nyquist frequency, the highest frequency that sampled data can represent, in Hz
- f is the true frequency of a signal component being sampled, in Hz
- f_alias is the false frequency that appears in the data when f is above f_Nyq, in Hz
- n is the integer multiple of f_s closest to f
The three calculator modes map directly to these three formulas. The first mode takes the highest frequency in your signal and returns the Nyquist rate, plus a practical rate with an oversampling factor applied. The second mode takes a sampling rate you already have and returns the Nyquist frequency it can support. The third mode takes a signal frequency and a sampling rate together, checks the Nyquist criterion, and predicts the ideal folded frequency magnitude if the rate is too low.
Standard Sampling Rates and Their Nyquist Limits
These are illustrative sampling rates and associated Nyquist boundaries. Actual acquisition requirements, usable bandwidth, filtering, and clinical specifications depend on the system and application.
| Application | Sampling rate | Nyquist frequency | Content it protects |
|---|---|---|---|
| Telephone voice | 8 kHz | 4 kHz | Speech band up to 3.4 kHz |
| Audio CD | 44.1 kHz | 22.05 kHz | Hearing range to 20 kHz |
| Professional audio and video | 48 kHz | 24 kHz | 20 kHz audio with wider filter margin |
| High resolution audio | 96 kHz | 48 kHz | Ultrasonic content, relaxed filtering |
| Clinical ECG | 500 Hz | 250 Hz | Diagnostic waveform detail to 150 Hz |
| Clinical EEG | 256 Hz | 128 Hz | Brain activity bands to about 100 Hz |
| Machine vibration analysis | 25.6 kHz | 12.8 kHz | Bearing and gear fault bands to 10 kHz |
Notice that every standard rate is more than twice its protected bandwidth. The gap is the guard band, and the ratio between the sampling rate and the protected bandwidth is the sampling factor used here (f_s / f_max). Other references may define oversampling ratio as f_s / (2 × bandwidth). Anti-aliasing filters cannot cut off instantly, so the guard band gives the filter room to roll off between the highest wanted frequency and the Nyquist frequency. This is why CD audio runs at 44.1 kHz instead of exactly 40 kHz, a factor of 2.205 rather than 2, and why vibration analyzers use a factor of 2.56 as an industry standard.
| Oversampling factor | Rate for a 20 kHz signal | When to use it |
|---|---|---|
| 2.0 (theoretical minimum) | 40 kHz | Theoretical boundary; equality does not generally preserve amplitude and phase |
| 2.2 to 2.56 (guard band) | 44 to 51.2 kHz | Illustrative guard-band choices; check the actual filter and converter specification |
| 5 to 10 | 100 to 200 kHz | When waveform shape in the time domain matters, such as transient capture |
| 10 to 20 | 200 to 400 kHz | Control loops and derivative estimation, where lag and noise must stay low |
Nyquist Sampling Rate Example Problems
Example 1: minimum rate for full range audio. You need to digitize audio containing frequencies up to 20 kHz. The Nyquist rate is f_N = 2 * 20,000 = 40,000 Hz, so 40 kHz is the theoretical minimum sampling rate. A real converter needs a guard band, so applying an oversampling factor of 2.205 gives 2.205 * 20,000 = 44,100 Hz, which is exactly the CD standard. Enter 20 kHz in the first calculator mode and it returns the 40 kHz Nyquist rate along with the practical rate for the factor you choose.
Example 2: predicting an alias. A 6 kHz tone is sampled at 10 kHz. The Nyquist frequency is 10,000 / 2 = 5,000 Hz, and 6 kHz is above it, so the tone will alias. The nearest integer multiple is n = 1 because 6,000 / 10,000 rounds to 1. The alias is f_alias = |6,000 - 1 * 10,000| = 4,000 Hz. The data will show a 4 kHz tone that does not exist in the original signal. The third calculator mode reproduces this check and reports that you would need a rate greater than 12 kHz, plus suitable filtering, to reconstruct the tone reliably.
Nyquist Sampling Rate FAQ
What is the difference between the Nyquist rate and the Nyquist frequency? They are two sides of the same rule and are easy to confuse. The Nyquist rate is a property of the signal: it equals twice the highest frequency in the signal and tells you the minimum sampling rate you must use. The Nyquist frequency is a property of the sampling system: it equals half the sampling rate and tells you the highest frequency that system can represent. If your signal's highest frequency sits below the system's Nyquist frequency, the Nyquist criterion is satisfied.
Why do real systems sample faster than twice the highest frequency? The factor of 2 assumes an ideal filter that passes everything below f_max and blocks everything above it instantly. Real anti-aliasing filters roll off gradually, so designers add a guard band between the highest wanted frequency and the Nyquist frequency for the filter to work in. That is why practical systems use factors like 2.205 for CD audio or 2.56 for vibration analysis, and why rates of 5 to 20 times f_max are common when waveform shape or control stability matters.
Can aliasing be removed after the data is recorded? No. Once a frequency folds into the band below the Nyquist frequency, it is mathematically indistinguishable from a real signal at the alias frequency, so no amount of processing can separate the two. Aliasing has to be prevented before sampling, either by low-pass filtering the analog signal ahead of the converter or by raising the sampling rate above twice the highest frequency actually present, including noise and harmonics.
