Building a Square Wave

Corners from smooth curves

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Square wave $= \tfrac{4}{\pi}\left(\sin x + \tfrac13\sin 3x + \tfrac15\sin 5x + \dots\right)$: odd harmonics only.
• More terms give a closer fit, but a ~9% overshoot persists at jumps (Gibbs).
• Symmetries of a signal decide which harmonics appear.

Steps

1. More and more terms
2. The fifth harmonic
3. Missing even harmonics
4. Gibbs overshoot
5. The seventh harmonic
6. A surprising sum