There was an old macintosh application that had an interactive FFT. I remember seeing it at the Haus Der Musik in Vienna and have always looked for it.
An interactive guide to the Fourier transform (2012)
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Re: An interactive guide to the Fourier transform (2012)
#32It is super to easy to find resources on HOW the FT works. But I find it very difficult to find resources on WHY we need it and WHERE it is useful to have. Does anybody have some good sources that explain the practical applications and how it is useful on real world usage?
Your cell phone wouldn't exist without the Fourier Transform, or the discrete fourier transform, to be correct. Image compression is another application, albeit, a 2-dimensional version. Software defined radios, or SDR, are completely dependent on the DFT. Radar processing. Earthquake analysis. The list goes on and on. Basically, our technological society would not exist in its current form without the fourier transf…
We went from mostly fearing vibrations in a mechanical world to harnessing them in a chemical and electrical world. It entirely reshaped our relationship to the world and perhaps so fast that our brains haven't entirely caught up yet. I wonder how many well controlled oscillations of mostly electrical charges happened between me swyping this and you hearing it in your head. We rely on them instead of fearing them now.
Re: An interactive guide to the Fourier transform (2012)
#33Earlier quoted context omitted.
From TFA: Stop. Here's where most tutorials excitedly throw engineering applications at your face. Don't get scared; think of the examples as "Wow, we're finally seeing the source code (DNA) behind previously confusing ideas". If earthquake vibrations can be separated into "ingredients" (vibrations of different speeds & amplitudes), buildings can be designed to avoid interacting with the strongest ones. If sound wave…
Yes I saw that. But it still hides the "meaty" part. Where are the articles that explain those processes? For example I would love to find an article that starts with "let's make a wav file smaller". And then somewhere in the middle it just says "and here we will use FT to achieve X".
It's not an article, but I found The Scientist and Engineer's Guide to
Digital Signal Processing book to be very comprehensive. There are a couple of chapters on applications, but not much code.
Re: An interactive guide to the Fourier transform (2012)
#34Just seen this, was playing with complex Fourier series yesterday. Here https://gist.github.com/kaleidawave/bdaf8649e7917152b6cdd624...
Mind blowing that there might be a solar system out there with a collection of orbiting objects whose final object could have square orbit
Re: An interactive guide to the Fourier transform (2012)
#35Just a nit. The pair of equations the author showed at the beginning of the article are not the equations of the Fourier Transform and its inverse. The Transform is a continuous function operating on an infinite input. The equation for the Transform involves the use of the integral taken over +/- infinity. What is shown, using the summation operator, is a discrete form on the Transform where the input is a limited ti…
Fourier transform can be defined for any locally compact abelian group. Integers modulo n is one such.
Given that monads are monoids in the category of endofunctors, I'm perfectly fine with it.
Re: An interactive guide to the Fourier transform (2012)
#36Earlier quoted context omitted.
From TFA: Stop. Here's where most tutorials excitedly throw engineering applications at your face. Don't get scared; think of the examples as "Wow, we're finally seeing the source code (DNA) behind previously confusing ideas". If earthquake vibrations can be separated into "ingredients" (vibrations of different speeds & amplitudes), buildings can be designed to avoid interacting with the strongest ones. If sound wave…
Yes I saw that. But it still hides the "meaty" part. Where are the articles that explain those processes? For example I would love to find an article that starts with "let's make a wav file smaller". And then somewhere in the middle it just says "and here we will use FT to achieve X".
Here's a pretty good lecture on exactly that subject:
https://www.rose-hulman.edu/~bryan/invprobs/jpegtalk2.pdf
Honestly though, the Fourier transform is useful anywhere that it's easy to think in terms of frequency instead of time. 90% of use cases in the real world come about from trying to do something that's that's easy to express in terms of signal frequency and difficult to express in terms of signal time/space. You can just use the Fourier transform to convert between them efficiently.
To give a morbid, but non-obvious example, I once did some work with a charity that was concerned with migrant deaths. They wanted to figure out where the migrants most at risk were working to approach the farmers hiring them directly. We got a dataset of when migrant bodies were found and did an FFT (among other processing) to find the periodicity of the crops that they were coming to harvest. There's only a few major crops and they tend to have distinct growing periods, so this is enough information to pinpoint certain crops like strawberries.
Re: An interactive guide to the Fourier transform (2012)
#37Just seen this, was playing with complex Fourier series yesterday. Here https://gist.github.com/kaleidawave/bdaf8649e7917152b6cdd624... Mind blowing that there might be a solar system out there with a collection of orbiting objects whose final object could have square orbit
A square orbit would require infinite acceleration. Specifically, a dirac delta.
Re: An interactive guide to the Fourier transform (2012)
#38Earlier quoted context omitted.
Minor point, it's not a projection because you don't lose dimensions.
nit: the identity matrix is a projection P^2 = P is the definition. not losing dimensions
True, but missing the point as you would not normally refer to the identity matrix as a projection.
Re: An interactive guide to the Fourier transform (2012)
#39Earlier quoted context omitted.
Minor point, it's not a projection because you don't lose dimensions.
nit: the identity matrix is a projection P^2 = P is the definition. not losing dimensions
Nit to your nit, which is incorrect w.r.t. parent reply:
A Fourier transform is not a projection, it's a change of basis represented by a unitary transformation.
Re: An interactive guide to the Fourier transform (2012)
#40Earlier quoted context omitted.
Your cell phone wouldn't exist without the Fourier Transform, or the discrete fourier transform, to be correct. Image compression is another application, albeit, a 2-dimensional version. Software defined radios, or SDR, are completely dependent on the DFT. Radar processing. Earthquake analysis. The list goes on and on. Basically, our technological society would not exist in its current form without the fourier transf…
> Your cell phone wouldn't exist without the Fourier Transform, or the discrete fourier transform, to be correct Yes that is great to know. But where can I find an article that explains how exactly FT helps in my cell phone? What exactly do we do with FT in a cell phone?
For example, to create a JPEG, part of the process is removing the "high frequency" parts of your image, since those take the most information to store. Here, high frequency refers to noise, or any large difference between neighboring pixels, as opposed to low frequency parts, which are averages over larger groups of pixels. So, at the extreme, if you average the whole image, you only have to store one pixel, so it's obviously less data to store, vs storing info about every single pixel. JPEG (and other lossy compression formats) tries to find a good middle ground between storing every pixel perfectly and storing just one pixel.
So, how do you remove the high frequency parts? You apply a Fourier transform to the image (in this case, it's a "Discrete Cosine Transform", which is extremely similar to a DFT and has no differences for the purpose of this explanation) and get a 2d array. This 2d array has the low frequency parts in the upper left corner, and the high frequency parts everywhere else (to the right and down for horizontal and vertical frequencies). So your compression algorithm will simply zero out the high frequency parts so you don't have to store them. In the simplest case, this is equivalent to a simple blur of the image, but there are some heuristics about how much to remove (zero out) to minimize image degradation.
To decode the image, you take the 2d array and apply the inverse FT to get the original image (now slightly blurry because it's been compressed).
More details here: https://en.wikipedia.org/wiki/JPEG#Discrete_cosine_transform