Finding the Recipe

Orthogonality and integrals

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

• $\sin mx$ and $\sin nx$ are orthogonal: their product averages 0 unless $m = n$.
• $b_n = \dfrac{1}{\pi}\int_0^{2\pi} f(x)\sin nx\,dx$ picks out one harmonic.
• This works like a filter tuned to one frequency.

Steps

1. Multiply and average
2. Filtering out one harmonic
3. Compute b₁ for the square wave
4. A coefficient detector
5. Average of a square
6. The third harmonic