Fourier Series Calculator 2026
Decompose any periodic waveform into its harmonic components. Compute Fourier coefficients, visualise partial sum approximations, and explore the frequency spectrum of sine, square, sawtooth, and triangular waves.
Waveform Parameters
Select your waveform type and define period, amplitude, and harmonics
Choose the periodic function to decompose into harmonics
Peak amplitude of the waveform
Full cycle duration of the periodic signal
More harmonics = closer approximation (1–50 terms)
Constant term added to the series (average value)
Series Output
Coefficients, waveform visualisation & formula
Choose a waveform, set your parameters, then click Compute Fourier Series to see the harmonic breakdown, coefficients table, and live waveform plot.
Standard Waveforms & Their Series
Classic periodic functions and their closed-form Fourier Series representations. All assume amplitude A and period T = 2π (ω = 1).
| Waveform | Non-zero Terms | Convergence Rate | Key Property |
|---|---|---|---|
| Square Wave | Odd harmonics only (1, 3, 5 …) | Slow — O(1/n) | Gibbs overshoot ≈ 9% at edges |
| Sawtooth Wave | All harmonics (1, 2, 3 …) | Slow — O(1/n) | All integer harmonics present |
| Triangular Wave | Odd harmonics only (1, 3, 5 …) | Fast — O(1/n²) | No discontinuities; smooth |
| Half-Wave Rectified | DC + even harmonics | Medium — O(1/n²) | Non-zero DC component |
| Full-Wave Rectified | DC + even harmonics | Fast — O(1/n²) | Higher DC value; faster decay |
Fourier Series FAQ
Understand harmonic analysis, coefficient computation, and practical applications of the Fourier Series in signal processing and mathematics.
A Fourier Series is a way to represent a periodic function as an infinite sum of sine and cosine terms. Each term corresponds to a harmonic frequency, allowing complex waveforms to be broken down into simple sinusoidal components.
Fourier coefficients (a₀, aₙ, bₙ) are the amplitudes of the constant, cosine, and sine terms in the series. They are calculated by integrating the product of the function with the corresponding sinusoidal basis function over one period.
For smooth waveforms, 5–10 harmonics typically give an excellent approximation. For sharp-edged waves like square or sawtooth waves, you may need 20–50 or more harmonics to adequately capture the steep transitions, though the Gibbs phenomenon will always cause a small overshoot near discontinuities.
The Gibbs phenomenon is the characteristic overshoot that appears near a jump discontinuity in a Fourier series approximation. Even as the number of harmonics increases to infinity, the overshoot remains approximately 9% of the jump height and never fully disappears.
A Fourier Series decomposes a periodic function into a discrete set of harmonics. A Fourier Transform extends this to non-periodic functions, producing a continuous frequency-domain spectrum. For periodic signals, the Fourier Series is the appropriate tool.