This is the discrete Fourier transform (DFT) not the Fourier transform.
The Fourier Transform, explained in one sentence (2014)
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Re: The Fourier Transform, explained in one sentence (2014)
#32The best explanation I’ve ever seen of the Fourier transform is from 3Blue1Brown: https://m.youtube.com/watch?v=spUNpyF58BY&vl=en
The key insight for me on this topic also came from 3Blue1Brown, but in a different video: it’s that e^x is NOT best thought of as repeated multiplication, but instead as the function exp(x) = 1 + x + x^2/2 + X^3/6 + x^4/24 + …
After being relieved of the burden of that misconception, I was finally able to understand the role of complex numbers in the Fourier Transform.
Re: The Fourier Transform, explained in one sentence (2014)
#33I think the best semi-intuitive, non rigorous explanation I've seen of the Fourier transform is still one that first explained signal correlation in the time domain and then described the transform as basically performing correlation on the signal for all the possible sines at different frequencies. Essentially you're just testing for the presence of individual sine waves (of different frequencies) within the signal.…
> best semi-intuitive, non rigorous explanation [...] signal correlation in the time domain [...] basically performing correlation on the signal for all the possible sines at different frequencies
That's only going to make sense for someone who understands your jargon usage of "correlate", and who groks that integrals of sine curves are orthogonal under addition. That's precisely the hard part the linked explanation is trying to explain.
If you've already gotten that far then the DFT is just calculator math.
Re: The Fourier Transform, explained in one sentence (2014)
#343Blue1Brown video that actually animates this: https://youtu.be/spUNpyF58BY
This is very cool.
Re: The Fourier Transform, explained in one sentence (2014)
#35This reminds me of an old joke in the Haskell community, where people who struggled to understand Monads would finally get it after a while, and would assume that whatever the last sentence they heard was the only necessary one for the explanation.
In that spirit, my pet "a monad is a monoid in the category of endofunctors, duh"-style one-line explanation of the Fourier transform is that it's just the decomposition in the common (Hilbert) eigenbasis for all translation operators. It makes it surprisingly clear (to some) why it is a both natural and important construction.
Re: The Fourier Transform, explained in one sentence (2014)
#36This reminds me of an old joke in the Haskell community, where people who struggled to understand Monads would finally get it after a while, and would assume that whatever the last sentence they heard was the only necessary one for the explanation.
If you lose something, you always find it in the last place you search... because why would you keep searching after you found it
Re: The Fourier Transform, explained in one sentence (2014)
#37This reminds me of an old joke in the Haskell community, where people who struggled to understand Monads would finally get it after a while, and would assume that whatever the last sentence they heard was the only necessary one for the explanation.
I’ve understood monads multiple times in my life, but each time that understanding was so fragile that it crumbled when I tried explaining to someone else.
I’m currently in a phase where I don’t understand them.
Re: The Fourier Transform, explained in one sentence (2014)
#38I’m a layman, but I always think of the Fourier Transform as “an algorithm that converts amplitude over time into frequency intensities”. I guess that’s more of a What than a How, but it still seems good enough for a single sentence.
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Re: The Fourier Transform, explained in one sentence (2014)
#39This reminds me of an old joke in the Haskell community, where people who struggled to understand Monads would finally get it after a while, and would assume that whatever the last sentence they heard was the only necessary one for the explanation.
https://byorgey.wordpress.com/2009/01/12/abstraction-intuiti...
well, it’s so simple: they’re just burritos!
Re: The Fourier Transform, explained in one sentence (2014)
#40I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that.
In high school physics some problems got way easier by redefining coordinates as x' and y', solving for those, them going back to x and y. That's what the Fourier transform does, but for functions.
Looking at its formula, we can see it looks like we are projecting a function into a series of complex exponentials, just like we need to project a vector in x' and y' for a change of basis in the euclidian space.
Then, the Fourier coefficients can be thought as "how much my function looks like a complex exponential of this given frequency". Compare the formulas for the Pearson Correlation and Convolution, they are almost the same.
Since the Fourier transform is just a particular case of changing basis, there are many, many other integral transforms that might be useful in different contexts. We can even create our own. But unis teach about the Fourier transform because it is really good for solving differential equations, which turns out to be a popular way of modeling problems in engineering.