Any waveform can be exactly modeled by combining sine waves

Evidence

Fourier further theorized that if he combined enough sine waves together, he could exactly model any waveform in existence. This allowed him to develop the Fourier transform pair—two equations that provide a means of transforming the time domain into the frequency domain and back. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 443

Categorization

• DEFINE: Defines the Fourier transform pair as the mathematical bridge between time domain and frequency domain.
• XREF: connects to: Any waveform can be approximated by summing orthogonal sine waves — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440; Any complex waveform can be decomposed into simple sine waves — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 439; Any complex waveform can be built from independent orthogonal sine waves (+10 more)

Broader Connections

• A square wave is built from odd-multiple sine components #exemplifies (0.95) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 449
• Any complex waveform can be decomposed into simple sine waves #corroborates (0.90) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 439
• Fourier analysis recovers constituent sinusoids from any summed waveform #expands (0.90) — Eric J. Heller, Why You Hear What You Hear- An…, 324
• Accurate square waves require infinitely many harmonics #exemplifies (0.90)
• Any complex waveform can be built from independent orthogonal sine waves #corroborates (0.85)
• Sine waves of varying frequency, amplitude, and phase can construct any waveform #corroborates (0.85) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 446
• Any complex tone is decomposable into sine wave components displayed as a spectrum #expands (0.85) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 449
• The Fourier transform measures how a signal matches each sinusoid pair #expands (0.85) — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 369
• Any waveform can be approximated by summing orthogonal sine waves #corroborates (0.80) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440
• Superposition lets complex waveforms be solved as sums of sine wave solutions #expands (0.80) — David Howard, Jamie Angus, coustics and Psychoacoustics, 439
• A waveform’s sharpness dictates how many high-frequency sine waves it contains #qualifies (0.80)
• Only a sinusoid has a single frequency; every other shape contains multiple frequencies #qualifies (0.75)
• Standing waves occur at all integer multiples of a fundamental frequency #expands (0.70) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 396