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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)

#51

Post by ndriscoll »

mrfox321 wrote: post #39
nh23423fefe wrote: post #30

Earlier quoted context omitted.

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.

The w Fourier coefficient F(w) is the dot product of f with an exponential function, `e_w • f`, and is in that sense a projection. The inverse Fourier transform writes the original function as a sum of the projected components: `f = sum_w (e_w • f) e_w = sum_w F(w) e_w`. This is exactly how writing an "arrow" style 2- or 3-D vector as a sum of orthogonal projections works.

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

#52

Post by anvuong »

amelius wrote: post #15
sorenjan wrote: post #10

The short explanation of the DFT that I like the most is that it projects the signal vector on to the vector room with the exponential functions as basis vectors. Then you can see how much of the signal that each basis vector, corresponding to a frequency, can explain. It's intuitive to see if you start with a 2D vector space (the regular euclidian plane) and a 2D vector, and then you can expand the definition of vec…

Minor point, it's not a projection because you don't lose dimensions.

Is it a math terminology? Because the way I think it's still a projection, just with 0 residuals?

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

#53

Post by dang »

alberto_ol wrote: post #2

previous submission with 79 comments https://news.ycombinator.com/item?id=27229836

Thanks! Macroexpanded:

An Interactive Guide to the Fourier Transform (2012) - https://news.ycombinator.com/item?id=27229836 - May 2021 (79 comments)

An Interactive Guide to the Fourier Transform - https://news.ycombinator.com/item?id=10635075 - Nov 2015 (18 comments)

An Interactive Guide To The Fourier Transform - https://news.ycombinator.com/item?id=4948082 - Dec 2012 (25 comments)

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

#54

Post by 2muchcoffeeman »

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.

Don’t quote me on that I need to double check the precise conditions.

But you can send laser light through an image printed on transparency and manually apply filters to clean up small artefacts.

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

#55

Post by nextaccountic »

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 rotation). and from this arises the fractional fourier transform, which is a halfway change: if the fourier transform is a 180 degrees rotation, a fractional transform is something in between

https://en.m.wikipedia.org/wiki/Fractional_Fourier_transform

and this also explains why the fourier transform is the inverse of itself: two 180 degrees rotations gets you to the same place you were before

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

#56

Post by jameshart »

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.

3b1b’s video is excellent, agreed.

But I also want to give a shout out to this video: https://youtu.be/ToMyB5Hk06w?si=yJLDbb82JireH4Q9

It does a really solid job of building the intuition for the way the underlying mathematical machinery works, using ‘inner products’ as its mode of thinking rather than complex analysis - I think it’s a great complement to Grant Sanderson’s explanation and for some people I think it’s probably more useful.

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

#57

Post by nicwilson »

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…

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.

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

#58

Post by earlymodernlvr »

duped wrote: post #11
earlymodernlvr wrote: post #5

Request for someone to make an intuitive explanation of why the Fourier Transform is (almost) it's own inverse. I know the math proof from taking analysis, but the formula is too pretty and symmetrical for the explanation to be so technical. Same for why it preserves L2 norm.

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.

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

#59

Post by dr_dshiv »

One of the guys he thanks is Steve Lehar, who is a hero of mine. Steve had incredible intuitions about Harmony and Resonance and was crushed down by normies in psychology. But, indeed, the world is made of waves and resonance/harmony does govern pretty much everything! He says it better than me:

https://scholar.google.com/citations?user=UmJanU0AAAAJ&hl=en

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

#60

Post by meindnoch »

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 fourier transform is the inverse of itself

Not exactly. Almost.

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