Fourier Transformations

So what was Fourier’s discovery, and why is it useful? Imagine playing a note on a piano. When you press the piano key, a hammer strikes a string that vibrates to and fro at a certain fixed rate (440 times a second for the A note). As the string vibrates, the air molecules around it bounce to and fro, creating a wave of jiggling air molecules that we call sound. If you could watch the air carry out this periodic dance, you’d discover a smooth, undulating, endlessly repeating curve that’s called a sinusoid, or a sine wave. (Clarification: In the example of the piano key, there will really be more than one sine wave produced. The richness of a real piano note comes from the many softer overtones that are produced in addition to the primary sine wave. A piano note can be approximated as a sine wave, but a tuning fork is a more apt example of a sound that is well-approximated by a single sinusoid.)

Now, instead of single key, say you play three keys together to make a chord. The resulting sound wave isn’t as pretty—it looks like a complicated mess. But hidden in that messy sound wave is a simple pattern. After all, the chord was just three keys struck together, and so the messy sound wave that results is really just the sum of three notes (or sine waves).

Fourier’s insight was that this isn’t just a special property of musical chords, but applies more generally to any kind of repeating wave, be it square, round, squiggly, triangular, whatever. The Fourier transform is like a mathematical prism—you feed in a wave and it spits out the ingredients of that wave—the notes (or sine waves) that when added together will reconstruct the wave.

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To summarize, the Fourier transform tells you how much of each ingredient “note” (sine wave or circle) contributes to the overall wave. Here’s why Fourier’s trick is useful. Imagine you were talking to your friend over the phone and you wanted to get them to draw this squarish wave. The tedious way to do this would be to read out a long list of numbers that represent the height of the wave at every instant in time. With all these numbers, your friend could patiently stitch together the original wave. This is essentially how old audio formats like WAV files worked. But if your friend knew Fourier’s trick, you could do something pretty slick: You could just tell them a handful of numbers—the sizes of the different circles in the picture above. They can then use this circle picture to reconstruct the original wave.

And this isn’t just some obscure mathematical trick. The Fourier transform shows up nearly everywhere that waves do. The ubiquitous MP3 format uses a variant of Fourier’s trick to achieve its tremendous compression over the WAV (pronounced “wave”) files that preceded it. An MP3 splits a song into short segments. For each audio segment, Fourier’s trick reduces the audio wave down to its ingredient notes, which are then stored in place of the original wave. The Fourier transform also tells you how much of each note contributes to the song, so you know which ones are essential. The really high notes aren’t so important (our ears can barely hear them), so MP3s throw them out, resulting in added data compression. Audiophiles don’t like MP3s for this reason—it’s not a lossless audio format, and they claim they can hear the difference.

It's like prism that breaks apart the components of a sound wave or image into it's smaller parts.