Calculator Inputs
Example Data Table
| Case | Maximum Signal Frequency | Sampling Frequency | Nyquist Rate | Alias Outcome | Interpretation |
|---|---|---|---|---|---|
| Audio channel | 20 kHz | 48 kHz | 40 kHz | Negligible for in-band content | Comfortable oversampling |
| Boundary case | 5 kHz | 10 kHz | 10 kHz | Critical edge | Exactly at Nyquist |
| Undersampled tone | 4 kHz | 6 kHz | 8 kHz | Aliasing likely | Unsafe for clean reconstruction |
| Instrumentation | 250 Hz | 2 kHz | 500 Hz | Very low | Robust acquisition margin |
Formula Used
Baseband sampling theorem: for a band-limited signal with highest frequency component fmax, accurate reconstruction requires fs ≥ 2fmax.
Nyquist rate: fN = 2fmax
Nyquist interval: TN = 1 / (2fmax)
Actual sampling period: Ts = 1 / fs
Oversampling ratio: OSR = fs / (2fmax)
Sampling margin: ((fs - 2fmax) / 2fmax) × 100
Alias frequency: falias = |f - round(f / fs) × fs|
Normalized frequency: f / fs cycles per sample
Digital angular frequency: Ω = 2πf / fs rad/sample
These relations help evaluate whether a chosen sample rate preserves the original signal or folds higher components into lower apparent frequencies.
How to Use This Calculator
- Enter the highest frequency present in the original signal.
- Enter the actual sampling frequency used by the acquisition system.
- Set a test tone frequency for alias analysis and the waveform graph.
- Choose amplitude, phase, and capture duration for the plotted signal.
- Click Calculate to show results above the form.
- Review the verdict, Nyquist rate, sampling margin, and alias frequency.
- Use the CSV or PDF buttons to export the current result set.
- Inspect the Plotly graph to compare continuous and sampled waveforms.
FAQs
1. What does the sampling theorem state?
It states that a band-limited signal can be reconstructed from samples when the sampling frequency is at least twice the highest signal frequency. That minimum threshold is the Nyquist rate.
2. What is the Nyquist rate?
The Nyquist rate equals two times the highest frequency present in the signal. Sampling below this level can cause spectral overlap and incorrect reconstruction.
3. Why does aliasing happen?
Aliasing happens when the sample rate is too low to separate higher-frequency components. Those components fold into lower apparent frequencies inside the sampled data.
4. Is sampling exactly at twice the frequency enough?
It is the theoretical minimum for ideal band-limited signals. Real systems usually sample faster to allow filter roll-off, timing tolerance, and better numerical stability.
5. What does oversampling improve?
Oversampling increases design margin, reduces alias risk, eases anti-alias filter requirements, and often improves digital processing robustness for noisy or practical signals.
6. What does alias frequency mean?
Alias frequency is the folded frequency observed after undersampling. It is the misleading lower-frequency component that appears in place of the true tone.
7. Why include a test tone frequency?
The test tone lets you examine how one specific component behaves under the chosen sampling rate. It also drives the waveform graph and alias calculation.
8. Can this calculator help with real acquisition systems?
Yes. It is useful for checking converters, instruments, controllers, and digital signal chains where frequency content, timing, and alias risk matter.