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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)

#61

Post by meindnoch »

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.

90 degree rotation? That would imply that the fourier transform is orthogonal to the original function.

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

#62

Post by meindnoch »

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.

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

#63

Post by keithalewis »

penguin_booze wrote: post #35
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.

Given that monads are monoids in the category of endofunctors, I'm perfectly fine with it.

There is no Royal Road to mathematics. There is also no shortage of people convincing themselves they understand some particular aspect of it by watching a YouTube video.

It is rare when a mathematician, or anyone, can write a book on a topic that no other expert in the field can top. Walter Rudin did that with Fourier analysis.

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

#64

Post by ska »

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…

> the best analogy is a basis change in a vector space:

It's closer than an analogy. FT is essentially a map of basis into a dual space, but this dual is non uniquely isomorphic with the original so given an arbitrary choice of isomorphism you can turn it into "just" a change of basis.

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

#65

Post by drtgh »

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".

FT breaks down a cyclic signal into the frequencies present, and their intensity. The longer the sample of the signal, the more precise becomes detecting the real frequencies present in the signal.

Mainly is tool for to obtain information that will feed latter other algorithms. Also can be used as a signal filter with the Inverse FT.

If this does help to understand, it is homologous to use several band pass filters repeatedly for trying to obtain the root signals, but without adjustments requirements, highly faster to compute, simpler, and with better result.

So your question is, What are the uses of knowing the main elements that make up a signal?

That information is useful for analysis of sounds or images, for to detect the presence of elements composing the signal out of the expected range.

Also for pattern detection, as you may give the data a signal form of your own, and analyze the frequencies peaks for example.

For compression, the above two, as knowing the main cyclic elements and their intensity allows to determine if some ones may be latter omitted (for loss compression), or what elements latter should be replaced as parameters that the decompression algorithm will use for to reconstruct the signal.

Also it is important what said the other answer. It does not give information about the time/space moment in with is produced the element, it only tells the element is present (the frequency in the signal).

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

#66

Post by ndriscoll »

earlymodernlvr wrote: post #58
duped wrote: post #11

Earlier quoted context omitted.

Same answer for both. It's an orthogonal transform, aka a change of basis. You're conceptually rotating the function/series to an equivalent one that's orthogonal to the original. The magnitude/energy hasn't changed (Parseval's theorem is a more succinct definition). And to perform the inverse transform you need to conceptually rotate it back to the original, which should mirror the original transform very nicely. If…

I really appreciate this reply, since this is something I've always been curious about. If you have time, I would really appreciate it if you could elaborate on this point (maybe with some equations), but you've already given me a ton to think about, thank you! Also, is the new basis orthogonal to the original, or just another orthonormal basis? I don't see why it would be orthogonal to the original.

It's an orthonormal basis. See my comment here[0] for more on why the forward and inverse transforms look similar (I've written e_w to mean e^iwt, but we want to think of it as a vector). The forward direction is doing a dot product with an exponential with a specific frequency to get the transform at that frequency (i.e. the function's projection/component at that frequency), which is a sum over the entire time basis. The inverse transform sums over all components/projections of the function at each frequency to rebuild the function. This is how you do an expansion in terms of orthogonal projections in any dimension.

This also explains why the forward transform has a minus and the inverse doesn't: a complex dot product sums over complexConjugate(v_i)*w_i, while the inverse/reconstruction just sums over the basis exponentials scaled by the Fourier coefficient for that frequency.

You can also prove that the sum over all imaginary exponentials is Dirac delta[1], so you could think of the inverse transform as being a dot product with delta to get the function at a specific time, and that dot product is a sum over the frequency basis.

[0] https://news.ycombinator.com/item?id=38658312

[1] https://math.stackexchange.com/questions/1343859/why-does-in...

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

#67

Post by nicwilson »

meindnoch wrote: post #61
nicwilson wrote: post #57

Earlier quoted context omitted.

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.

90 degree rotation? That would imply that the fourier transform is orthogonal to the original function.

It is, because wavenumber and position are distinct variables and are orthogonal to each other. FT turn position into wavenumber (positional frequency) and wavenumber into negative position:

[ 0 1] [x] [ ω]

          =
[-1 0] [ω] [-x]

see also https://en.wikipedia.org/wiki/Linear_canonical_transformatio...

the rotation matrix [[ 0 1], [-1 0]] is a 90 degree rotation.

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

#68

Post by Espressosaurus »

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".

If you understand the fourier transform, you should have some idea of its inverse. And that's exactly how (one part) of compressing an image or audio works.

You take a signal, take its fourier transform, and then cut off the frequencies above some threshold you don't care about (I'm ignoring certain complicating factors for simplicity). Let's say your original signal has frequencies up to 22KHz. If you're only interested in moderate fidelity human voices talking, maybe you cut off everything below 100 Hz and above 3.4 KHz. You just toss that information. Then on the receiving end, you do an inverse DFT and reconstruct the signal.

JPG and MP3 do something like this, along with a pile of other tricks.

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

#69

Post by ndriscoll »

keithalewis wrote: post #28
sizzzzlerz wrote: post #18

Just 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.

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.

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

#70

Post by meindnoch »

nicwilson wrote: post #67
meindnoch wrote: post #61

Earlier quoted context omitted.

90 degree rotation? That would imply that the fourier transform is orthogonal to the original function.

It is, because wavenumber and position are distinct variables and are orthogonal to each other. FT turn position into wavenumber (positional frequency) and wavenumber into negative position: [ 0 1] [x] [ ω] = [-1 0] [ω] [-x] see also https://en.wikipedia.org/wiki/Linear_canonical_transformatio... the rotation matrix [[ 0 1], [-1 0]] is a 90 degree rotation.

Ok, now I see it!
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