Calculator Inputs
Example Data Table
Illustrative sample using a square waveform, amplitude 1, period 2π, and 5 Fourier terms.
| x | Exact Signal | Partial Sum S5(x) | Error |
|---|---|---|---|
| 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 0.785398 | 1.000000 | 0.991780 | -0.008220 |
| 1.570796 | 1.000000 | 1.063054 | 0.063054 |
| 2.356194 | 1.000000 | 0.991780 | -0.008220 |
| 3.141593 | 0.000000 | 0.000000 | 0.000000 |
Formula Used
The calculator evaluates the truncated Fourier series:
SN(x) = a0/2 + Σn=1..N[ancos(nω0x) + bnsin(nω0x)]
The fundamental angular frequency is: ω0 = 2π / T
The harmonic amplitude is: cn = √(an2 + bn2)
For built-in waveforms, the page loads known coefficients automatically. For example, the square-wave model uses only odd sine terms, while the triangle model uses only odd cosine terms.
In custom mode, you provide a0, an, and bn directly, so the calculator can evaluate any compatible finite Fourier sum.
How to Use This Calculator
- Select a waveform preset or choose custom coefficients.
- Enter amplitude, period, number of terms, evaluation point, and graph window.
- Use custom mode only when you want direct control of a0, an, and bn.
- Press Calculate Fourier Sum to show the result section above the form.
- Review the computed value, error metrics, coefficient table, and Plotly charts.
- Export the sampled dataset as CSV or save the result panel as a PDF report.
Frequently Asked Questions
1) What does the Fourier sum represent?
It represents a finite approximation of a periodic signal using sine and cosine harmonics. As you increase the number of terms, the partial sum typically tracks the target waveform more closely across the chosen interval.
2) Why does the square wave overshoot near jumps?
That behavior is the Gibbs phenomenon. Finite Fourier sums often overshoot near discontinuities. Adding more terms narrows the oscillation region, but the peak overshoot does not completely disappear.
3) What is the meaning of a₀?
The coefficient a₀ controls the average level of the series. Because the standard form uses a₀/2, a nonzero value shifts the whole approximation upward or downward.
4) When should I use custom mode?
Use custom mode when you already know the Fourier coefficients from theory, measurement, or another tool. It is helpful for checking partial sums of signals that are not included among the built-in waveform presets.
5) Why do some presets use only sine or cosine terms?
Symmetry determines which coefficients vanish. Odd functions usually keep sine terms, even functions often keep cosine terms, and many harmonics become zero because the waveform structure cancels them in the Fourier integrals.
6) What does the RMS value tell me?
RMS measures the energy-like magnitude of the sampled approximation over the selected interval. It is useful when comparing waveform strength, power-related behavior, or the effect of adding more harmonics.
7) Does the graph range affect the computed series?
The coefficients do not depend on the graph window, but the displayed samples do. Changing x minimum, x maximum, and sample count affects the plotted view and the numerical summaries calculated over that sampled range.
8) What is included in the CSV export?
The CSV file contains the sampled x values, Fourier approximation values, exact waveform values when available, and pointwise error values. That makes it convenient for audits, reports, and further spreadsheet analysis.