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Ever saw those EQ lines jumping around as the time goes by and music sounds? That is the direct result of drawing the information the FFT gives. There are many interesting applications, for example, voice recognition https://towardsdatascience.com/understanding-audio-data-four... It is really amazing
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.
[1]: https://byorgey.wordpress.com/2009/01/12/abstraction-intuiti...
Earlier quoted context omitted.
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…
Or even more simply, if you know that e^x on the complex plane rotates you around the origin.
Later in that video, we see a visualization of the rotation. I was able to grasp how the exp function could yield rotation where I’d never been able to understand why e*e*e*e… did.
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.
I'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 se…
It's exactly a change of basis
`x(t) = sum_k w_k sin(2pik*t)`
Then you just ask the question: given x, solve for w.
The next question is "why sinusoids", and the answer is that because for any linear, time invariant system acting on x:
y(t) = F[x](t)
that system diagonalizes over (complex) sinusoids:
y(f) = F(f)x(f)
I'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 se…
Sine waves aren't the only basis with which you can make this transformation. Your basis can be an arbitrary set of periodic functions as long as it meets certain requirements. Decomposition into wavelet functions is commonly used in seismic signal analysis, for example.
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.
This is very much called out in the Monad Burrito Tutorial Fallacy[1]. [1]: https://byorgey.wordpress.com/2009/01/12/abstraction-intuiti...
Earlier quoted context omitted.
If you lose something, you always find it in the last place you search... because why would you keep searching after you found it
It took me an embarrassingly long time to realize that it was a joke when people said "It's always the last place you look." Like well into my teens. But ever since I figured it out, I always look at least one more place after finding something.
Earlier quoted context omitted.
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.
A monad is a computational context, where the nature of that context is determined by two things: the shape of the data structure corresponding to it, and the definition of (>>=) which handles sequencing of two computations in that context. Anything more specific than that should be handled case-by-case until you build an intuition for how any given monad will behave.