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The unreasonable effectiveness of the Fourier transform

joshuawise.com

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Re: The unreasonable effectiveness of the Fourier transform

#41

Post by RossBencina »

ted_dunning wrote: post #36
petermcneeley wrote: post #28

I dont like the Fourier Transform. It is infinite which makes it coarse and rough and it it gets everywhere.

Anybody who does anything in the real world with Fourier transforms uses the fast Fourier transform operating on windowed data. This eliminates all of that infinite support and infinite resolution of frequencies.

To be more precise, when working with sampled data with uniform sample rate you use the Discrete Time Fourier Transform (DTFT), not the Fourier Transform!. None the less, you still end up with an approximate spectrum which is the signal spectrum convolved with the window function's spectrum.

In my view the Fourier Transform is still useful in the real world. For example you can use it to analytically derive the spectrum of a given window.

But I think the parent is hinting at wavelet basis.

Re: The unreasonable effectiveness of the Fourier transform

#42

Post by lutusp »

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 property.

Re: The unreasonable effectiveness of the Fourier transform

#44

Post by [deleted] »

brcmthrowaway wrote: post #31
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…

How was Gauss so productive with 6 children?

[deleted]

Re: The unreasonable effectiveness of the Fourier transform

#45

Post by libraryofbabel »

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.

One of the things I admire about many top mathematicians today like Terence Tao is that they are clearly excellent mentors to a long list of smart graduate students and are able to advance mathematics through their students as well as on their own. You can imagine a half-formed thought Terence Tao has while driving to work becoming a whole dissertation or series of papers if he throws it to the right person to work on.

In contrast, Gauss disliked teaching and also tended to hoard those good ideas until he could go through all the details and publish them in the way he wanted. Which is a little silly, as after a while he was already widely considered the best mathematician in the world and had no need to prove anything to anyone - why not share those half-finished good ideas like Fast Fourier Transforms and let others work on them! One of the best mathematicians who ever lived, but definitely not my favorite role model for how to work.

Re: The unreasonable effectiveness of the Fourier transform

#47

Post by rbonvall »

pizza wrote: post #24
seba_dos1 wrote: post #14

Earlier quoted context omitted.

The unreasonable effectiveness of considering something harmful.

Lies, Damned lies, and Unreasonable Effectiveness

Lies, Damned lies, and Unreasonable Effectiveness For Fun and Profit

Re: The unreasonable effectiveness of the Fourier transform

#48

Post by Chrupiter »

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.

Re: The unreasonable effectiveness of the Fourier transform

#50

Post by moralestapia »

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.

You can do it with infrared and webcams see some of it, but I'm not sure if they're sensitive enough for that.
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