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An interactive guide to the Fourier transform (2012)

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Re: An interactive guide to the Fourier transform (2012)

#71

Post by ndriscoll »

kkapelon wrote: post #26
CaptainFever wrote: post #16

Earlier 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".

Take a look at [0] to get a quick visual demonstration of how that works. As others have said, lossy compression often throws away high frequency data (for sound, it corresponds to high pitches that you can barely hear. For images, high frequency corresponds to fast changing parts of a picture, so think crisp edges, fine detail (e.g. hair or stubble), and noise/film grain). So a basic lossy compression algorithm might use a FT and then only store the lower frequencies. When you do the inverse transform without the high frequency data, you lose details.

Note that the spectrograms in [0] use what's called a short time fourier transform[1]. Basically, split your song into windows (e.g. 1 second each), and do a fourier transform on each window. This lets you get a picture of the spectrum over time.

[0] https://sound.stackexchange.com/questions/38709/320kbps-mp3-...

[1] https://en.wikipedia.org/wiki/Short-time_Fourier_transform

Re: An interactive guide to the Fourier transform (2012)

#72

Post by 2muchcoffeeman »

meindnoch wrote: post #62
2muchcoffeeman wrote: post #54

I find all these Fourier explanations bad. The basic idea is very simple: Fourier takes a function, and converts it into the reciprocal domain. So if your X axis is time, t. Then Fourier gives you 1/t. What’s 1/t if t was some unit time? Frequency. If you want a nice intuitive example of this, (hand waving begins) a lens will give you a Fourier transform on its back focal plane if all the light coming in are parallel…

4F Optical Correlator: https://youtu.be/wcRB3TWIAXE A simple optical computer.

Yep. It’s a great experiment because you can have spatial frequency as well which gives you a physical demonstration of a Fourier transform.

Then you can do things like create physical filters and reform the original image to see if your filter worked.

Re: An interactive guide to the Fourier transform (2012)

#73

Post by dustingetz »

nextaccountic wrote: post #55
jameshart wrote: post #41

I don’t like the casting of the frequency domain view as the ‘recipe’ and the time domain view as the ‘product’. The point of Fourier is that you can switch between these perspectives losslessly - they contain equivalent information. The ‘smoothie’ metaphor of ‘unmixing’ the smoothie to get the ingredients, and then blending it to get the smoothie back conjures the impression that Fourier transformation is some sort…

the best analogy is a basis change in a vector space: you can have the same data (the same vector) viewed in different ways if you look at it using different a base. for example, in physics, the numerical value of the coordinates of an object change in different frames of reference and this is not just an analogy: in the vector space of functions the fourier transform is indeed a basis change (to be more precise, a r…

fractional FT is the uncertainty principle then? you can know position locally, or momentum locally, or you can know them both blurrily with a rotated basis in between

Re: An interactive guide to the Fourier transform (2012)

#74

Post by euroderf »

gnarlouse wrote: post #46

https://youtu.be/spUNpyF58BY?si=dM8J8Df5U7DTV9ls This is the definitive, the last Fourier transform guide you’ll ever need. It’s so intuitive and simple to understand. I watched this video *once* six years ago, and I can still rebuild the Fourier formula from memory. If only this had been around during my digital signal processing coursework in undergrad.

> If only this had been around during my digital signal processing coursework in undergrad.

I hear that.

At my school the course in Fourier Series & Boundary Value Problems was better known to students as "Mystery Math". This because there was typically a single moment of enlightenment.. where suddenly... the entire content of the course became pud.

If this moment of mathematical satori came before the final exam, you were in luck.

Re: An interactive guide to the Fourier transform (2012)

#75

Post by nextaccountic »

dustingetz wrote: post #73
nextaccountic wrote: post #55

Earlier quoted context omitted.

the best analogy is a basis change in a vector space: you can have the same data (the same vector) viewed in different ways if you look at it using different a base. for example, in physics, the numerical value of the coordinates of an object change in different frames of reference and this is not just an analogy: in the vector space of functions the fourier transform is indeed a basis change (to be more precise, a r…

fractional FT is the uncertainty principle then? you can know position locally, or momentum locally, or you can know them both blurrily with a rotated basis in between

Yes, but that's because the momentum is the fourier transform of the position and vice versa

So there's an uncertainty principle for signal processing, entirely because of the fourier transform and not because of quantum mechanics

See this video https://www.youtube.com/watch?v=MBnnXbOM5S4

Re: An interactive guide to the Fourier transform (2012)

#76

Post by nextaccountic »

nicwilson wrote: post #57
nextaccountic wrote: post #55

Earlier quoted context omitted.

the best analogy is a basis change in a vector space: you can have the same data (the same vector) viewed in different ways if you look at it using different a base. for example, in physics, the numerical value of the coordinates of an object change in different frames of reference and this is not just an analogy: in the vector space of functions the fourier transform is indeed a basis change (to be more precise, a r…

FT is a 90 degree rotation not a 180, the FT of the FT of a function is the mirror image about the origin, not the function itself.

Ok you're right! I originally wrote down 90 degrees but then I had a conflicting view about being the inverse and then reasoned it must be 180 degrees

So the fourier transform of the fourier transform isn't the same as the inverse fourier transform? (ignoring the scaling bits that can be normalized I think), so I've been lied to?

Anyway here is a funny pair of questions

https://math.stackexchange.com/questions/1472528/why-is-the-...

https://math.stackexchange.com/questions/3922412/why-isnt-th...

Re: An interactive guide to the Fourier transform (2012)

#77

Post by keithalewis »

ndriscoll wrote: post #69
keithalewis wrote: post #28

Earlier quoted context omitted.

Fourier transform can be defined for any locally compact abelian group. Integers modulo n is one such.

You actually don't need abelian. e.g. the group of 1d affine transformations T(a,b)(x) = ax+b gives a variation of a wavelet transform. But you no longer have 1-D irreducible representations, so your Fourier coefficients become operators instead of numbers or something like that.

What is Haar measure in this case?

Re: An interactive guide to the Fourier transform (2012)

#78

Post by ndriscoll »

keithalewis wrote: post #77
ndriscoll wrote: post #69

Earlier quoted context omitted.

You actually don't need abelian. e.g. the group of 1d affine transformations T(a,b)(x) = ax+b gives a variation of a wavelet transform. But you no longer have 1-D irreducible representations, so your Fourier coefficients become operators instead of numbers or something like that.

What is Haar measure in this case?

Apparently the left invariant measure is 1/a^2 dadb, and the right invariant measure is 1/a dadb. The book I have just sticks to left measures, and is an engineering book so it doesn't really get into details that a math book would (the whole group theory chapter is 31 pages out of a 1500 page book and the wavelet portion is ~2 pages, though it also has an 8 page section on wavelets in a previous chapter). It's very much a "what are groups, what are representations, and why do they matter for physics and signal processing" kind of thing. For reference it's Barrett & Myers Foundations of Image Science.

Re: An interactive guide to the Fourier transform (2012)

#79

Post by nh23423fefe »

amelius wrote: post #38
nh23423fefe wrote: post #30

Earlier quoted context omitted.

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.

What was your point? An incorrect characterization of projections?
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