What Is the Fourier Transform?

As we listen to a piece of music, our ears perform a calculation. The high-pitched flutter of the flute, the middle tones of the violin, and the low hum of the double bass fill the air with pressure waves of many different frequencies. When the combined sound wave descends through the ear canal and into the spiral-shaped cochlea, hairs of different lengths resonate to the different pitches, separating the messy signal into buckets of elemental sounds.

It took mathematicians until the 19th century to master this same calculation.

In the early 1800s, the French mathematician Jean-Baptiste Joseph Fourier discovered a way to take any function and decompose it into a set of fundamental waves, or frequencies. Add these constituent frequencies back together, and you’ll get your original function. The technique, today called the Fourier transform, allowed the mathematician — previously an ardent proponent of the French revolution — to spur a mathematical revolution as well.

Out of the Fourier transform grew an entire field of mathematics, called harmonic analysis, which studies the components of functions. Soon enough, mathematicians began to discover deep connections between harmonic analysis and other areas of math and physics, from number theory to differential equations to quantum mechanics. You can also find the Fourier transform at work in your computer, allowing you to compress files, enhance audio signals and more.

“It’s hard to overestimate the influence of Fourier analysis in math,” said Leslie Greengard of New York University and the Flatiron Institute. “It touches almost every field of math and physics and chemistry and everything else.”

Flames of Passion

Fourier was born in 1768 amid the chaos of prerevolutionary France. Orphaned at 10 years old, he was educated at a convent in his hometown of Auxerre. He spent the next decade conflicted about whether to dedicate his life to religion or to math, eventually abandoning his religious training and becoming a teacher. He also promoted revolutionary efforts in France until, during the Reign of Terror in 1794, the 26-year-old was arrested and imprisoned for expressing beliefs that were considered anti-revolutionary. He was slated for the guillotine.

Before he could be executed, the Terror came to an end. And so, in 1795, he returned to teaching mathematics. A few years later, he was appointed as a scientific adviser to Napoleon Bonaparte and joined his army during the invasion of Egypt. It was there that Fourier, while also pursuing research into Egyptian antiquities, began the work that would lead him to develop his transform: He wanted to understand the mathematics of heat conduction. By the time he returned to France in 1801 — shortly before the French army was driven out of Egypt, the stolen Rosetta stone surrendered to the British — Fourier could think of nothing else.

If you heat one side of a metal rod, the heat will spread until the whole rod has the same temperature. Fourier argued that the distribution of heat through the rod could be written as a sum of simple waves. As the metal cools, these waves lose energy, causing them to smooth out and eventually disappear. The waves that oscillate more quickly — meaning they have more energy — decay first, followed eventually by the lower frequencies. It’s like a symphony that ends with each instrument fading to silence, from piccolos to tubas.

The proposal was radical. When Fourier presented it at a meeting of the Paris Institute in 1807, the renowned mathematician Joseph-Louis Lagrange reportedly declared the work “nothing short of impossible.”

What troubled his peers most were strange cases where the heat distribution might be sharply irregular — like a rod that is exactly half cold and half hot. Fourier maintained that the sudden jump in temperature could still be described mathematically: It would just require adding infinitely many simpler curves instead of a finite number. But most mathematicians at the time believed that no number of smooth curves could ever add up to a sharp corner.

Today, we know that Fourier was broadly right.

“You can represent anything as a sum of these very, very simple oscillations,” said Charles Fefferman, a mathematician at Princeton University. “It’s known that if you have a whole lot of tuning forks, and you set them perfectly, they can produce Beethoven’s Ninth Symphony.” The process only fails for the most bizarre functions, like those that oscillate wildly no matter how much you zoom in on them.

So how does the Fourier transform work?

A Well-Trained Ear

Performing a Fourier transform is akin to sniffing a perfume and distinguishing its list of ingredients, or hearing a complex jazzy chord and distinguishing its constituent notes.

Mathematically, the Fourier transform is a function. It takes a given function — which can look complicated — as its input. It then produces as its output a set of frequencies. If you write down the simple sine and cosine waves that have these frequencies, and then add them together, you’ll get the original function.

To achieve this, the Fourier transform essentially scans all possible frequencies and determines how much each contributes to the original function. Let’s look at a simple example.

Consider the following function:

The Fourier transform checks how much each frequency contributes to this original function. It does so by multiplying waves together. Here’s what happens if we multiply the original by a sine wave with a frequency of 3:

There are lots of large peaks, which means the frequency 3 contributes to the original function. The average height of the peaks reveals how large the contribution is.

Now let’s test if the frequency 5 is present. Here’s what you get when you multiply the original function by a sine wave with the frequency 5:

There are some large peaks but also large valleys. The new graph averages out to around zero. This indicates that the frequency 5 does not contribute to the original function.

The Fourier transform does this for all possible frequencies, multiplying the original function by both sine and cosine waves. (In practice, it runs this comparison on the complex plane, using a combination of real and imaginary numbers.)

In this way, the Fourier transform can decompose a complicated-looking function into just a few numbers. This has made it a crucial tool for mathematicians: If they are stumped by a problem, they can try transforming it. Often, the problem becomes much simpler when translated into the language of frequencies.

If the original function has a sharp edge, like the square wave below (which is often found in digital signals), the Fourier transform will produce an infinite set of frequencies that, when added together, approximate the edge as closely as possible. This infinite set is called the Fourier series, and — despite mathematicians’ early hesitation to accept such a thing — it is now an essential tool in the analysis of functions.

Encore

The Fourier transform also works on higher-dimensional objects such as images. You can think of a grayscale image as a two-dimensional function that tells you how bright each pixel is. The Fourier transform decomposes this function into a set of 2D frequencies. The sine and cosine waves defined by these frequencies form striped patterns oriented in different directions. These patterns — and simple combinations of them that resemble checkerboards — can be added together to re-create any image.

Any 8-by-8 image, for example, can be built from some combination of the 64 building blocks below. A compression algorithm can then remove high-frequency information, which corresponds to small details, without drastically changing how the image looks to the human eye. This is how JPEGs compress complex images into much smaller amounts of data.

In the 1960s, the mathematicians James Cooley and John Tukey came up with an algorithm that could perform a Fourier transform much more quickly — aptly called the fast Fourier transform. Since then, the Fourier transform has been implemented practically every time there is a signal to process. “It’s now a part of everyday life,” Greengard said.

It has been used to study the tides, to detect gravitational waves, and to develop radar and magnetic resonance imaging. It allows us to reduce noise in busy audio files, and to compress and store all sorts of data. In quantum mechanics — the physics of the very small — it even provides the mathematical foundation for the uncertainty principle, which says that it’s impossible to know the precise position and momentum of a particle at the same time. You can write down a function that describes a particle’s possible positions; the Fourier transform of that function will describe the particle’s possible momenta. When your function can tell you where a particle will be located with high probability — represented by a sharp peak in the graph of the function — the Fourier transform will be very spread out. It will be impossible to determine what the particle’s momentum should be. The opposite is also true.

The Fourier transform has spread its roots throughout pure mathematics research, too. Harmonic analysis — which studies the Fourier transform, as well as how to reverse it to rebuild the original function — is a powerful framework for studying waves. Mathematicians have also found that harmonic analysis has deep and unexpected connections to number theory. They’ve used these connections to explore relationships among the integers, including the distribution of prime numbers, one of the greatest mysteries in mathematics.

“If people didn’t know about the Fourier transform, I don’t know what percent of math would then disappear,” Fefferman said. “But it would be a big percent.”

Editor’s note: The Flatiron Institute is funded by the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage. More information about the relationship between Quanta Magazine and the Simons Foundation is available here.