# Nyquist frequency In [[Signal_processing|signal processing]] the **Nyquist frequency** of a sampler is half its sampling rate, `fN = fs/2`: the highest frequency that a sequence of samples taken `fs` times per second can represent unambiguously. It is a property of the sampler, not of any signal passing through it, and it is named after Harry Nyquist, whose 1928 analysis of telegraph signalling established that a channel of bandwidth `B` carries at most `2B` independent pulses per second.[^nyq28] A signal component below the Nyquist frequency is represented exactly by its samples; a component above it is recorded as the samples of some lower frequency, the error known as [[Aliasing|aliasing]], and the Nyquist frequency is the line about which those higher frequencies fold, which is why it is also called the folding frequency. The [[Nyquist–Shannon_sampling_theorem]] states the same boundary from the signal's side: a signal is recoverable from its samples when its highest frequency lies below the Nyquist frequency of the sampler, so that [[Sampling_(signal_processing)|sampling]] at more than twice the highest frequency of interest is the design rule, and an [[Anti-aliasing_filter|anti-aliasing filter]] ahead of the sampler enforces it. The related but distinct Nyquist rate, twice the bandwidth of a given signal, is the lowest sampling rate whose Nyquist frequency clears that signal's content. The page's primary microsim, *Nyquist frequency*, lets the reader set a signal frequency and a sample rate and watch the tone's marker walk along a frequency axis creased at every multiple of half the sample rate, while its folded image bounces inside the baseband and its spectral images cross the amber Nyquist line. ## Folding frequency Sampling a signal at `fs` samples per second replicates its spectrum at every multiple of the sampling rate. Tiberius and Mulder write the sampled spectrum as `X_s(f) = sum_k X(f - k*fs)`, having scaled the sampling comb by the sampling interval;[^tib108] Dyer and Davis leave the comb unscaled and obtain the same sum multiplied by `1/Ts`, the form the sketch's header quotes as `fs * sum_k X(f - k*fs)`.[^dyer47] The two conventions differ only by that constant, and the sketch draws line positions rather than amplitudes, so the difference does not appear on screen. What matters is the spacing: every tone at `f` acquires images at `f - k*fs` for every integer `k`, together with the images of its negative-frequency twin at `-f - k*fs`, and the samples cannot say which member of the family was present. The band `[-fN, fN]`, one sampling rate wide and centred on zero, contains exactly one representative of each family. That band is the baseband, and its edge, `fN = fs/2`, is the Nyquist frequency. The sampling theorem, `fs > 2*fh` for a signal whose spectrum vanishes above `fh`, is the statement that the whole of the signal lies inside the baseband, with the inequality strict unless the spectrum is already zero at `fh`.[^tib111] The representative that lands in the baseband is the apparent frequency, `f_a = |f - fs*round(f/fs)|`, which always lies between zero and `fN`; it is the number the sketch prints beside its orange marker. Its geometry gives the name folding frequency. Cut the positive frequency axis into zones each `fN` wide, zone 1 being `[0, fN]`, zone 2 `[fN, 2fN]` and so on; sampling maps every zone onto zone 1, but with alternating orientation, the odd zones straight through and the even zones reversed, so that the axis creases like a paper fan at each multiple of the Nyquist frequency. A tone at 30 kHz sampled at 40 kHz, where `fN` is 20 kHz, sits in zone 2 and folds to `40 - 30 = 10 kHz`. The sketch's default tone of 27 Hz sampled at 20 Hz, where `fN` is 10 Hz, sits in zone 3 and passes straight through to `27 - 20 = 7 Hz`, which its DIAGNOSTICS panel reports as zone #3 (fwd) and a folded image of 7.00 Hz. Dyer and Davis express the same fold as a two-step rule, an odd-or-even zone test `N = mod(int(f/(fs/2)), 2)` followed by `f_a = N*fs/2 + (-1)^N*mod(f, fs/2)`,[^dyer49] and tabulate it at a sampling rate of 1,000 Hz, where `fN` is 500 Hz: 700 Hz folds to 300 Hz, 1,200 Hz to 200 Hz and 3,400 Hz to 400 Hz.[^dyer50] The [[Frequency_domain|frequency-domain]] picture explains why the fold begins exactly at `fN` and not before. In the sketch's SAMPLED SPECTRUM panel the [[Sine_wave|tone]] is a cyan line at `+f` and `-f`, its images are grey lines at `±f - k*fs`, and the amber Nyquist lines stand at `±fN`. While `f` is below `fN`, the nearest image, at `fs - f`, lies above `fN` and is removed by the reconstruction [[Low-pass_filter|low-pass filter]] that keeps only the baseband. As `f` rises toward `fN`, the image at `fs - f` descends toward it from the other side, and the two meet at the Nyquist line; push `f` past it and the image enters the baseband, where the filter keeps it, and the orange marker now sits at `fs - f` instead of `f`. Tiberius and Mulder's example is an 11 Hz cosine: sampled at 30 Hz, so that `fN` is 15 Hz, its images at 19, 41, 49 and 71 Hz all lie outside the baseband and the reconstruction is perfect;[^tib112] sampled at 10 Hz, so that `fN` is 5 Hz, the tone sits in zone 3 and its image at 1 Hz is what the reconstruction returns.[^tib113] A laboratory exercise in Fonseca and Dickerson makes the same point with a 20 Hz tone at 50 Hz, whose images appear at 30 and 70 Hz, outside a 25 Hz Nyquist frequency,[^fons228] and then with a 20 kHz rate, where `fN` is 10 kHz, so that a 15 kHz tone folds to 5 kHz and an 18 kHz tone to 2 kHz.[^fons232] Each of these can be set on the sketch's sliders at the stated rates or, for the kilohertz cases, at a thousandth of them. The Nyquist frequency itself is a boundary rather than a safe operating point. A tone exactly at `fN` is sampled twice per cycle, and depending on its phase the two samples per cycle may fall on its peaks, on its zero crossings or anywhere between, so its amplitude cannot be recovered even though it does not fold; this is the reason the theorem's inequality is strict.[^tib111] Practical systems therefore keep a margin. Audio sampled at 44.1 kHz, the rate of the examples in Downey's Think DSP, where a 10 ms segment holds 441 samples,[^downey70] has a Nyquist frequency of 22.05 kHz, and the rate of compact-disc audio is the same 44.1 kHz,[^cd] leaving the [[Anti-aliasing_filter|anti-aliasing filter]] a band of roughly 2 kHz above the conventional 20 kHz limit of hearing in which to fall from pass to stop. When that margin is too narrow for an affordable analog filter, the [[Analog-to-digital_converter|converter]] samples faster than the application needs, a practice called [[Oversampling|oversampling]], and a [[Digital_signal_processing|digital]] filter does the steep part of the job before the rate is reduced. In every case the Nyquist frequency of the final sample rate is the frequency above which nothing survives, and the folding rule says where anything that does survive will appear. ## Other meanings The word Nyquist attaches to two different numbers, and the two are often confused. The Nyquist frequency is half the sampling rate and belongs to the sampler; it does not change when a different signal is applied. The Nyquist rate is twice the bandwidth of a particular signal and belongs to that signal; it is the lowest sampling rate at which the signal can be sampled without aliasing, and it does not change when a different [[Analog-to-digital_converter|converter]] is used. The two coincide only at the design point where a sampler runs at exactly the Nyquist rate of its input, so that the input's highest frequency sits on the sampler's Nyquist frequency. Fonseca and Dickerson state the design condition in the rate form, `fs ≥ 2B`,[^fons227] and Dyer and Davis in the frequency form, `fs > 2*fmax`;[^dyer48] both describe the same requirement, that the sampler's Nyquist frequency must exceed the signal's highest frequency. | Quantity | Belongs to | Definition | Example | |---|---|---|---| | Nyquist frequency | the sampler | `fN = fs/2` | a 20 Hz sample rate has `fN = 10 Hz` | | Nyquist rate | the signal | `2B` for a signal of bandwidth `B` | a signal reaching 27 Hz has a Nyquist rate of 54 Hz | The sketch's tip line makes the second row concrete: with the default 27 Hz tone it advises raising the sample rate above 54 Hz, the tone's Nyquist rate, so that the sampler's Nyquist frequency clears the tone. Some publications use Nyquist frequency for the second quantity as well, calling twice the signal bandwidth the signal's Nyquist frequency; the usage is a minority one, and the number it names is otherwise called the Nyquist rate.[^usage] The rate sense is closer to Nyquist's own result, which concerned how many pulses per second a telegraph channel of bandwidth `B` could carry, namely `2B`, a signalling rate rather than a frequency.[^nyq28] [[Claude_Shannon|Claude Shannon]]'s 1949 paper turned the same number around into the sampling theorem, the statement that `2B` samples per second suffice to describe a signal of bandwidth `B`,[^shan49] and it is the frequency `fs/2` of that theorem that carries Nyquist's name. In [[Discrete_time_and_continuous_time|discrete time]] the Nyquist frequency is a fixed number in normalised units. Stiber, Stiber and Larson measure frequency in radians per sample, where a tone at `w0` becomes the digital frequency `w_hat = w0*Ts` and every distinct frequency lies in `[-π, π]`, so the Nyquist frequency is `π` radians per sample;[^stiber56] Johnson measures it in cycles per sample, where the distinct frequencies fill `(-1/2, 1/2)` and the Nyquist frequency is one half.[^john25] The [[Discrete-time_Fourier_transform]] of a sampled sequence is periodic with period `fs`, so its half-period is the Nyquist frequency, and in an `N`-point [[Discrete_Fourier_transform]] of a real sequence the bin at index `N/2` is the Nyquist bin, the last independent one before the spectrum mirrors. The concept also applies to sampling in space rather than time. A [[Digital_image_processing|digital image]] sensor with pixel pitch `p` samples the scene `1/p` times per unit length and therefore has a Nyquist frequency of `1/(2p)` cycles per unit length; a 5 µm pitch gives 100 cycles per millimetre, and finer detail is recorded as moiré. The same arithmetic gives the half-wavelength element spacing of a [[Radar|radar]] or [[Sonar|sonar]] array that must resolve directions of arrival without ambiguity. ## Microsims The primary microsim, *Nyquist frequency*, draws the fold line for a sampler the reader configures. SIGNAL FREQUENCY sets the true tone from 0.5 to 60 Hz, default 27 Hz, and SAMPLE RATE sets the sampler's rate from 4 to 50 Hz, default 20 Hz, with `fN = fs/2` printed beside it. The NYQUIST ZONES strip shows the true frequency axis shaded in alternate zones each `fN` wide, with amber dividers labelled fN = fs/2, 1·fs, 3·fN and so on; a cyan marker sits at the true frequency and a dashed connector drops it to an orange marker at its folded position on the baseband axis, labelled "folds to" the apparent frequency. The SAMPLED SPECTRUM panel plots the tone at `±f`, its images at `±f - k*fs` for `k` up to ±2 (the images control turns them off) and the folded baseband line, with the baseband `[-fN, fN]` shaded and the Nyquist lines in amber. The DIAGNOSTICS panel reports fs, fN, f, the Nyquist zone with its orientation (fwd or rev), the folded image `f_a`, samples per cycle, and a verdict banner reading BELOW NYQUIST or ABOVE NYQUIST with the folded value. The sweep control drives the signal frequency slowly up and down the axis for a hands-free demonstration. The model is ILLUSTRATIVE: it shows a single pure [[Sine_wave|sine]] and the positions of its spectral lines, not their amplitudes, and it has no noise, no [[Quantization_(signal_processing)|quantization]] and no filter. *Try:* Set SAMPLE RATE to 20 Hz and drag SIGNAL FREQUENCY slowly upward from 5 Hz; watch the orange image climb with the tone until they meet at the bold amber fN = 10 Hz line, then reverse and descend while the tone keeps rising, reaching zero at 20 Hz and climbing again in zone 3 to "folds to 7.00 Hz" at the 27 Hz default, where the banner reads ABOVE NYQUIST and the tip asks for a sample rate above 54 Hz. A three.js companion renders this concept in depth. The sketch runs live at https://editor.p5js.org/sciencenibber/full/hajBrucMj and can be forked at https://editor.p5js.org/sciencenibber/sketches/hajBrucMj . ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Nyquist_frequency) : [Wikitube](https://en.wikitube.io/wiki/Nyquist_frequency) Skeleton mirrored at revision 1371304274. Prose, emphasis and the microsims are Wikitube's own. ## Notes The sketch is ILLUSTRATIVE: it places the spectral lines of one pure tone and its images and does not draw their amplitudes, so the constant factor that separates the scaled and unscaled comb conventions never appears. Page numbers in the references are PDF pages of the open editions. ## References [^nyq28]: Nyquist, H. "Certain Topics in Telegraph Transmission Theory". *Transactions of the American Institute of Electrical Engineers*, vol. 47, 1928, pp. 617-644. [^shan49]: Shannon, C. E. "Communication in the Presence of Noise". *Proceedings of the IRE*, vol. 37, no. 1, 1949, pp. 10-21. [^tib108]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 108-110 (PDF pages): the sampling model with the comb scaled by the sampling interval, and the sum of spectral copies. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib111]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 111-113 (PDF pages): the sampling theorem and the strictness of the inequality. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib112]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 112-113 (PDF pages): an 11 Hz tone sampled at 30 Hz, images at 19, 41, 49 and 71 Hz, perfect reconstruction. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib113]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 113-115 (PDF pages): the same tone sampled at 10 Hz aliases to 1 Hz. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^dyer47]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 47-48 (PDF pages): the sampled spectrum with an unscaled comb, carrying the factor 1/Ts. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer48]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 48-49 (PDF pages): the sampling condition fs > 2 fmax. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer49]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, p. 49 (PDF page): the two-step alias-frequency algorithm. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer50]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, p. 50 (PDF page): at fs = 1,000 Hz, 700, 1,200 and 3,400 Hz fold to 300, 200 and 400 Hz. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^fons227]: Fonseca, A.; Dickerson, J. *Open Signals and Systems Laboratory Exercises*. 2nd ed., 2024, p. 227 (PDF page): the condition fs ≥ 2B. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/open-signals-and-systems-laboratory-exercises . CC BY-NC. [^fons228]: Fonseca, A.; Dickerson, J. *Open Signals and Systems Laboratory Exercises*. 2nd ed., 2024, p. 228 (PDF page): a 20 Hz tone sampled at 50 Hz shows images at 30 and 70 Hz. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/open-signals-and-systems-laboratory-exercises . CC BY-NC. [^fons232]: Fonseca, A.; Dickerson, J. *Open Signals and Systems Laboratory Exercises*. 2nd ed., 2024, pp. 232-233 (PDF pages): 15 kHz and 18 kHz tones at a 20 kHz rate, posed as questions; 5 kHz and 2 kHz are the folding rule's answers. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/open-signals-and-systems-laboratory-exercises . CC BY-NC. [^downey70]: Downey, A. *Think DSP: Digital Signal Processing in Python*. 2012, p. 70 (PDF page): a 0.01 s segment at 44.1 kHz holds 441 samples. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/think-dsp-digital-signal-processing-in-python . CC BY-NC. [^cd]: Citation needed: the 44.1 kHz sampling rate of compact-disc audio is fixed by the Compact Disc Digital Audio standard (IEC 60908); a page reference to that standard, or to a Portal Book that states the rate, would settle it. [^stiber56]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, pp. 56-57 (PDF pages): digital frequency in radians per sample and its range [−π, π]. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^john25]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 25-26 (PDF pages): discrete-time frequency is unique only on (−1/2, 1/2) cycles per sample. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^usage]: Citation needed: the minority usage of "Nyquist frequency" for twice the signal bandwidth; a textbook page that defines the term that way, set beside one that uses "Nyquist rate", would settle it. ## See also - [[Anti-aliasing_filter]] - [[Discrete-time_Fourier_transform]] - [[Nyquist–Shannon_sampling_theorem]] - [[Aliasing]] - [[Sampling_(signal_processing)]] - [[Oversampling]] - [[Low-pass_filter]] - [[PORTAL_Signal_Processing]] <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->