Calculator Inputs
Example Data Table
This example represents one period of a smooth periodic signal sampled from 0 to 2π.
| x | f(x) |
|---|---|
| 0.00000000 | 5.00000000 |
| 1.57079633 | 3.50000000 |
| 3.14159265 | 1.00000000 |
| 4.71238898 | 3.50000000 |
| 6.28318531 | 5.00000000 |
Paste the full sample list above, or replace it with your own evenly or unevenly spaced periodic measurements.
Formula Used
For a period T over x ∈ [x0, x0 + T]: f(x) = a0/2 + Σ from n=1 to N [ an cos(2πn(x - x0)/T) + bn sin(2πn(x - x0)/T) ] a0 = (2/T) ∫ f(x) dx an = (2/T) ∫ f(x) cos(2πn(x - x0)/T) dx bn = (2/T) ∫ f(x) sin(2πn(x - x0)/T) dx Amplitude_n = √(an² + bn²) Phase_n = atan2(bn, an) This calculator evaluates the integrals numerically with the trapezoidal rule, so it works with measured sample points, laboratory data, imported logs, and custom periodic observations.
How to Use This Calculator
- Enter the signal period T and the starting x location.
- Choose how many harmonics you want in the approximation.
- Set the reconstruction point count and decimal precision.
- Paste one x,y sample pair on each line for exactly one full period.
- Submit the form to compute a0, an, bn, amplitudes, phases, and error metrics.
- Inspect the waveform graph, spectrum graph, and comparison table.
- Download the coefficient report as CSV or PDF when needed.
Why This Fourier Coefficients Tool Helps
An online Fourier coefficients calculator is useful when a periodic signal must be reduced into interpretable harmonic components. Engineers use Fourier coefficients to isolate dominant frequencies, compare waveform quality, and reconstruct signals with a smaller number of terms. Students use the same expansion to understand how trigonometric basis functions model complex repeating patterns. Analysts can also inspect phase shifts, amplitude distribution, and mean offset without rewriting a long derivation by hand.
This page accepts sampled data rather than a symbolic expression alone. That makes it practical for measured voltage traces, rotating machinery data, vibration logs, acoustic wave snapshots, and classroom datasets. Since the coefficients are estimated numerically over one full period, the tool can work with uneven spacing as long as the sample range covers the chosen period. The reconstruction error table helps you judge whether the selected number of harmonics is sufficient for your signal.
The waveform graph compares the original observations with the reconstructed series, while the spectrum graph highlights the strongest harmonic amplitudes. Together, these outputs make it easier to validate periodic assumptions, detect dominant modes, and create cleaner summaries for reports. Export features also help when results must be shared, archived, or checked in another application.
FAQs
1. What data should I paste into the calculator?
Paste one x,y pair per line for one full period. The first x should match the chosen start value. The final endpoint may be added automatically.
2. Can I use unevenly spaced sample points?
Yes. The calculator uses trapezoidal numerical integration, so uneven spacing is acceptable when the data still covers one complete period properly.
3. What does a0 represent?
a0 represents twice the average value over the selected period. The constant or DC component used in reconstruction is a0 divided by two.
4. What do an and bn mean?
an measures cosine content for harmonic n. bn measures sine content for harmonic n. Together they describe each harmonic contribution completely.
5. Why do I see amplitude and phase too?
Amplitude and phase provide a compact harmonic view. They transform cosine and sine coefficients into a more interpretable magnitude and shift description.
6. Why is my reconstruction error not zero?
Error appears when the selected harmonic count is limited, the samples are noisy, or the period does not align well with the pasted data.
7. What does Parseval partial mean here?
It estimates average signal power captured by the displayed coefficients. Comparing it with actual power helps judge how much energy the truncation retains.
8. Can I export results for reports?
Yes. The page includes CSV export for tables and PDF export for a clean summary containing settings, coefficients, and reconstruction values.