Signals & Systems
Sampling & Aliasing Calculator
From a signal frequency and a sample rate: the Nyquist rate, whether the tone is aliased, and the exact folded (apparent) frequency the samples appear to contain.
About this calculator
The sampling theorem says a signal whose highest frequency is f_max is fully recoverable from its samples only when the sample rate exceeds twice f_max — the Nyquist rate. Sample any slower and frequencies above the folding frequency f_s/2 masquerade as lower ones: aliasing. This calculator takes a tone frequency and a sample rate and tells you, concretely, the Nyquist rate, whether the criterion is met, and the exact alias frequency the sampled data would appear to hold.
The alias frequency is the tone folded into the band 0 to f_s/2: f_alias = |f − f_s·round(f/f_s)|. It is the frequency an ideal reconstruction would produce — the higher tone and its alias are genuinely indistinguishable from the samples alone, which is why anti-alias filtering before sampling matters. The plot overlays the original tone with the alias tone so you can see the two curves pass through the same sample instants.
This is the same phenomenon behind the wagon-wheel effect in film and the reason audio is sampled at 44.1 kHz for a 20 kHz audible band. Slide the sample rate down through 2·f to watch a correctly sampled tone suddenly fold into a lower alias.
Design notes & common mistakes
- Nyquist RATE = 2·f_max (a property of the signal); Nyquist/folding FREQUENCY = f_s/2 (a property of the sampler). Keep the two straight.
- Aliasing is irreversible: once a tone folds, no post-processing can recover the original — you must band-limit before sampling.
- Sampling exactly at 2·f is the boundary case; a tone at exactly f_s/2 can vanish or beat depending on phase, so always keep a guard band.
- The wagon-wheel effect and the 44.1 kHz audio rate (for a 20 kHz band) are everyday instances of this same folding rule.
Assumptions
- A single real sinusoid is sampled uniformly at rate f_s; ideal (instantaneous) sampling.
- The alias is computed by folding into the first Nyquist zone [0, f_s/2].
- Reconstruction is assumed ideal (sinc interpolation); practical filters need a guard band beyond the bare Nyquist rate.
- No quantization is modeled — this is about rate, not bit depth.
When to use this calculator
Appropriate for
- Checking whether a chosen sample rate satisfies Nyquist for a given tone
- Computing the apparent frequency a tone folds to when undersampled
- Teaching the sampling theorem, folding, and the wagon-wheel effect
Not suitable for
- Designing anti-alias or reconstruction filters
- Broadband or multi-tone aliasing budgets
- Bandpass (intentional undersampling) system design without further analysis
What this calculator does not cover
- Single-tone analysis — a real broadband signal aliases every component above f_s/2 simultaneously.
- Ideal sampling and reconstruction; aperture effect, jitter, and non-ideal filters are not modeled.
- No anti-alias filter design — it reports the folding, it does not size the filter you would need.
- Amplitude is normalized; aliasing does not change amplitude here (only apparent frequency).
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
What is the difference between the Nyquist rate and the Nyquist frequency?
The Nyquist rate is 2·f_max — the minimum sample rate needed to capture a signal whose highest frequency is f_max. The Nyquist (or folding) frequency is f_s/2 — the highest frequency a given sample rate can represent. One is a property of the signal, the other of the sampler.
How do I compute the alias frequency?
Fold the tone into the band 0 to f_s/2: f_alias = |f − f_s·round(f/f_s)|. For example, 600 Hz sampled at 1000 Hz folds to 400 Hz, and 1100 Hz folds to 100 Hz.
Can aliasing be removed after sampling?
No. Once a frequency folds into the baseband it is indistinguishable from a genuine tone at the alias frequency, so no post-processing can separate them. Aliasing must be prevented before sampling with an anti-alias (low-pass) filter.
References
- Oppenheim, A. V. & Willsky, A. S., Signals and Systems, 2nd ed., Ch. 7 (sampling theorem, aliasing)
- Proakis, J. G. & Manolakis, D. G., Digital Signal Processing, 4th ed., Ch. 1 (sampling of analog signals)
- Shannon, C. E., 'Communication in the presence of noise,' Proc. IRE, 1949 (the sampling theorem)
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