I don't really like this explanation, tbh. "spin your function" doesn't really make sense. If I didn't already know what they were trying to say, I'd have no idea.
The Fourier Transform, explained in one sentence (2014)
151–160 of 171 posts
Re: The Fourier Transform, explained in one sentence (2014)
#152It's useful to test explanations on people unfamiliar with it. So in this spirit, I want to share: the sentence didn't explain it to me. After watching3Blue1Brown's video, I get it. So I can share with you what confused me: It's this part of the sentence: "average a bunch of points along that path". I know what averaging it, but I don't know what averaging "along a path" means. Also I was (wrongly) expecting the outp…
> I just expect "energy" to be a number. Instead, here what we're doing is averaging complex numbers so the output is a complex number. What a complex energy means is left as an exercise to the reader.
They shouldn't be using the term energy, really. That should be reserved for the magnitude of the complex number (so re(z)^2+im(z)^2). You can write a complex number either as a real and imaginary part, or as a amplitude r (the distance you're referencing) and phase (the rotation from r+0im).
> "To find the energy at a particular frequency, spin your signal around a circle at that frequency, and find the center of mass of the plot."
As someone who's spent a lot of time with the subject, I like this better than the original.
Re: The Fourier Transform, explained in one sentence (2014)
#153Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes…
As someone who still doesn't understand the Fourier Transform, this explanation doesn't help, nor does the article's. As a complete noob, your explanation shows off what you know but doesn't help newcomers learn it.
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Re: The Fourier Transform, explained in one sentence (2014)
#154IME explanations of the Fourier transform focus far too much on the specific mechanics of it without motivating the reason. I find people often don't even realize the time space and frequency space functions are the same function! I dive in like so "There are many different ways to write down the number 5. Tally marks, the sigil 5, 4+1, 2.5 times 2, 10/2. Each of them is more or less useful in different circumstances…
I like this explanation. It would be good to include a little bit about why we even care to do frequency-domain analysis, and what a "transform" is.
this is a wonderful explanation and illustrates the importance of fourier transforms in a simple way anyone can grasp.
coupled with voidhorse's comment, it appears like "transform" obscures the concept whereas "translation" would be more appropriate and clearer.
Re: The Fourier Transform, explained in one sentence (2014)
#155Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes…
As someone who still doesn't understand the Fourier Transform, this explanation doesn't help, nor does the article's. As a complete noob, your explanation shows off what you know but doesn't help newcomers learn it.
Imagine you have a voice recording. You are looking at it a long squiggly line with no uniformity.
Zoom way in. If you go far enough it’ll just look like a curved line.
That curved line can be estimated down into a sin wave, or more accurately, a few sin waves that combine to make almost the same wave you have.
FFT is a way to take a complex wave and reduce it down to the sin wave components that would all combine to make it up.
To practically apply this, your voice recording is very high resolution and has a lot of bits at a high sample rate. If you instead used sin waves to represent the data, it could be almost as good sounding while being a lot less data to store.
… Watch a video. It’s something many people need to see to get.
Re: The Fourier Transform, explained in one sentence (2014)
#156Earlier quoted context omitted.
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.
Is computational context another way of saying scope?
That's more or less how I think of it. Each bind/flatMap introduces a new value into scope and can consume data in the form of the result of a further bind/flatMap or final return.
It's essentially a mini interpreter for tiny functionally pure segments of a program.
One tricky part of the interpretation is understanding the most familiar monadic data types in this paradigm. For example, the list monad representing discrete nondeterminism.
Writing about this so tersely is probably only going to confuse anyone who isn't already 90% of the way to understanding....
Re: The Fourier Transform, explained in one sentence (2014)
#157Earlier quoted context omitted.
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.)
Laplace is a generalisation of Fourier. Fourier can in principle only represent periodic signals. To circumvent this limitation, they invented the windowing of consecuive parts of signals and do the transform on each window. Laplace adds the transient part, that's what makes it more complicated.
That's not true. The Fourier series represents periodic functions. The Fourier transform applies to all real functions. They need not be periodic, although I'm not sure it's a particularly useful representation of aperiodic functions.
It's been a long time since I've thought about the Laplace transform, but IIRC, it has some properties that are preferable the Fourier transform as a math tool.
The Fourier transform better maps to an intuitive concept of frequency, which is why it is the basis for the transforms we actual use in practice in engineering, such as the discrete Fourier transform, which is what the famous FFT algorithm computes.
Re: The Fourier Transform, explained in one sentence (2014)
#158Earlier quoted context omitted.
Or even more simply, if you know that e^x on the complex plane rotates you around the origin.
People told me that over and over but it didn’t help — because it didn’t make sense why repeated multiplication would cause rotation! 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. https://www.youtube.com/watch?v=ZxYOEwM6Wbk&t=2178s
The best explanation I've seen, which 3blue1brown probably avoided because it requires calculus, is that exp is the unique function whose value at 0 is 1 and whose derivative is itself. Then by the chain rule the derivative of exp(ix) is i*exp(ix), meaning that the direction of motion is perpendicular to the current position vector. Which naturally leads to circular motion because you're always staying the same distance from the origin.
With that definition it's easy to derive the Taylor series expansion (every derivative at 0 is 1), and you can think of Euler's formula not as telling you how to evaluate exp(ix) (it's already perfectly well defined), but as an introduction of cos and sin as shorthand for its real and imaginary parts.
Re: The Fourier Transform, explained in one sentence (2014)
#159Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes…
As someone who still doesn't understand the Fourier Transform, this explanation doesn't help, nor does the article's. As a complete noob, your explanation shows off what you know but doesn't help newcomers learn it.
A visual like the one in this article helped make it click for me:
https://kinder-chen.medium.com/denoising-data-with-fast-four...
Re: The Fourier Transform, explained in one sentence (2014)
#160Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes…
To be honest I think your explanation misses the point in two important ways.
First, the Fourier transform does not just measure the 'energy' or amplitude of a single wave; it must also take into account its phase.
Second, complex exponentials are in no way an essential ingredient for defining the Fourier transform. We just work with them for computational simplicity, essentially because exp(ix) exp(iy) = exp(i(x+y)).
I also have trouble assigning meaning to your last paragraph. After all, complex exponentials can cancel just as much as sine waves (proof: sine waves are a combination of complex exponentials).