Fourier Series: Square Wave

Lesson notes

Overview

Add odd sine harmonics and a square wave appears — with a Gibbs overshoot at each edge. This topic is rated advanced and is number 29 of 93 in Engineering. Deeper models, formal results and harder reasoning.

What you are seeing

The panel above shows a signal built up from simple waves. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

A square wave can be built from sine waves: the odd harmonics with amplitudes 1, 1/3, 1/5, and so on, Σ sin((2k − 1)x)/(2k − 1). Adding more terms makes the corners sharper, but there is always an overshoot of about 9% near each jump, the Gibbs phenomenon, which does not vanish, only becomes narrower. It shows how a discontinuous signal needs infinitely many frequencies and why sharp edges in signals require wide bandwidth.

Reading the relationship

The expression "Σ sin((2k−1)x)/(2k−1)" is a compact summary of the idea. It uses the symbols Σ, k, x; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller, and a sum, so many small contributions are added together. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.

Try this

Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.

Common pitfalls

Always ask what the safety margin is and which assumption would break the design first. Idealised models hide losses, noise and limits.

How it connects

It builds on “Feedback Control Loops”, the topic before it in Engineering. Once you are comfortable, continue with “Fourier Series: Sawtooth Wave”.

Why it matters

Engineers use this idea to design structures, machines and systems that are safe, efficient and reliable.