# Aliasing **Aliasing** is the error that appears when a signal is [[Sampling_(signal_processing)|sampled]] too slowly for its content: a frequency component above half the sampling rate produces exactly the same samples as some lower frequency, and a reconstruction from those samples contains the lower frequency in place of the original. The substitute is the alias, and the boundary at which the ambiguity begins, half the sampling rate, is the [[Nyquist_frequency]]. Because the samples of the alias and of the original are the same numbers, nothing done after sampling can tell them apart; the only defences are to sample faster or to remove the offending components beforehand with an [[Anti-aliasing_filter|anti-aliasing filter]]. The word covers more than audio: a camera samples a scene at its pixel pitch and a film samples motion once per frame, so fine texture becomes moiré and a fast wheel seems to turn backwards; in [[Signal_processing|signal processing]] all of these obey the same arithmetic. The page's primary microsim, *Aliasing*, samples one [[Sine_wave|sine wave]] at a rate the reader sets and draws through the sample dots the sinusoid a converter would reconstruct; dragging the signal frequency past half the sample rate makes the reconstructed wave slow down while the true wave speeds up, and a folding plot traces the apparent frequency. ## Description A sampler records a continuous signal only at instants spaced by the sampling interval `Ts = 1/fs`. In the [[Frequency_domain|frequency domain]] the consequence is exact: sampling makes the spectrum a sum of copies of the original, `X_s(f) = sum_k X(f - k*fs)`, one centred on every multiple of the sampling rate.[^tib108] If the original spectrum is confined below half the sampling rate the copies sit side by side and a [[Low-pass_filter|low-pass filter]] that keeps only the copy around zero recovers the original exactly; if not, neighbouring copies overlap and a component of one copy cannot be told from a component of another. That overlap is aliasing, and the misplaced components are the aliases. The clearest case is a single tone. Tiberius and Mulder sample an 11 Hz cosine at 30 Hz: the sampled spectrum carries lines at 11 Hz and at 19, 41, 49 and 71 Hz, everything except the 11 Hz line lies above the 15 Hz half-rate, the filter removes it, and the reconstruction is perfect.[^tib112] Sampled at 10 Hz instead, the same cosine yields exactly the samples of a 1 Hz cosine, and 1 Hz is what any reconstruction produces.[^tib113] The microsim repeats this experiment for any pair of frequencies in its range: past half the sample rate the orange reconstructed wave is visibly slower than the grey true wave while still passing through every sample dot. The same arithmetic governs space: a [[Charge-coupled_device|CCD]] samples a scene at its pixel pitch, and a pattern finer than two pixels per cycle is recorded as the coarse moiré of [[Digital_image_processing|digital images]]. ## Bandlimited functions A signal is bandlimited if its [[Fourier_transform]] vanishes above some highest frequency `fh`. For such a signal the [[Nyquist–Shannon_sampling_theorem]] gives the condition under which the spectral copies do not overlap: the sampling rate must exceed twice the highest frequency, `fs > 2*fh`, with equality admissible only when the spectrum is already zero at `fh` itself.[^tib111] The strictness matters: a cosine at exactly half the sampling rate can be sampled at its zero crossings, so its amplitude and phase are lost. The textbook example is `sinc²(t)`, whose spectrum is a triangle that reaches zero at 1 Hz, so `fh = 1 Hz` and the copies overlap for any sampling rate below 2 Hz.[^tib107] Sampled at 1.5 Hz and reconstructed with the ideal low-pass filter, it comes back as `(3/4)*sinc(1.5*t) + (1/4)*sinc²(0.5*t)`, a function that passes through every sample but is not the original.[^tib119] The part of each neighbouring copy that spilled below the 0.75 Hz half-rate has been folded into the answer, and no [[Whittaker–Shannon_interpolation_formula|interpolation formula]] can fold it back out. No signal of finite duration is bandlimited, because a function that vanishes outside a finite interval has a spectrum without an upper bound, so some aliasing always occurs and the practical question is how much. A square wave is the standard warning: its harmonics decay only as `1/n`, so at any sampling rate some lie above the half-rate and fold back, and it must be band-limited before sampling rather than after.[^stiber58] That is the job of the [[Anti-aliasing_filter|anti-aliasing filter]]. ## Bandpass signals The copies only need to avoid one another, so a signal whose energy occupies a narrow band well above zero can be sampled far more slowly than twice its highest frequency. Fonseca and Dickerson state the requirement as `fs ≥ 2B`, where `B` is the bandwidth rather than the top frequency.[^fons227] For a real band between `fL` and `fH` the admissible rates fall in windows `2*fH/n ≤ fs ≤ 2*fL/(n - 1)`, for integers `n` no larger than `fH/B`, chosen so that no copy of the negative-frequency half of the spectrum lands on the positive band. The band is thereby translated down to a low frequency without a mixer, a technique called bandpass sampling or undersampling, the deliberate opposite of [[Oversampling|oversampling]]. The microsim shows the translation for a single tone. At its default settings a 21 Hz tone sampled at 16 Hz is read as 5 Hz, and the signed folded frequency the sketch computes, `f - fs*round(f/fs) = +5 Hz`, is positive, so a band of tones near 21 Hz would come through in its original order; had the fold been negative the band would have arrived inverted. Bandpass sampling is used in [[Superheterodyne_receiver|radio receivers]] and on [[Radar|radar]] returns, and it is safe only when everything outside the chosen band, including noise, has been removed first. ## Sampling sinusoidal functions A [[Sine_wave|sinusoid]] is the natural test signal for sampling because its samples form a sinusoidal sequence, and because the identity that makes aliasing exact is one line of algebra. ### Folding Sampling `sin(2π f t)` at instants `t = n*Ts` gives the sequence `sin(2π f n Ts)`. Replacing `f` by `f - k*fs` for any integer `k` changes the argument by `2π k n`, a whole number of cycles, so `sin(2π f n Ts) = sin(2π (f - k*fs) n Ts)`: the samples of the tone at `f` are, number for number, the samples of the tone at `f - k*fs`. A reconstruction chooses the candidate of smallest magnitude, which the sketch computes as `f_alias = |f - fs*round(f/fs)|`, always between zero and half the sample rate. Dyer and Davis give the same rule as a two-step algorithm on the half-rate, `N = mod(int(f/(fs/2)), 2)` and `f_alias = N*fs/2 + (-1)^N*mod(f, fs/2)`,[^dyer49] and tabulate it at a sampling rate of 1,000 Hz: 700 Hz appears at 300 Hz, 1,200 Hz at 200 Hz and 3,400 Hz at 400 Hz.[^dyer50] Scaled down a hundredfold, the same three cases read 3, 2 and 4 Hz in the sketch's DIAGNOSTICS panel. Plotted against the true frequency, the apparent frequency is a triangle wave that rises to `fs/2`, falls to zero at `fs` and rises again at `3fs/2`, as if the frequency axis were creased at every multiple of half the sample rate and folded onto its first segment. The FREQUENCY FOLDING panel draws this triangle, marks the creases in amber at fs/2, 1·fs, 3·fs/2 and 2·fs, and moves a marker along it with the signal frequency slider; the green diagonal below the first crease is the ideal case, and the [[Nyquist_frequency|Nyquist frequency]] is where the two curves part. ### Complex sinusoids For a complex exponential the sign of the frequency is meaningful, and aliasing then has a direction as well as a magnitude. Stiber, Stiber and Larson write the sampled phasor in terms of the digital frequency `w_hat = w0*Ts`, in radians per sample, which is unique only in `[-π, π]`; an [[Angular_frequency|angular frequency]] outside that interval is indistinguishable from one inside it, and the full set of apparent frequencies of a tone at `w0` is `±w0 + k*ws`.[^stiber56] Sampling at four times the tone's frequency gives `w_hat = π/2`, inside the interval, while sampling at only four-thirds of it gives `w_hat = 3π/2`, the same angle as `-π/2`, an apparent frequency of `-w0/3`: a phasor turning the wrong way at a third of the speed.[^stiber56] Johnson states the same uniqueness in cycles per sample, on `(-1/2, 1/2)`.[^john25] For a real cosine the sign is invisible, because `cos(-x) = cos(x)`, and the reversal survives only as a change of phase.[^stiber56] The sketch keeps the sign anyway: it draws the reconstruction as `sin(2π f_fold t)` with `f_fold = f - fs*round(f/fs)`, so that when the fold is negative the orange wave is the mirror image of a slow sine and still passes through every dot. With the animate control on, the true wave drifts steadily one way and, whenever the fold is negative, the aliased wave drifts the other way. ### Sample frequency Choosing the sample rate is choosing where the crease lies. The rule is `fs > 2*fmax` for the highest frequency to be preserved,[^dyer48] and the DIAGNOSTICS panel states it as a count, samples per cycle `fs/f`, flagged red below 2 and green at 2 or above. The default 21 Hz tone at 16 Hz gives 0.76 samples per cycle and the banner "21.0 Hz is read as 5.0 Hz", with the advice that the sample rate must rise above 42 Hz. Fonseca and Dickerson's laboratory runs the arithmetic both ways: a 20 Hz tone sampled at 50 Hz shows spectral images at 30 and 70 Hz, above the 25 Hz half-rate and removable,[^fons228] whereas at a 20 kHz rate a 15 kHz tone folds to 5 kHz and an 18 kHz tone to 2 kHz.[^fons232] In a pulse [[Radar|radar]] the rate is the pulse repetition frequency, and Doppler shifts above half of it fold in the same way, the velocity ambiguity of pulse-Doppler processing. ## Historical usage The phenomenon was understood before it had a name. Harry Nyquist's 1928 analysis of telegraph signalling established that a channel of bandwidth `B` carries at most `2B` independent pulses per second,[^nyq28] the number that, read the other way, says a signal of bandwidth `B` needs `2B` samples per second; [[Claude_Shannon|Claude Shannon]]'s 1949 paper on communication in the presence of noise stated the sampling theorem in its modern form.[^shan49] The word is usually credited to Ralph Blackman and John Tukey, whose 1958 treatment of power-spectrum estimation in the Bell System Technical Journal used alias for a frequency that a sampled record cannot distinguish from another and aliasing for the resulting folding of a [[Spectral_density|spectral estimate]].[^blackman58] Radio engineering had met the same ambiguity under another name. In a [[Superheterodyne_receiver|superheterodyne receiver]], introduced by Edwin Armstrong at the end of the First World War,[^arm21] a wanted station at `f_LO + f_IF` and an unwanted one at `f_LO - f_IF` both mix with the local oscillator to the same intermediate frequency `f_IF`, and once mixed they cannot be separated. Radio engineers call the unwanted station the image and reject it with a tuned filter ahead of the mixer, precisely the role of an [[Anti-aliasing_filter|anti-aliasing filter]] ahead of a sampler, which behaves as a mixer with every harmonic of `fs` as a local oscillator at once. ## Angular aliasing The sampled quantity need not be time. When it is an angle, either the direction from which a wave arrives or the direction from which a scene is viewed, the ambiguity is called angular aliasing. An array of [[Sensor|sensors]] spaced `d` apart samples an arriving [[Plane_wave|plane wave]] in space: a wave from an angle `θ` off broadside advances by `d*sin θ` from one element to the next, a phase step of `2π (d/λ) sin θ`, which is unambiguous over all directions only when it cannot exceed `π`, that is when `d ≤ λ/2`. A half-wavelength spacing is the spatial form of sampling at twice the highest frequency, the [[Nyquist_frequency|Nyquist]] condition; with wider spacing two directions produce identical phase steps, and the spurious responses, called grating lobes, are the array's aliases. A lenticular print or a multi-view screen samples the other kind of angle: it stores a finite number of views and shows each over a wedge of viewing directions, so an observer moving sideways sees the same picture for a while and then a jump to the next, the continuous parallax of a real object sampled at the angular pitch of the views. An ordinary photograph is the extreme case, one view for all angles, in which every angular frequency has been aliased to zero. [[Loudspeaker|Loudspeaker]] arrays that synthesise a sound field meet the same limit once their spacing exceeds half a wavelength. ## More examples ### Audio example Stiber, Stiber and Larson sample a 330 Hz tone at 300 samples per second and obtain the samples of a 30 Hz tone, more than three octaves below the note played.[^stiber54] In the microsim, 33 Hz at a 30 Hz sample rate gives the same result, 3.00 Hz. A tone swept upward through the half-rate is heard to rise, turn and fall, and every harmonic of a rich [[Sound|sound]] that crosses the crease comes back down at a pitch unrelated to the fundamental, so aliased audio sounds not merely dull but wrong. A square wave produces such ghosts at any sampling rate unless it is filtered first.[^stiber58] In an [[Audio_signal_processing|audio system]] the [[Anti-aliasing_filter|anti-aliasing filter]] keeps the ghosts below the noise; because a real filter needs some width to fall from pass to stop, practical systems sample somewhat faster than twice the highest frequency they keep. ### Direction finding A [[Radar|radar]] or [[Sonar|sonar]] that finds a target's bearing by comparing the phase of its echo at two or more antennas is sampling the wavefront in space. Two receivers a distance `d` apart measure a phase difference `2π (d/λ) sin θ`; if `d` exceeds half a wavelength the measured phase corresponds to more than one bearing, and the system reports one of several ghosts. Interferometric direction finders accept the ambiguity deliberately, because a wide baseline gives a precise bearing, and resolve it with a second, shorter baseline whose coarse but unambiguous answer selects the right ghost. ## Microsims The primary microsim, *Aliasing*, samples one pure [[Sine_wave|sine]] at a rate the reader sets. SIGNAL FREQUENCY sets the true tone from 0.5 to 60 Hz, default 21 Hz, and SAMPLE RATE sets the converter's rate from 2 to 50 Hz, default 16 Hz, so the [[Nyquist_frequency|Nyquist frequency]] is fs/2. The TIME DOMAIN panel shows a one-second window with the true signal in grey, the samples as cyan dots at every Ts and, when reconstruction is ON, the orange sinusoid that interpolates the dots, marked FOLDED when the tone is above Nyquist. The FREQUENCY FOLDING panel plots apparent against true frequency, 0–60 Hz across and 0 to fs/2 up; the DIAGNOSTICS panel prints f, fs, fs/2, samples per cycle and the apparent frequency, and ends in a verdict banner. The animate control adds a slow phase drift so that a folded wave can be seen moving the wrong way. The model is ILLUSTRATIVE: a single noiseless tone, a reconstruction that is simply the lowest-frequency sinusoid through the dots, and no [[Quantization_(signal_processing)|quantization]]. *Try:* Leave SAMPLE RATE at 16 Hz and drag SIGNAL FREQUENCY from 5 Hz up through 8, 16 and 21 Hz; watch the orange reconstruction flatten to a line at 16 Hz and the banner change from "NO ALIASING" to "21.0 Hz is read as 5.0 Hz", then raise SAMPLE RATE past 42 Hz and watch the alias vanish. A three.js companion renders this concept in depth. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Aliasing) : [Wikitube](https://en.wikitube.io/wiki/Aliasing) Skeleton mirrored at revision 1372423745. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Anti-aliasing_filter]] - [[Nyquist_frequency]] - [[Sampling_(signal_processing)]] - [[Nyquist–Shannon_sampling_theorem]] - [[Oversampling]] - [[Spectral_leakage]] - [[Superheterodyne_receiver]] - [[Whittaker–Shannon_interpolation_formula]] - [[PORTAL_Signal_Processing]] ## References [^tib108]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 108-110 (PDF pages): the sampling model 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, copies at 19, 41, 49 and 71 Hz. 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. [^tib107]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, p. 107 (PDF page): the triangular spectrum of sinc²(t) with highest frequency 1 Hz. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib119]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 119-120 (PDF pages): example 10.1, sinc²(t) sampled at 1.5 Hz and its aliased reconstruction. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^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 alias 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. [^stiber56]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, pp. 56-57 (PDF pages): digital frequency, the set of apparent frequencies, the examples at four and four-thirds times the tone frequency, and the sign of the alias for complex phasors. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^stiber54]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, pp. 54, 58 (PDF pages): a 330 Hz tone sampled at 300 Hz appears at 30 Hz. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^stiber58]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, p. 58 (PDF page): a square wave aliases at any sampling rate and must be band-limited first. 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). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^fons227]: Fonseca, A.; Dickerson, J. *Open Signals and Systems Laboratory Exercises*. 2nd ed., 2024, p. 227 (PDF page): the condition fs ≥ 2B in terms of bandwidth. 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; the lab poses these as questions, and 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. [^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. [^blackman58]: Blackman, R. B.; Tukey, J. W. "The Measurement of Power Spectra from the Point of View of Communications Engineering", Parts I and II. *Bell System Technical Journal*, vol. 37, 1958. Citation needed: the passage in which the words alias and aliasing are introduced, or an earlier printed use in sampling, would settle whether the coinage is theirs. [^arm21]: Armstrong, E. H. "A New System of Short Wave Amplification". *Proceedings of the Institute of Radio Engineers*, vol. 9, no. 1, 1921, pp. 3-11. ## Further reading - Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory* (2026). CC BY. https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory - Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain* (2020). CC BY-SA. https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain - Fonseca, A.; Dickerson, J. *Open Signals and Systems Laboratory Exercises*, 2nd ed. (2024). CC BY-NC. https://open.umn.edu/opentextbooks/textbooks/open-signals-and-systems-laboratory-exercises ## External links - Aliasing microsim, live sketch: https://editor.p5js.org/sciencenibber/full/tHGSVAofH - Fork the sketch in the p5.js editor: https://editor.p5js.org/sciencenibber/sketches/tHGSVAofH <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->