The Fourier Transform and its kin operate by analyzing an input waveform into a series of sinusoidal waves of various frequencies and amplitudes. This is called a Fourier Series. (If you are not comfortable with sine waves, frequency, phase, and the like, Sine Wave Basics will help get you rolling.)

It turns out that any waveform can be built from only sine and cosine waves of various amplitudes, and hence any waveform can be broken down or "analyzed" into these same components. This is not due to any magical abilities of sinusoids... you could actually use practically any wave shape, no matter how silly (waves that look like the profile of a '57 Chevy or Lincoln's face, for instance). But sinusoids are useful because we can relate them to meaningful physical phenomena like vibrations: The harmonics of a voice or a violin string, the resonance of a structure during an earthquake, the buzz of insect wings, the whine of a defective gear or bearing in a machine.

The classic example of sine wave synthesis is a square wave, built from a sine wave at the fundamental frequency, plus another at 3 times that frequency (known as the "3rd harmonic") but 1/3 the amplitude, plus a 5th harmonic at 1/5 the amplitude, and so on for all odd harmonics. You can try your hand at this sort of synthesis with the Daqarta Generator... see Making Waves for details.

See also Spectrum (Fourier Transform) Theory

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• Ahead to Sine Wave Analysis For Fourier Transforms
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