Over at Quanta Magazine [Shalma Wegsman] asks What Is the Fourier Transform?
[Shalma] begins by telling you a little about Joseph Fourier, the French mathematician with an interest in heat propagation who founded the field of harmonic analysis in the early 1800s.
Fourier’s basic insight was that you can represent everything as a sum of very basic oscillations, where the basic oscillations are sine or cosine functions with certain parameters. [Shalma] explains that the biology of our ear can do a similar thing by picking the various notes out from a tune which is heard, but mathematicians and programmers work without the benefit of evolved resonant hairs and bone, they work with math and code.
[Shalma] explains how frequency components can be discovered by trial and error, multiplying candidate frequencies with the original function to see if there are large peaks, indicating the frequency is a component, or if the variations average to zero, indicating the frequency is not a component. [Shalma] tells how even square waves can be modeled with an infinite set of frequencies known as the Fourier series.
Taking a look at higher-dimensional problems [Shalma] mentions how Fourier transforms can be used for graphical compression by dropping the high frequency detail which our eyes can barely perceive anyway. [Shalma] gives us a fascinating look at the 64 graphical building blocks which can be combined to create any possible 8×8 image.
[Shalma] then mentions James Cooley and John Tukey and the development of the Fast Fourier Transform in the 1960s. This mathematical tool has been employed to study the tides, to detect gravitational waves, to develop radar and magnetic resonance imaging, and to support signal processing and data compression. Even quantum mechanics finds use for harmonic analysis, and [Shalma] explains how it relates to the uncertainty principle. The Fourier transform has spread through pure mathematics and into number theory, too.
[Shalma] closes with a quote from Charles Fefferman: “If people didn’t know about the Fourier transform, I don’t know what percent of math would then disappear, but it would be a big percent.”
If you’re interested in the Fourier transform and want to dive deeper we would encourage you to read The Fastest Fourier Transform In The West and Even Faster Fourier Transforms On The Raspbery Pi Zero.
Header image: Joseph Fourier, Attributed to Pierre-Claude Gautherot, Public domain.
The 3blue1brown YouTube channel has an amazing video titled “But what is the Fourier Transform” that really nailed home my intuition for it.
https://youtu.be/spUNpyF58BY?feature=shared
Excellent. Thank you!
Hackaday once featured an awesome interactive web page on Fourier that it looks this one might be based on https://www.jezzamon.com/fourier/index.html
Thanks for the link. But, oh man, I’m gonna need some sleep before I take this one on!
Just don’t get it confused with the Furrier transform, which only works during a full moon. Awooooooooo!
werewolf in london
i wouldn’t know, despite having used it many times a day for many years
How I got past Calc 2 is still a mystery. If I was my professor, I would’ve failed myself
If you feel like trying again there’s a really good chapter on DFT and FFT in Introduction to Algorithms. I have 3ed but I haven’t read it as closely as what I would have liked! https://www.amazon.com/dp/026204630X
Seriously. I went through all sorts of very high level math. Multivariable Calc, Differential Equations, Fluid Dynamics and others….
Still don’t understand it all (also not an engineer). But FFT I use often for SDR stuff. I wish I had seen practical applications of the math I was learning way back when. I would probably use a lot more of it.
The first time I tried to post this, it dumped it; so I’m trying again. Fortunately I kept a copy so I wouldn’t have to re-type it all.
Math books always put this stuff in math shorthand, making it much too sterile for me. (Apparently it’s also too sterile for a lot of other people who have a lot more formal education than I, and I’ve found myself explaining it to them.) But when I wanted to understand it, I had to finally push the books aside and just think about what had to happen, and try various ideas. What I came up with, when cleaned up, looked awfully familiar. I went back to the books, and doggone—it was the same thing! But now I understood it. I have to learn and figure things out my own way. I have not been able to figure out the Cooley-Tukey method though. I did however take someone’s algorithm for it and write a Forth program for it to run on my home-made workbench computer, and it ran correctly on first try.
In 1992 I worked for the SETI project of Mountain View CA. They created a 15 million point FFT RF spectrometer that broke down a 20MHz band RF spectrum into 15 million .7Hz wide bins looking for signals coming from intelligent civilations. We never were able to confirm a signal, but we got a lot of (WOWs). Wows are transient signals where you look at the data, and go WOW! But they have never been confirmed. So, they are added to the list of WOWs. Fourier Analysis allows a system to look down through oceans of noise to uncover very very faint signals burried in this noise. Fourier Analysis is amazing.
There’s a movie that comes to mind. Something about a natural number that never repeats and using a drill to relieve the pressure.
Or, rather, an irrational number that appears in nature.
What good is the Fourier transform going to you in 2025?
But college professor love Fourier transforms as well at the Bolzano Weierstras-theorem?
Did Bill Gates drop out of Harvard rather than listen to college professor tell him how computer should work?
The first time I tried to post this, it dumped it. Let’s see if this one works.
Not as recent as 2025, but I’ve used it for audio spectrum analysis. One thing I’ve used it for is taking the impulse response of a speaker and getting the amplitude and phase versus frequency, in the design of active noise-cancelling headsets, for work. A simple spectrum analyzer won’t usually give you phase response.
I tried my hand at a shiny app to explain using Fourier transforms to model soil temperatures. Looking at it now (oof, it takes some time to load), it’s a little much. But I did enjoy the coding and there are some interactive bits that might help with also understanding what Fourier transforms can do?
https://erksome.shinyapps.io/Fourier_and_Soil_Temps/
An approach for building intuition uses filters. Imagine there is a signal possible buried in noise and you know what the signal should look like. The optimal filter for that signal is formed by multiplying the incoming noisy source by your copy of the signal. The hard part is you don’t know the phase and have to multiply by all possible phases of your copy. When your copy and the incoming signal line up, you get only positive products of the signal while the noise is randomly positive and negative. You sum the results for each shift in phase and look for a peak.
The analytical Fourier Transform that uses the calculus does this over a continuous space. In the digital domain this is very very math intensive. The FFT makes it practical. It also is used to do convolution and deconvolution practical, which is needed for the optimal filter ideas. It essentially can do all the frequencies or all the phases at once.
I think of the FFT as doing the optimal filter operation with the reference signals being all possible frequencies and phases of the sine function.
The from used in SDR is usually ignored in the FFT descriptions. In order to get phase information SDR uses the complex form, as in complex numbers. SDR uses I/Q sampling where the input is multiplied by a sine signal and the same signal shifted by 90 degrees. The unshifted output goes to the “real’ part of the complex number and the shifted part to the “imaginary” part. This preserves the phase. Then the whole FFT/Convolution stuff is done in the complex domain.
FYI, the other thing that makes SDR front-end circuitry so simple is also based on Fourier Analysis. If look for the Sin/Cos references for I/Q sampling you won’t find it in modern systems since the mid 1990’s. In stead you see a switch that chops the inputs with square waves. How does this work? The Fourier Transform of a square wave is sine waves at all the odd harmonics of the square wave frequency. This means that if you filter the chopped signal to block anything higher that the chopping frequency you get the input multiplied by a sine at the chopping frequency! You just don’t need the analog circuitry that used non-linear response of some parts to get a product of two signals. IIRC this comes from some work at Stanford in the 1990’s. First the chopping trick, then the I/Q and SDR a few years later and soon it was fully digital. I did not get the real implications until the Hack-RF One https://greatscottgadgets.com/hackrf/one/
please see “series and products in the history of mathematics” (2021), published by cambridge university press, written by RANJAN ROY.
Another article that mentions FFT and omits to mention the DFT.
Have to tried running a DFT on 4k samples?