The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face


Over a decade ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn’t a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, MP3 and JPEG compression, X-ray crystallography, voice recognition, and PET or MRI scans. This mathematical tool—named the Fourier transform, after 18th-century French physicist and mathematician Joseph Fourier—was even used by James Watson and Francis Crick to decode the double helix structure of DNA from the X-ray patterns produced by Rosalind Franklin. (Crick was an expert in Fourier transforms, and joked about writing a paper called, “Fourier Transforms for birdwatchers,” to explain the math to Watson, an avid birder.)

You probably use a descendant of Fourier’s idea every day, whether you’re playing an MP3, viewing an image on the web, asking Siri a question, or tuning in to a radio station. (Fourier, by the way, was no slacker. In addition to his work in theoretical physics and math, he was also the first to discover the greenhouse effect.)

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.)[…]

Source: The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face

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About agogo22

Director of Manchester School of Samba at http://www.sambaman.org.uk
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