What Is the Fourier Transform?
87 comments
·September 4, 2025abetusk
mitthrowaway2
As mentioned by other commenters, a reason for the FT's dominance in particular is because sine, cosine, and complex exponentials are the eigenfunctions of the derivative operator. Since so many real-world systems are governed by differential equations, the Fourier Transform becomes a natural lens to analyze these systems. Sound waves are one (of many) examples.
And there's another good reason why so many real-world signals are sparse (as you say) in the FT domain in particular: because so many real-world systems involve periodic motion (rotating motors, fly's wings as you noted, etc). When the system is periodic, the FT will compress the signals very effectively because every signal has to be harmonic of the fundamental frequency.
abdullahkhalids
The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion".
Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.
yatopifo
As you noted, it’s about what’s important to us. The physical function may or may not be sparse, but our brain model is guaranteed to be sparse. A note played on a violin is anything but a sine function, yet our brains associate it with a single idealized tone. Our world model is super compressed.
anyfoo
If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform).
This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.
armanj
Years ago, I often struggled to choose between Amazon products with high ratings from a few reviews and those with slightly lower ratings but a large volume of reviews. I used the Laplace Rule of Succession to code a browser extension to calculate Laplacian scores for products, helping to make better decisions by balancing high ratings with low review counts. https://greasyfork.org/en/scripts/443773-amazon-ranking-lapl...
segfault99
When I did EE, didn't have access to any kind of computer algebra system. Have 'fond' memories of taking Laplace transform transfer functions and converting to z-transform form. Expand and then re-group and factor. Used a lot of pencil, eraser and line printer fanfold paper for doing the very basic but very tedious algebra. Youngsters today don't know how lucky.. (ties onion to belt, etc., etc.)
artyom
Essentially that's what electrical/electronics engineering is about.
ptzz
For lossy compression, turns out a sinusoidal (typically DCT) composition maximizes energy compaction and compress ability. A proof that this is true for AR-processes was a key realization for me. That you get a nice and intuitive domain to work with (modify frequencies) is a nice bonus on top :)
yshklarov
As everyone in this thread is sharing links, I'm gonna pitch in, too.
This lecture by Dennis Freeman from MIT 6.003 "Signals and Systems" gives an intuitive explanation of the connections between the four popular Fourier transforms (the Fourier transform, the discrete Fourier transform, the Fourier series, and the discrete-time Fourier transform):
https://ocw.mit.edu/courses/6-003-signals-and-systems-fall-2...
mallowdram
Excellent! Thanks!
_kb
For those interested in an extension from this article, https://howthefouriertransformworks.com/ has a beautifully kitsch set of videos on the topic.
They do a really good job at breaking down the fundamental knowledge needed to build an understanding.
chamomeal
So weird, I was just reading this article yesterday. I did an undergrad in physics and really miss this stuff. Ended up getting nostalgic and watching 3 blue 1 brown videos while drinking tequila.
laszlokorte
Shameless plug: If you are interested in Fourier Transform and signal processing you might enjoy my somewhat artistic 3D visualisation of the fourier transform as well as the fractional fourier transform [1]
(Fractional fourier transform on the top face of the cube)
And for short time fourier transform showing how a filter kernel is shiftes across the signal. [2]
nblgbg
Thanks a lot for all of this ! https://tools.laszlokorte.de/
laszlokorte
I am glad you are enjoying it! :)
hovden
If I might also plug ‘the Atlas of Fourier Transforms’. If your interested in understanding building intuition of symmetry and phase in fourier space, the book illustrates many structures.
laszlokorte
Looks amazing! Thank you
yshklarov
I love the visualization! Thanks for sharing.
How do you compute the fractional FT? My guess is by interpolating the DFT matrix (via matrix logarithm & exponential) -- is that right, or do you use some other method?
laszlokorte
I am glad you like it!
Yes the simplest way to think of it is to exponentiate the dft matrix to an exponent between 0 and 1 (1 being the classic dft). But then the runtime complexity is O(n^2) (vector multiplied with precomputed matrix) or O(n^3) opposed to the O(n log n) of fast fourier transform. There are tricks to do a fast fractional fourier transform by multiplying and convolving with a chirp signal. My implementation is in rust [1] compiled to web assembly, but it is based on the matlab of [2] who gladly answered all my mails asking many questions despite already being retired.
[1]: https://github.com/laszlokorte/svelte-rust-fft/tree/master/s...
xphos
I made a cool rust fft tui a long time ago too
mallowdram
Fantastic! Thanks!
tzury
This is a great playlist by 3b1b about the subject:
https://www.youtube.com/watch?v=spUNpyF58BY&list=PL4VT47y1w7...
fracus
Learning about the FT in engineering and how it can represent pretty much any repeating signal was mind blowing. It was the culmination of so much learning in mathematics that brought me to that wow moment.
Terr_
IANAMathematician, but I've tried to come up with a metaphor which I hope isn't too wrong:
1. You've start with a signal fluctuating going up and down, and it's on a strip of little LEDs labeled from -1 to +1.
2. You mount that strip to a motor, and spin it at a certain rate. After a while the afterimages make a blob-shape.
3. For each rotation rate, measure how much the shape appears off-center.
In this way you can figure out how much the underlying signal does (or doesn't) harmonize with a given rotation hertz.
astrange
> 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.
I slightly object to this. Removing small details = blurring the image, which is actually quite noticeable.
For some reason everyone really wants to assume this is true, so for the longest time people would invent new codecs that were prone to this (in particular wavelet-based ones like JPEG-2000 and Dirac) and then nobody would use them because they were blurry. I think this is because it's easy to give up on actually looking at the results of your work and instead use a statistic like PSNR, which turns out to be easy to cheat.
antimora
I think the best explanation and understanding I received about Fourier Transform was from studying linear algebra.
The Fourier Transform equation essentially maps a signal from the time domain onto orthogonal complex sinusoidal basis functions through projection.
And the article does not even mention this. =)
chamomeal
Best explanation I got was from 3 blue 1 brown! It’s what you said, but with nice visuals
esalman
Almost every paragraph of this article says the same thing but in different ways.
I have a pet theory that the reason why the FT, and other transforms (generating functions, Mellin/Laplace/Legendre/Haar), are so useful is because many real world functions are sparse and lend themselves to compressed sensing.
The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the problem. Why? Pet theory: because many functions that look complex are actually composed of simpler building in the transformed space.
Take the sound wave of a fly and it looks horribly complex. Pump it through the FT and you find a main driver of the wings beating at a single frequency. Take the sum of two sine waves and it looks a mess. Take the FT and you see the signal neatly broken into two peaks. Etc.
The use of the FT (or DCT or whatever) for JPEG, MP3 or the like, is basically exploiting this fact by noticing the signal response for human hearing and seeing it's not uniform, and so can be "compressed" by throwing away frequencies we don't care about.
The "magic" of the FT, and other transforms, isn't so much that it transforms the signal into a set of orthogonal basis but that many signals we care about are actually formed from a small set of these signals, allowing the FT and cousins to notice and separate them out more easily.