Fourier Series Approximation of a Square Wave

Overview

A brief visual demonstration of how a square wave is progressively approximated by adding more odd‑harmonic sine terms from its Fourier series, illustrating convergence toward the ideal waveform.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3 s	Title fades in at the top. The ideal square wave (black, dashed) appears faintly as a reference.
2	First Term (n=1)	~4 s	The first sine term $\frac{4}{\pi}\sin(x)$ is drawn in blue, forming the initial approximation. The partial‑sum equation $S_1(x)=\frac{4}{\pi}\sin x$ appears at the bottom.
3	Add Third Harmonic (n=3)	~4 s	The next odd term $\frac{4}{3\pi}\sin(3x)$ fades in (green) and adds to the existing wave, updating the visual to the new sum. Equation updates to $S_2(x)=\frac{4}{\pi}\bigl(\sin x+\frac{1}{3}\sin 3x\bigr)$.
4	Add Fifth Harmonic (n=5)	~4 s	The fifth term $\frac{4}{5\pi}\sin(5x)$ appears (red) and the combined waveform updates. Equation becomes $S_3(x)=\frac{4}{\pi}\bigl(\sin x+\frac{1}{3}\sin 3x+\frac{1}{5}\sin 5x\bigr)$.
5	Convergence Overview (up to N terms)	~5 s	A rapid loop adds successive odd terms up to a chosen $N$ (e.g., 9 terms). Each new term briefly flashes a different color, the waveform smooths, and the equation updates accordingly.
6	Outro	~2 s	The final approximation (with $N$ terms) stays on screen while the reference square wave becomes solid. Title fades out, leaving a caption "Fourier series converges to the square wave as more odd harmonics are added."

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA                    │
├─────────────────────────────────────────────┤
│                 MAIN VISUAL                 │
│               (centered plot)               │
├─────────────────────────────────────────────┤
│                BOTTOM AREA                  │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Fourier Series Approximation of a Square Wave"	Fades in during Intro, fades out in Outro
Main Visual	Plot showing the reference square wave (dashed) and the current Fourier partial sum (solid, colored per term)	Central focus; new sine term added with a smooth fade‑in and additive overlay
Bottom	Current partial‑sum equation $S_k(x)$ displayed in LaTeX	Updates each phase; small but readable font

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Notes

• Only odd harmonics are used: term index $n = 1,3,5,\dots$.
• Each new sine component should be introduced with a brief highlight (e.g., a brighter outline) before blending into the cumulative waveform.
• The reference square wave remains constant throughout, serving as a visual benchmark.
• Keep total runtime under 30 seconds; the rapid convergence loop (Phase 5) should be fast enough to stay within this limit.
• No textual narration is required; the visual progression and equations convey the concept.