Analysis and calculus · Maths EE idea · Solid
Building a square wave from sines
A research question to start from
How well do partial Fourier sums approximate a square wave, and why does the overshoot near the jump not disappear?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A precise phenomenon (Gibbs) that you can compute, explain and measure.
Mathematics you would need
- Fourier coefficients by integration
- Orthogonality of sines and cosines
- Partial sums
- Limits
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Derive the Fourier series of a square wave.
- Plot and measure partial sums near the jump.
- Show the overshoot tends to a fixed proportion.
- Discuss convergence away from the jump.
Scope and difficulty
Solid. Solid.
Pitfalls
- Graphs replacing analysis.
- Stating the Gibbs constant without computing it.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Gibbs phenomenon square wave overshoot; Fourier series orthogonality. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).