Fourier Series Visualization

Any periodic function can be represented as a sum of sine functions of different frequencies

f(t) = a₀ + Σ(aₙcos(nt) + bₙsin(nt))
Epicycles View
Drawing Area

Fourier Coefficients

Draw or select a waveform to view coefficients

What is Fourier Series?

Fourier Series is a powerful tool in mathematics that shows any periodic function can be represented as an infinite series sum of sine and cosine functions of different frequencies.

Key Concepts:

Epicycles

Multiple rotating circles connected head-to-tail, each representing a term in the Fourier series. The circle's radius corresponds to amplitude, and rotation speed corresponds to frequency.

Discrete Fourier Transform (DFT)

Converts time-domain signals to frequency-domain representation, calculating the amplitude and phase of each frequency component. Performing DFT on hand-drawn curves yields Fourier coefficients.

Approximation & Convergence

More terms used leads to better approximation. However, some discontinuities may exhibit Gibbs phenomenon.

Applications:

How to Use:

  1. Draw any closed curve in the drawing area on the right with your mouse, or select preset waveforms (square wave, sawtooth wave)
  2. Adjust the "Number of Terms N" slider to control how many circles are used to approximate your shape
  3. Observe how the epicycle animation on the left reproduces your shape through rotating circles
  4. View the Fourier coefficients displayed at the bottom to understand each frequency component's contribution