▲An Interactive Guide to the Fourier Transform(betterexplained.com)
168 points by pykello 5 days ago | 11 comments
▲freshtake 5 hours ago
If you're only interested in the gist of the concept and how it can be applied to compression, without the mathematical rigor, here is my go to: https://bertolami.com/index.php?engine=blog&content=posts&de...
▲dmd 5 hours ago
The absolute best teaching of the Fourier transform I've ever encountered is the extremely bizarre book "Who is Fourier?"

https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...

▲arialdomartini 4 hours ago
Brilliant!

I would just suggest the author to replace the sentence “99% of the time, it refers to motion in one dimension” with “most of the time” since this is a mathematical article and there’s no need to use specific numbers when they don’t reflect actual data.

▲kens 10 hours ago
If you're dealing with computer graphics, audio, or data analysis, I highly recommend learning Fourier transforms, because they explain a whole lot of things that are otherwise mysterious.
▲devanshp 3 hours ago
It's quite interesting that our ears implement a better-than-Fourier-like algorithm internally: https://arxiv.org/pdf/1208.4611
▲gsf_emergency_6 2 hours ago
Article on how this might work (nonlinearity)

https://jontalle.web.engr.illinois.edu/Public/AllenSpeechPro...

Note the two electric circuit models figs 3.2 & 3.8

▲seam_carver 6 hours ago
If anyone wants to learn about the 2D DFT, the best explanation I've ever read was the relevant chapter in Digital Image Processing by Nick Efford.

If anyone wants to see my favorite application of the 2D DFT, I made a video of how the DFT is used to remove rainbows in manga on Kaleido 3 color eink on Kobo Colour:

https://youtu.be/Dw2HTJCGMhw?si=J6dUYOj2IRX1nPRF

▲brad0 5 hours ago
In the video you show a 2D mask to blur diagonal lines. How is that mask applied to the DFT? Is the mask also converted to a DFT and the two signals get combined?
▲seam_carver 2 hours ago
Just remove anything under the mask basically, similar to a low pass filter.
▲analog31 11 hours ago
My only quibble is that the article is about the discrete Fourier transform.
▲shash 8 hours ago
It’s usually easier to explain the dft. and easier to do a periodic function than a totally arbitrary sequence.
▲krackers 2 hours ago
I've actually found the opposite, it's easier to conceptually understand the continuous FT, then analyze the DTFT, DFT, and Fourier Series as special cases of applying a {periodic summation, discrete sampling} operator before the FT.
▲biophysboy 7 hours ago
My favorite application of the Fourier transform is converting convolution into pointwise multiplication. This is used to speed up multiple sequence alignment in bioinformatics.
▲zkmon 10 hours ago
It is more about the duality between the amplitude and frequency spaces and conversion between them. A bit similar to Hadamard gate for transforming a quantum state from computational basis to diagonal basis.
▲kuharich 11 hours ago
Past comments: https://news.ycombinator.com/item?id=38652794
▲constantcrying 11 hours ago
>The Fourier Transform is one of deepest insights ever made.

No, it is not. In fact it is quite a superficial example of a much deeper theory, behind functions, their approximations and their representations.

▲fedsocpuppet 10 hours ago
The Fourier transform predates functional analysis by a century. I don't see the point in downplaying its significance just because 'duh it's simply a unitary linear operator on L2'.
▲NewsaHackO 9 hours ago
But is it the deepest insights ever made?
▲badlibrarian 8 hours ago
The Fourier Transform isn't even Fourier's deepest insight. Unless we're now ranking scientific discoveries based on whether or not they get a post every weekend on HN.

The FFT is nifty but that's FINO. The Google boys also had a few O(N^2) to O(N log N) moments. Those seemed to move the needle a bit as well.

But even if we restrict to "things that made Nano Banana Pro possible" Shannon and Turing leapfrog Fourier.

▲lispisok 6 hours ago
>Unless we're now ranking scientific discoveries based on whether or not they get a post every weekend on HN.

Glad I'm not the only one who noticed there is a weekly (or more) post on what Fourier transform is.