The jet engine growls to life. The seat beneath you vibrates. But all you hear are the dulcet tones of a Bach cantata. Thank you, noise-canceling headphones. And thank you, Fourier transform. This miracle of calm was brought to you by the same equation that can compress image files and predict the tides and motion of the planets. It all comes down to waves. Sound is created by changes in air pressure, rising and falling like swells in the ocean. The faster a sound wave oscillates—that is, the greater its frequency—the higher the sound’s pitch. The taller the wave is (its amplitude), the louder the sound. That engine roar is like an ocean in a storm: Waves pile on top of one another, traveling at varying speeds in varying directions. The Fourier transform tames the chaos, separating the sound into its simple constituent waves. (It can perform a similar trick on nonwave phenomena—analyzing images, for instance, into patterns of light and dark lines.) Noise-canceling headphones use this formula to break external noise into simple waves, then emit identical waves that are shifted out of phase so that the peaks of each new one correspond to the troughs of its counterpart. These “antiwaves” wipe out the unwanted sounds while leaving your Bach cantata intact. Ahhh! Here’s how it works.
The signal, which could be a sound, an image, or even an electrocardiogram. In the case of sound, f(t) is the air pressure at any given time t.
The integral sign, which tells you to evaluate the formula inside it for every frequency k between 0 and infinity (∞) and then add them all up.
The cosine function’s graph takes the form of a sinusoidal wave. The rest of the formula tweaks this basic shape into a variety of specific simple waves.
The amplitude of the wave at frequency k. Multiplying the cosine by this factor turns its prototypical waveform into one with the correct height.
The phase shift. Waves that are in phase reinforce one another, creating bigger peaks and valleys; those that are out of phase cancel each other out.
Illustration: Toby and Pete
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