The Fourier Transform

Excerpt from Science & Mathematics nautil.us

From article by Aatish Bhatia is a recent physics Ph.D. working at Princeton University to bring science and engineering to a wider audience. He writes the award-winning science blog Empirical Zeal and is on Twitter as @aatishb.

 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.

If this sounds a little abstract, here are a few different ways of 

visualizing Fourier’s trick. The first one comes to us from

 Lucas V. Barbosa, a Brazilian physics student who volunteers 

his time to make incredible math and science animations for 

Wikipedia, where he goes by “LucasVB.”

the Fourier transform is a recipe—it tells you exactly how 

much of each note you need to mix together 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.

This is also how the smartphone app Shazam can recognize a 

song. It splits the music into chunks, then uses Fourier’s trick to 

figure out the ingredient notes that make up each chunk. It then 

searches a database to see if this “fingerprint” of notes matches 

that of a song they have on file. Speech recognition uses the same 

Fourier-fingerprinting idea to compare the notes in your speech 

to that of a known list of words.

You can even use Fourier’s trick for images. Here’s a great 

video that shows how you can use circles to draw Homer Simpson’s

 face. The online encyclopedia Wolfram Alpha uses a similar idea 

to draw famous people’s faces. This might seem like a trick you’d 

reserve for a very nerdy cocktail party, but it’s also used to 

compress images into JPEG files. In the old days of Microsoft 

Paint, images were saved in bitmap (BMP) files which were a long 

list of numbers encoding the color of every single pixel. JPEG is 

the MP3 of images. To build a JPEG, you first chunk your image 

into tiny squares of 8 by 8 pixels. For each chunk, you use the same

 circle idea that reconstructs Homer Simpson’s face to 

reconstruct this portion of the image. Just as MP3s throw out the 

really high notes, JPEGs throw out the really tiny circles. The 

result is a huge reduction in file size with only a small reduction in 

quality, an insight that led to the visual online world that we all 

love (and that eventually gave us cat GIFs).

How is Fourier’s trick used in science? I put out a call on 

Twitter for scientists to describe how they used Fourier’s idea

 in their work. The response astounded me. The scientists who 

responded were using the Fourier transform to study the 

vibrations of submersible structures interacting with fluids, to 

try to predict upcoming earthquakes, to identify the ingredients 

of very distant galaxies, to search for new physics in the heat 

remnants of the Big Bang, to uncover the structure of proteins from 

X-ray diffraction patterns, to analyze digital signals for NASA, 

to study the acoustics of musical instruments, to refine models 

of the water cycle, to search for pulsars (spinning neutron stars),

 and to understand the structure of molecules using nuclear 

magnetic resonance. The Fourier transform has even been used to

 identify a counterfeit Jackson Pollock painting by 

deciphering the chemicals in the paint.

Whew! That’s quite the legacy for one little math trick.