Fourier Coefficient Calculator

Example Data Table

The following example represents one period of a simple square-wave style input with T = 1. This is also the default sample set.

Formula Used

For a periodic signal with period T, the calculator estimates the Fourier series coefficients numerically from the supplied samples.

a₀ = (2/T) ∫ f(x) dx

aₙ = (2/T) ∫ f(x) cos(2πnx/T) dx

bₙ = (2/T) ∫ f(x) sin(2πnx/T) dx

f_N(x) = a₀/2 + Σ[aₙ cos(2πnx/T) + bₙ sin(2πnx/T)]

The amplitude for harmonic n is √(aₙ² + bₙ²). The phase is atan2(bₙ, aₙ). Because entered samples may be discrete rather than symbolic, the calculator applies the trapezoidal rule across the supplied period to estimate each integral.

How to Use This Calculator

1. Enter the signal period in the Period field.
2. Select how you want to provide data: Y-Only samples or X,Y pairs.
3. Paste the sampled signal values into the Sample Data box.
4. Choose how many harmonics you want in the approximation.
5. Press the calculate button to generate coefficients and the reconstructed signal.
6. Review the summary, coefficient table, error table, and graph.
7. Use the export buttons to save CSV or PDF output.

Frequently Asked Questions

1. What does this calculator return?

It returns a₀, each aₙ and bₙ coefficient, amplitude, phase, reconstructed values, and the root mean square reconstruction error for the selected harmonic count.

2. Can I paste only y-values?

Yes. In Y-Only mode, the calculator assumes those values are evenly spaced across one full period. It automatically builds the matching x-grid.

3. Can I use unevenly spaced sample locations?

Yes. Choose X,Y mode and paste one x,y pair per line. Every x-value must stay within 0 and the chosen period.

4. Why do my coefficients change when I add more samples?

More samples usually improve the numerical integration estimate. Sparse or irregular samples can miss waveform details and shift the estimated harmonic content.

5. How many harmonics should I choose?

Start with a small value like 5 or 10. Increase it when you want a closer reconstruction. More harmonics usually improve detail but can emphasize sharp transitions.

6. What does the phase column mean?

The phase shows the angular shift for each harmonic component. It helps describe where that sinusoidal term is positioned relative to the start of the period.

7. Why is the reconstruction error not zero?

The signal may need more harmonics, more samples, or both. Numerical integration and abrupt waveform changes also create approximation error.

8. Can I export the output?

Yes. After calculation, use the CSV button for spreadsheet-friendly output and the PDF button for a compact report you can save or share.