A Fourier series says that a complex periodic signal can be assembled from simple sine and cosine waves. Each harmonic has an amplitude and phase, which determine how strongly that frequency appears and where it begins in the cycle.
Choose the original, or target, function f(t) below. Then add reconstruction terms f1(t), f2(t), and so on. The Decompose f(t) action calculates the coefficients of the selected target so you can compare your work with its Fourier decomposition.
Fourier series playground
Build a signal from simple waves.
The solid orange reference curve is the original f(t). Each row is a term fn(t) in its reconstruction. Its coefficient controls how much of that frequency is added to the sum.
Use a zero-mean periodic expression with numbers, t, pi, sin(), cos(), +, -, *, /, ^, and parentheses. Example: cos(t) + 0.4 * sin(3 * t).
What to notice
- The coefficient is a weight. A larger absolute value gives that harmonic more influence over the sum.
- Higher harmonics add detail. Lower frequencies set the broad shape; higher frequencies make corners and fast changes possible.
- More terms are not always equal. The coefficient rule determines which harmonics matter and how quickly their effect fades.
Further reading
- Understanding Fourier series — the basic idea behind decomposition into harmonics.
- Fourier coefficients and phase — how amplitude and timing shape the reconstructed waveform.