The Fourier Transform — Narrated Remake, improving on my earlier Fourier video

An improved, narrated remake of my earlier Fourier transform video — this time with a synced voice track and a light music bed over the same cinematic animation. Given only a messy wiggle, how do you recover the pure tones hiding inside it? This is a narrated, animated explainer built with Manim — the voice is Canopy Lab's open-source Orpheus TTS, running locally; the score is a light original ambient bed. Covered: One messy wiggle: two pure tones (2 Hz + 3 Hz) add into a jagged wave that hides them. Waves add up: a signal is a sum of sines — forward (add) is easy, the inverse is hard. Wind it around a circle: plot g(t)·e^(−2πi f t) and watch the wound "flower". Watch the center of mass: sweep the winding speed; spikes mark the hidden frequencies. Where the picture lies: the center of mass is complex (phase), spikes come in ± pairs, and a finite recording smears them (spectral leakage). The integral: grow the formula to ĝ(f) = ∫ g(t) e^(−2πi f t) dt. Reading the spectrum: the chord resolves to clean spikes at 2 and 3 Hz; an uncertainty teaser. Recap + the WIND mnemonic, and one last open question. Built with Manim, narrated, with a light original-style score. Music: original public-domain (CC0) ambient bed synthesized for this video.