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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#61

Post by laszlokorte »

hovden wrote: post #50
laszlokorte wrote: post #26

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] [1]: https://static.laszlokorte.de/frft-cube/ [2]: htt…

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.

Looks amazing! Thank you

Re: What Is the Fourier Transform?

#62

Post by nblgbg »

laszlokorte wrote: post #26

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] [1]: https://static.laszlokorte.de/frft-cube/ [2]: htt…

Thanks a lot for all of this ! https://tools.laszlokorte.de/

Re: What Is the Fourier Transform?

#63

Post by laszlokorte »

xphos wrote: post #52
laszlokorte wrote: post #37

Earlier quoted context omitted.

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 impleme…

I made a cool rust fft tui a long time ago too https://github.com/lquinn2015/FFT-tui

Look great! I was already thinking about reimplementing mine as TUI

Re: What Is the Fourier Transform?

#64

Post by 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.

Re: What Is the Fourier Transform?

#65

Post by artyom »

anyfoo wrote: post #33

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.

Essentially that's what electrical/electronics engineering is about.

Re: What Is the Fourier Transform?

#66

Post by laszlokorte »

nblgbg wrote: post #62
laszlokorte wrote: post #26

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] [1]: https://static.laszlokorte.de/frft-cube/ [2]: htt…

Thanks a lot for all of this ! https://tools.laszlokorte.de/

I am glad you are enjoying it! :)

Re: What Is the Fourier Transform?

#68

Post by artyom »

ThrowawayR2 wrote: post #47

Kind of a tangent but why are there so many articles and videos popularizing the Fourier transform and practically none for the Laplace transform? The first person to do a well done 3Blue1Brown style video that focuses on intuition and visualization would probably be an overnight sensation (well, among engineers at least).

My take on this is that Fourier is easier to understand, has more immediate and relatable practical applications, and can be explained to "common folk" with just-enough sophistication and some tangible analogies (like the wool picture from the article).

Explaining Laplace to solve differential equations with analogies it's a bit more... complicated.

Re: What Is the Fourier Transform?

#69

Post by segfault99 »

anyfoo wrote: post #33

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.

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.)
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