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The unreasonable effectiveness of the Fourier transform

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Re: The unreasonable effectiveness of the Fourier transform

#91

Post by 3abiton »

aquafox wrote: post #27
derektank wrote: post #23

My favorite story about the Fourier Transform is that Carl Friedrich Gauss stumbled upon the algorithm for the Fast Fourier Algorthim over a century before Cooley and Tukey’s publication in 1965 (which itself revolutionized digital signal processing).[1] He was apparently studying the motion of the asteroids Pallas and Juno and wrote the algorithm down in his notes but it never made it into public knowledge. [1] http…

There is a saying about Gauss: when another mathematician came to show him a new result, Gauss would remark that he had already worked on it, open a drawer in his desk, and pull out a pile of papers on the same topic.

> There is a saying about Gauss: when another mathematician came to show him a new result, Gauss would remark that he had already worked on it, open a drawer in his desk, and pull out a pile of papers on the same topic.

As if phd students need more imposter syndrom to deal with. Ona serious side, I wonder what conditions allow such minds to grow. I guess a big part is genetics, but I am curious if the "epi" is relevant and how much.

Re: The unreasonable effectiveness of the Fourier transform

#92

Post by bschwindHN »

Jun8 wrote: post #88
bschwindHN wrote: post #79

Earlier quoted context omitted.

> I was minimizing time so spent while she was minimizing number of washes I'm probably just slow, but I'm not following. Do you mean because you went fast, you had to run another cycle to clean everything properly? If you haven't already, you should watch the Technology Connections series on dishwashers. https://www.youtube.com/watch?v=jHP942Livy0

Since I’m rushing to load it as fast as possible the packing is not as good as hers so some dishes are left out. Overall this leads to more loads.

Ahhh that makes sense!

Re: The unreasonable effectiveness of the Fourier transform

#93

Post by btilly »

hammock wrote: post #83
Jun8 wrote: post #58

A signal cannot be both time and frequency band limited. Many years ago I was amazed when I read that this fact I learned in my undergraduate is equivalent to the Uncertainty Principle! On a more mundane note: my wife and I always argue whose method of loading the dishwasher is better: she goes slow and meticulously while I do it fast. It occurred to me we were optimizing for frequency and time domains, respectively,…

It’s literally the Heisenberg uncertainty principle, applied to signal processing.

For those who don't get this comment, the Heisenberg uncertainty principle applies to any two quantities that are connected in QM via a Fourier transform. Such as position and momentum, or time and energy. It is really a mathematical theorem that there is a lower bound on the variance of a function times the variance of its Fourier transform.

That lower bound is the uncertainty principle, and that lower bound is hit by normal distributions.

Re: The unreasonable effectiveness of the Fourier transform

#94

Post by 3eb7988a1663 »

Chrupiter wrote: post #48
hedgehog wrote: post #33

Once you start looking at the world through the lens of frequency domain a lot of neat tricks become simple. I have some demo code that uses fourier transform on webcam video to read a heartrate off a person's face, basically looking for what frequency holds peak energy.

I don't think pulsing skin (due to blood flow) is visible from a webcam though.

MIT was able to reconstruct voice by filming a bag of chips on a 60FPS camera. I would hesitate to say how much information can leak through.

https://news.mit.edu/2014/algorithm-recovers-speech-from-vib...

Re: The unreasonable effectiveness of the Fourier transform

#95

Post by slow_typist »

rcxdude wrote: post #56
Chrupiter wrote: post #48

Earlier quoted context omitted.

I don't think pulsing skin (due to blood flow) is visible from a webcam though.

It is, but there's a lot of noise on top of it (in fact, the noise is kind of necessary to avoid it being 'flattened out' and disappearing). The fact that it covers a lot pixels and is relatively low bandwidth is what allows for this kind of magic trick.

The frequency resolution must be pretty bad though. You need 1 minute of samples for a resolution of 1/60 Hz. Hopefully the heartrate is staying constant during that minute.

Re: The unreasonable effectiveness of the Fourier transform

#96

Post by derektank »

refulgentis wrote: post #78
ggm wrote: post #75

Earlier quoted context omitted.

But what he deemed being posited as true, was this really bad parenting? It could be to head off competition or it could be brutal realism to head off future depression. Nepotism existed since time immemorial but for a mathematical genius, what was the nepotistic deliverable for the child? A sinecure placement at university?

It’s unusual to tell others not to do something because you’re projecting they’re secretly doing it to compete with you, or that they’ll be depressed when they don’t do what you did. Doubly so when the rationale is “I’m so fucking awesome” Triply so when it’s something you’re passionate about, presumably inherently. Quadruply so when it’s your child. Its tough as a kid hearing your parents come up with elongated excu…

For what it’s worth, his children were quite successful by all accounts. Two of the boys became successful businessmen after emigrating to the US and one of the boys became a director of the railway network in Hannover. Seems as though they weren’t harmed by their upbringing.

Re: The unreasonable effectiveness of the Fourier transform

#97

Post by RossBencina »

lutusp wrote: post #42

Too bad -- the article doesn't mention Gauss. The Fourier transform is best presented to students in its original mathematical form, then coded in the FFT form. It also serves as a practical introduction to complex numbers. As to the listed patent, it moves uncomfortably close to being a patent on mathematics, which isn't permitted. But I wouldn't be surprised to see many outstanding patents that have this hidden pro…

Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.* There is a book "Math you Can't Use" that discusses this.

* not a legal definition, IANAL.

Re: The unreasonable effectiveness of the Fourier transform

#98

Post by wtetzner »

RossBencina wrote: post #97
lutusp wrote: post #42

Too bad -- the article doesn't mention Gauss. The Fourier transform is best presented to students in its original mathematical form, then coded in the FFT form. It also serves as a practical introduction to complex numbers. As to the listed patent, it moves uncomfortably close to being a patent on mathematics, which isn't permitted. But I wouldn't be surprised to see many outstanding patents that have this hidden pro…

Pretty sure in the USA you can patent mathematics if it is an integral part of the realisation of a physical system.* There is a book "Math you Can't Use" that discusses this. * not a legal definition, IANAL.

I would think you could only patent a particular usage of it.

Re: The unreasonable effectiveness of the Fourier transform

#99

Post by Certhas »

kingkongjaffa wrote: post #20
Certhas wrote: post #10

Earlier quoted context omitted.

I find it hard to parse the middle of your post. Are you saying Wigner's article, which is what all the "unreasonable effectiveness" titles reference, is silly? If that is what you are saying I suggest that you actually go back and read it. Or at least the Wiki article: https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness... By means of contrast: I think it's clear that mathematics is, for example, not unreas…

Yeah it's silly, I don't mean it in any mean spirited way. > Wigner's first example is the law of gravitation formulated by Isaac Newton. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations"[3] to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations." So despite 'very scan…

I am not discussing the FT case. But as regards Wigner's article, the core thing he points out is that while we are used to the effectiveness of maths, centuries after Newton, there in fact is not any prior grounds to expect this effectiveness.

And no, this is unrelated to whether math is invented or discovered. If anything this is related to the extreme success of reductionism in physics.

As a general point of reflection: If an influential article by a smart person seems silly to you, it's good practice to entertain the question if you missed something, and to ask what others are seeing in it that you're missing.

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