| Scenario | Period | a₀ | a₁ | b₁ | a₂ | b₂ | Terms | Purpose |
|---|---|---|---|---|---|---|---|---|
| Square Wave Approximation | 6.283185 | 0 | 0 | 1.273240 | 0 | 0 | 9 | Odd sine-only convergence demo |
| Triangle Wave Approximation | 6.283185 | 0 | 0.810569 | 0 | 0 | 0 | 7 | Even cosine-only smooth waveform |
| Custom Mixed Signal | 10 | 1.2 | 0.8 | 0.5 | -0.2 | 0.3 | 6 | General signal reconstruction |
Formula Used
The truncated Fourier series for a periodic signal with period T is evaluated as:
fN(x) = a₀/2 + Σ [aₙ cos(nω₀x) + bₙ sin(nω₀x)], for n = 1 to N
where ω₀ = 2π / T is the fundamental angular frequency. The harmonic amplitude is computed by Aₙ = √(aₙ² + bₙ²). A phase representation may be obtained from the coefficient pair.
Approximate signal energy of the truncated series is estimated using Parseval-style accumulation: E ≈ a₀²/4 + 1/2 Σ(aₙ² + bₙ²).
In built-in waveform mode, the page also compares the reconstructed series against a reference periodic waveform and reports absolute and RMS error metrics.
How to Use This Calculator
- Choose Manual coefficients or Built-in waveform.
- Enter the period, coefficient count, sample range, and precision.
- In manual mode, fill harmonic coefficients aₙ and bₙ.
- In built-in mode, select a waveform and optional scale factor.
- Press Evaluate Fourier Series to generate the reconstruction.
- Review the summary cards, coefficient table, sampled output table, and graph.
- Use Download CSV for spreadsheet analysis.
- Use Download PDF for a shareable report.
1) What does this Fourier series evaluator calculate?
It reconstructs a periodic signal from selected Fourier coefficients, computes partial sums, harmonic amplitudes, phase values, sampled outputs, and summary statistics for the chosen interval.
2) What is the difference between manual and built-in mode?
Manual mode uses your entered coefficients directly. Built-in mode automatically fills coefficients for standard waveforms like square, sawtooth, triangle, and half-wave rectified sine.
3) Why is a₀ divided by 2?
In the common Fourier series form, the constant term is written as a₀/2. That value represents the average or DC level of the periodic signal.
4) What does the harmonic amplitude mean?
It measures the strength of each harmonic component. Larger amplitudes indicate stronger frequency contributions within the reconstructed periodic signal.
5) Why do sharp waveforms show oscillations near jumps?
That behavior is linked to Gibbs-type overshoot. Finite partial sums approximate discontinuities imperfectly, so ringing appears near abrupt waveform transitions.
6) What are the error metrics in built-in mode?
The calculator compares the reconstructed series with a reference waveform and reports mean absolute error, maximum absolute error, and root mean square error.
7) How many harmonics should I use?
Use more harmonics for sharper or more detailed waveforms. Smooth signals often converge quickly, while discontinuous signals usually need more terms for closer visual agreement.
8) Can I export the calculation results?
Yes. The page includes CSV export for tabular analysis and PDF export for printable reporting after you evaluate the series.