Fourier analysis recovers constituent sinusoids from any summed waveform

Evidence

By construction, the complex vibration represented in equation 3.10 comprises two known sinusoids. Suppose, however, that only the trace y(t) of the net vibration had been presented, as a kind of puzzle, and you were asked to find the frequencies and amplitudes that together make up the signal y(f)—that is, to find the righthand side of equation 3.10 given only the lefthand side. This kind of puzzle solving can be done systematically and is called Fourier analysis, or Fourier decomposition; here the solution is simply equation 3.10. We could make up more difficult puzzles by adding together more sinusoids, but the solution would of course exist. All such explicit sums of sinusoids have solutions, however unrecognizable they might be to someone who had not seen their construction. The question is, can any function be represented this way? — Eric J. Heller, Why You Hear What You Hear- An…, 324

Categorization

• DEFINE: Introduces Fourier analysis/decomposition as puzzle-solving: recover the parts from the whole, with the completeness question posed as an open hook.
• XREF: connects to: Any waveform can be exactly modeled by combining sine waves — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 443