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Why is the Fourier transform so important?

Is a sound one wave, or many? Guess first, then see a note as a recipe of pure tones, why filters keep frequencies, and why the FFT is so fast.

The steps5 · 28 min
  1. 1Almost any wave is a sum of pure sine waves6 min
  2. 2The transform measures how much of each frequency is there7 min
  3. 3Sines stay sines through a filter, which makes filtering simple6 min
  4. 4Sharp in time means blurred in pitch6 min
  5. 5The FFT gets the same answer with far fewer steps5 min
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What it takes apart

Sound like you?

The ideas people most often get wrong on the way, in their own words. Each step starts from one of them.

“A sound has a single frequency, the pitch you hear, so a note on a violin is just one wave at one speed.”Taken apart in step 1Thinking a sound has one frequency“Turning a signal into its frequencies loses detail, like blurring a photo.”Taken apart in step 2Thinking a spectrum throws information away“Fourier analysis only applies to waves that repeat forever, like a held note.”Taken apart in step 2Thinking it only works for repeating waves“An equaliser or any filter changes a sound by adding new frequencies that weren't there.”Taken apart in step 3Thinking an equaliser adds new notes“I can measure exactly when a sound happened and exactly what frequency it has, at the same time.”Taken apart in step 4Thinking you can know exactly when and exactly what pitch“The FFT is a fast approximation of the real Fourier transform, so it loses some accuracy.”Taken apart in step 5Thinking the FFT is a rougher version of the transform
Signals

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The short answerRather read it than work it out?Show it (it gives the steps away)

A real sound is a recipe of pure tones: a fundamental that sets the pitch, plus harmonics at whole-number multiples, each with its own strength. A violin and a flute playing 220 Hz share the pitch but differ in the recipe. A square wave of height 1 is the sum of odd harmonics with strengths 4/pi, 4/(3 pi), and so on.

To measure one frequency, multiply the signal by a test sine of that frequency and average the result. A matching sine builds up a big average; other frequencies cancel out. Do this for every frequency and you have the spectrum. It rebuilds the signal exactly, and works on any signal, repeating or not.

Put a sine wave into a linear system such as an equaliser, a room or a speaker, and a sine of the same frequency comes out, scaled and shifted in time. So a filter can be described by what it does to each frequency. Splitting a signal into sines, scaling each, and adding back is the whole job. New frequencies only come from distortion.

Telling two frequencies apart needs time for them to drift apart by a cycle. A clip of T seconds can separate frequencies about 1 over T hertz apart: 1 Hz for a second, 20 Hz for 0.05 s, 100 Hz for 0.01 s. So a click has no clear pitch, and a spectrogram's window length trades timing against pitch.

The direct calculation of N frequencies from N samples takes about N squared steps. The FFT splits the job into halves repeatedly, sharing sub-results, and takes about N times log2 N. The answer is identical, apart from rounding. For about a million samples it is some 50,000 times faster, which is why audio, images and radio can use it live.

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