Two Great Tastes Processing Signals Together

March 16, 2023 11:11 PM

The Algorithm That Transformed The World - "The Fast Fourier Transform is used everywhere but it has a fascinating origin story that could have ended the nuclear arms race."[1,2,3,4]

• Researchers thought this was a bug (Borwein integrals) - "A curious pattern of integrals that all equal pi... until they don't."
• But what is a convolution? - "Discrete convolutions, from probability, to image processing and FFTs."
• But what is the Central Limit Theorem? - "A visual introduction to probability's most important theorem."
• The more general uncertainty principle, regarding Fourier transforms - "The meaning of the uncertainty principle in the context of Fourier transforms."[5,6]
• Pure Fourier - "Because why not?"

also btw...
Digital Light: Sampling Spacetime - "In the last six chapters we've become intimate with the three most important mathematical underpinnings of the digital world: Fourier waves, Kotelnikov samples, and Turing computations."[7,8,9]

posted by kliuless (11 comments total)

(good content, but that first video is shilling the 80000 Hours thing from the EA cult, be warned)
posted by scruss at 3:32 AM on March 17 [4 favorites]

Memories of my first job - part of which was coding the FFT algorithm in 80387 assembly code as it needed to be blazing fast - way faster than C version in the end. We used FFT to process high frequency sound waves for non-destructive materials testing, specifically the pressure vessel inside a nuclear power plant (Vuje in Slovakia). Fun times.
posted by whatevernot at 3:52 AM on March 17 [14 favorites]

How do you measure all the frequencies at once for IR spectra? . For NMR spectra?

Or how modern spectrometers use FFTs to speed up chemical measurements by orders of magnitude and drive modern science....
posted by lalochezia at 5:46 AM on March 17

The Fourier transform on groups is, imo, the coolest piece of math that basically no one knows about. The basic idea is that all of the Fourier theory still works for functions on a group. One of the best examples is the group of permutations... You can use the group fft for permutations to concisely analyse vote tallies, compare the properties of different voting systems, and deduce the number of shuffles needed to randomize a deck of cards. And a lot more.
posted by kaibutsu at 7:34 AM on March 17 [7 favorites]

("the EA cult" = "Effective Altruism", if like me you were wondering if Electronic Arts had really stepped up its corporate culture)
posted by thecaddy at 7:55 AM on March 17 [13 favorites]

I 'knew' Gauss had discovered the FFT, but as I watched the first linked video, I thought I must have misread my sources.

Gauss also anticipated the discovery of non-Euclidean Geometry, and didn’t publish it because he thought it would be rejected out of hand and he would face ridicule.

And he's known for making discoveries and then not publishing them until he could come up with completely unintuitive proofs in order to cover his tracks.

As great as his reputation was and is, it could have been so much greater — if he had been a better, more open person and not so jealous about that reputation? Perhaps.
posted by jamjam at 9:31 AM on March 17 [1 favorite]

3Blue1Brown is one of my very favorite YouTube channels.
posted by Ivan Fyodorovich at 12:09 PM on March 17 [1 favorite]

The video is from Veritasium, not 3Blue1Brown. It just uses the same Manin software. Both channels are great!
posted by dsquared at 7:13 PM on March 17

2 Fast 2 Fourier, that's it, that's my entire contribution to this conversation
posted by taquito sunrise at 4:53 AM on March 18 [4 favorites]

“The video is from Veritasium, not 3Blue1Brown.”

Five videos below the fold of the post are from 3Blue1Brown.
posted by Ivan Fyodorovich at 9:02 AM on March 18

Delia’s Tatty Green Lampshade - Effectrode has an example of audio FFT that's quite intuitive (via)
posted by scruss at 7:52 AM on March 24 [1 favorite]