Calculator Inputs
Use sampled spectrum values. The calculator numerically reconstructs the signal over your selected time interval.
Example data table
This sample spectrum represents a smooth Gaussian-shaped frequency profile with zero phase. It is suitable for testing the reconstruction workflow.
| Frequency f | Magnitude | Phase (rad) | Equivalent real part | Equivalent imaginary part |
|---|---|---|---|---|
| -3 | 0.011109 | 0 | 0.011109 | 0 |
| -2 | 0.135335 | 0 | 0.135335 | 0 |
| -1 | 0.606531 | 0 | 0.606531 | 0 |
| 0 | 1.000000 | 0 | 1.000000 | 0 |
| 1 | 0.606531 | 0 | 0.606531 | 0 |
| 2 | 0.135335 | 0 | 0.135335 | 0 |
Formula used
Ordinary frequency form:
x(t) = ∫ X(f)ei2πft df
Angular frequency form:
x(t) = (1 / 2π) ∫ X(ω)eiωt dω
Numerical approximation used here:
x(t) ≈ c Σ [ X(νk)eiανkt ] Δν
For trapezoidal integration, consecutive weighted endpoints are averaged over each interval. For ordinary frequency, α = 2π and c = 1. For angular frequency, α = 1 and c = 1 / 2π.
Because the calculator works from finite sampled data, the output is a numerical reconstruction, not a symbolic closed-form expression.
How to use this calculator
- Select whether your spectrum is entered as magnitude-phase or real-imaginary values.
- Choose ordinary frequency f or angular frequency ω to match your source formula.
- Paste frequency samples and matching spectrum values in the text boxes.
- Set the time range and the number of output points you want to reconstruct.
- Pick trapezoidal integration for smoother results on sampled data.
- Press Calculate inverse transform to place the result above the form.
- Review the summary metrics, time-domain graph, spectrum graph, and output table.
- Use the CSV or PDF buttons to export the computed results.
Frequently asked questions
1) What does this calculator return?
It numerically reconstructs a time-domain signal from sampled frequency-domain data. The output includes real part, imaginary part, magnitude, phase, summary metrics, a results table, and two interactive plots.
2) What is the difference between ordinary and angular frequency?
Ordinary frequency uses cycles per unit and the kernel ei2πft. Angular frequency uses radians per unit and the kernel eiωt with the 1/2π scaling factor.
3) Can I enter nonuniformly spaced frequencies?
Yes. The calculator sorts the samples and integrates across the actual spacing between consecutive frequency points, so equally spaced input is not required.
4) Why can the reconstructed signal be complex?
A general inverse Fourier transform may produce complex values when the spectrum lacks conjugate symmetry. Many physical signals become purely real only when the spectrum satisfies the correct symmetry conditions.
5) Why is this not a symbolic inverse transform tool?
This page is designed for sampled numerical spectra. It estimates the inverse transform directly from data points, which makes it useful for analysis, experiments, and frequency samples generated by other software.
6) How many time points should I choose?
Use more points for smoother curves and better export detail. Moderate values like 200 to 400 often balance clarity and speed, while very large grids increase computation time.
7) Why might I see ringing or edge effects?
Ringing can appear when the sampled spectrum is sharply truncated, sparsely sampled, or missing important high-frequency content. Expanding the spectrum range or refining the sampling usually improves the reconstruction.
8) What do the export buttons include?
CSV export saves the full reconstructed table. PDF export includes key settings, summary values, a results table, and the plotted time-domain chart when the browser allows chart image capture.