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I’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.
It's close enough. But then, how is it different from Laplace transform? (I have to admit, I actually learned about both during my uni time. But now I totally forgot them all.)
This 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.
Anything more specific than that should be handled case-by-case until you build an intuition for how any given monad will behave.
I 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…
At the totally basic level, another thing that I think can really trip people up and that a lot of texts are not clear about is that the transform is about moving between representations and that we're not actually transforming anything. Not enough maths texts take the time to explain how the representation connects up with the objects we wish to study and how there are multiple possible ways to represent them, and in some cases, like Fourier, useful ways to "translate" between different representations.
This is the discrete Fourier transform (DFT) not the Fourier transform.
I’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.
It's close enough. But then, how is it different from Laplace transform? (I have to admit, I actually learned about both during my uni time. But now I totally forgot them all.)
The 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. https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=4…
This 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
There are many interesting applications, for example, voice recognition https://towardsdatascience.com/understanding-audio-data-four...
It is really amazing
I 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.…