An animated introduction to the Fourier Transform [video]
11–20 of 33 posts
Re: An animated introduction to the Fourier Transform [video]
#12He touches on it - but I’d love to see an intuitive explanation of why the response of each frequency to the input function is linearly independent. i.e the fact that Fourier transform of the sum is equal to the sum of the Fourier transforms. This is “why it works” - it’s what makes the frequency space an orthonormal basis - but it’s never been intuitively obvious to me. Otherwise, there would be more than one way of…
Here's how I think about it.
You can play the individual notes of a chord on one piano or several, but they still come together to produce the same chorus of frequencies.
The Fourier series representation of a waveform is itself a sum, since trigonometric functions are waveforms as well. Thus, a combination thereof should be commutative because a sum of two sums retains the properties of addition.
Re: An animated introduction to the Fourier Transform [video]
#13This fourier transform simulation example from shadertoy is good.
https://www.shadertoy.com/view/ltKSWD
Re: An animated introduction to the Fourier Transform [video]
#14This fourier transform simulation example from shadertoy is good.
Re: An animated introduction to the Fourier Transform [video]
#15This fourier transform simulation example from shadertoy is good. https://www.shadertoy.com/view/ltKSWD
The link kills my MacBook Pro. Makes the system unresponsive.
Re: An animated introduction to the Fourier Transform [video]
#16He touches on it - but I’d love to see an intuitive explanation of why the response of each frequency to the input function is linearly independent. i.e the fact that Fourier transform of the sum is equal to the sum of the Fourier transforms. This is “why it works” - it’s what makes the frequency space an orthonormal basis - but it’s never been intuitively obvious to me. Otherwise, there would be more than one way of…
It actually follows from the 'centre of mass' explanation.
If you take the centre of mass as described of f + g you get the centre of mass of f plus the centre of mass of g.
One way to explain this is to just say the + can be moved out of the integral.
Alternatively, consider the centre of mass only over the horizontal axis. Now say we only look at the 'contribution' of f + g at time t (ignoring the issues of that contribution being infinitesimal). That contribution is (f(x) + g(x)) * sin (theta) where theta is the angle of our point. Clearly this equals f(x) * sin (theta) + g(x) * sin (theta). These are the separate contributions of f(x) and g(x). The same argument holds for the centre of mass over the vertical axis (replacing sin with cos).
If we were to make the alternative explanation formal, we get back to the + being able to move outside the integral. Note that our decomposition into the horizontal and vertical part is an alternative way to de the fourier transform without complex numbers. The vertical part here is essentially the imaginary part of the fourier transform.
Re: An animated introduction to the Fourier Transform [video]
#173blue1brown’s videos are excellent. They build intuition in a calm and friendly way with an appropriate amount of useful animation. This is how we make mathematics accessible. I’m currently considering moving back into academia and there are a lot of topics in my field that I know students often struggle with that would be greatly helped by some simple animations. Fortunately I’m pretty competent with blender and I r…
Do you have any links showing how to do that with blender?
Re: An animated introduction to the Fourier Transform [video]
#183blue1brown’s videos are excellent. They build intuition in a calm and friendly way with an appropriate amount of useful animation. This is how we make mathematics accessible. I’m currently considering moving back into academia and there are a lot of topics in my field that I know students often struggle with that would be greatly helped by some simple animations. Fortunately I’m pretty competent with blender and I r…
Do you have any links showing how to do that with blender?
Anything in particular? I was thinking of things like viscosity and stress analysis for fluid mechanics. It would be easy enough to animate a stress tensor and show how each term behaves when prodded. Similarity the basic concepts behind laminar boundary layers would be equally straight forward.
Mimicking 3b1b’s style would be trickier since he uses a lot of 2D plots. Of course you can run python directly from blender so you never know.
Re: An animated introduction to the Fourier Transform [video]
#19I really wish this stuff existed when I was learning about FFTs - this video describes the theory far better and in far less time than my broken-english college professors ever could.
Re: An animated introduction to the Fourier Transform [video]
#203blue1brown’s videos are excellent. They build intuition in a calm and friendly way with an appropriate amount of useful animation. This is how we make mathematics accessible. I’m currently considering moving back into academia and there are a lot of topics in my field that I know students often struggle with that would be greatly helped by some simple animations. Fortunately I’m pretty competent with blender and I r…
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