Calculator
Physics • Sampling TheoryChoose a mode, enter known values, and compute Nyquist limits, minimum sampling rates, or aliased frequencies.
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| Time | Mode | fs (Hz) | Nyquist (Hz) | fin (Hz) | Alias (Hz) | fmax (Hz) | fs min (Hz) | Notes |
|---|---|---|---|---|---|---|---|---|
| No calculations yet. Run the calculator to build a history. | ||||||||
Example Data Table
Use these examples to verify your inputs and understand typical outcomes.
| Scenario | Inputs | Expected output |
|---|---|---|
| Audio sampling | fs = 44.1 kHz | Nyquist = 22.05 kHz |
| Sensor design | fmax = 5 kHz | fs minimum = 10 kHz |
| Aliasing check | fin = 30 kHz, fs = 40 kHz | Alias = 10 kHz (folds into baseband) |
Formula Used
How to Use This Calculator
- Select a mode: Nyquist, minimum sampling, or aliasing.
- Choose sampling input as rate (fs) or period (Ts).
- Enter values with appropriate units and set decimals.
- Click Calculate to view results above the form.
- Use CSV/PDF buttons to export your session history.
Sampling rate and bandwidth planning
Sampling rate sets how often a sensor, ADC, or model captures values each second. If a channel must preserve content up to 2.5 kHz, target fs above 5 kHz, then add margin for filter roll‑off and transients. Many teams choose 6.4 kHz or 8 kHz to simplify FFT sizing and decimation. Converting between sampling period and rate early helps when timing budgets are written in microseconds.
Nyquist frequency in common systems
Nyquist frequency equals half the sampling rate and acts as the highest unambiguous sinusoid in an ideal model. At 44.1 kHz audio, the limit is 22.05 kHz; at 48 kHz, it is 24 kHz; at 96 kHz, it is 48 kHz. In RF telemetry, 1 MS/s provides a 500 kHz limit, often paired with an analog cutoff near 450 kHz. In imaging, higher pixel clock increases the Nyquist spatial frequency along each axis.
Aliasing patterns and folding intuition
When fin exceeds Nyquist, the sampled data appears at a different, lower frequency. Sampling a 30 kHz tone at 40 kHz produces a 10 kHz alias because spectra repeat every fs and mirror around fs/2. This creates believable “ghost” peaks in FFTs. Alias mode predicts where they land before you interpret a plot.
Anti-alias filtering and guard bands
Real filters have finite transition bands, so engineers keep a guard band between the highest wanted frequency and Nyquist. A practical guideline is placing the analog cutoff at 0.8×Nyquist or lower, depending on filter order and stopband needs. If you require strong attenuation by Nyquist, increase fs or reduce usable bandwidth.
Oversampling benefits and tradeoffs
Sampling above the minimum relaxes filtering and can improve noise performance after digital processing. Oversampling by 4× can deliver about one extra effective bit after decimation when noise is well behaved. The cost is higher data rate, more storage, and additional compute for filtering and analysis.
Practical validation with this calculator
Use Nyquist mode to confirm limits from fs or Ts, sampling mode to compute the minimum rate from fmax, and alias mode to test suspected interferers. Export CSV or PDF to document units, rounding, and assumptions for design reviews and lab reports.
FAQs
1) What is Nyquist frequency?
Nyquist frequency is half the sampling rate. It is the highest sinusoidal frequency that can be uniquely represented by ideal uniform samples without ambiguity.
2) How do I choose a safe sampling rate?
Start with fs ≥ 2×fmax after filtering, then add guard band for real filter roll‑off. Many designs use 2.5× to 5× fmax to simplify filters and improve robustness.
3) Why does aliasing create “wrong” peaks?
Sampling replicates spectra every fs. Components above fs/2 fold back into baseband and can look like legitimate low‑frequency content. Without an anti‑alias filter, the sampled signal cannot distinguish originals from folded copies.
4) Is fs = 2×fmax always sufficient?
It is the theoretical minimum for band-limited signals. In practice, you need margin for filter transition bands, noise, and non‑idealities, so fs is typically higher than 2×fmax.
5) How is the alias frequency computed?
The calculator maps fin into the 0 to fs/2 range using a modulo fold around fs/2. The result matches the observed tone frequency in sampled data under ideal uniform sampling.
6) Which units should I use?
Use any consistent unit. Enter Hz, kHz, MHz, or GHz based on your domain, and pick the output unit for readability. The calculator converts internally to Hz to avoid rounding issues.