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The Fourier Transform, explained in one sentence (2014)

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Re: The Fourier Transform, explained in one sentence (2014)

#101

Post by Snowbird3999 »

prof-dr-ir wrote: post #69

Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…

What works for me is learning a single concept in many different ways. A textbook, a youtube video, a friend explaining it to me... Each next source of material (however casual or formal it is) is more likely make the concept "click", or allow me to understand the concept more efficiently.

To that extent, this blog post seems valuable to me. The author is using a language and colors to convey an intuition that I have not precisely seen before.

Re: The Fourier Transform, explained in one sentence (2014)

#102

Post by becquerel »

wkjagt wrote: post #37
sebzim4500 wrote: post #14

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.

Understanding monads isn't particularly useful unless you also understand functors, which are a relatively much easier concept to understand long-term. They're really just functors with a few extra rules that make them a bit more useful.

Re: The Fourier Transform, explained in one sentence (2014)

#103

Post by teolandon »

tel wrote: post #16

Honestly, 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…

This is good, but I think it's cyclical (heh). We want to compare our signal with a "pure tone". What's a pure tone? A sine wave. Why is a sine wave a pure tone? Because when we compare it with a pure tone, it's identical.

Re: The Fourier Transform, explained in one sentence (2014)

#104

Post by evanwise »

prof-dr-ir wrote: post #69

Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…

I would tend to agree. The only people I saw really struggle with learning Fourier series and the Fourier transform were people who had struggled with more basic concepts and failed to internalize them. This kind of explanation might be helpful to build someone's intuition up a bit but is not a substitute for a proper education on the subject and definitely not some magic key to understanding.

Re: The Fourier Transform, explained in one sentence (2014)

#105

Post by fifilura »

prof-dr-ir wrote: post #69

Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…

I wish my university courses had started with the intuition behind and utility of the functions we were about to learn. Be it linear algebra, differential equations or Fourier transforms.

This is what all these blog posts excel with.

Of course I got it along the way, but I do believe it would have been easier if the background was something like this.

Re: The Fourier Transform, explained in one sentence (2014)

#106

Post by georgeburdell »

prof-dr-ir wrote: post #69

Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…

I agree. Way back when I learned the FT, I happened to be taking a signals and statistics class in the same semester and rationalizing the Fourier transform as the correlation between a function and a sinusoid was an incredibly natural mental leap and didn’t require me to hold much additional knowledge in my head

Re: The Fourier Transform, explained in one sentence (2014)

#107

Post by hackinthebochs »

trenchgun wrote: post #75
TedDoesntTalk wrote: post #60

Earlier quoted context omitted.

Is computational context another way of saying scope?

Not exactly. Computational context in this sense is like, is this computation of the type that can either fail or succeed? (Maybe monad). Or is this computation of the type that can produce multiple values of the same type? (List monad). Or maybe this computation can produce value of one type or another (Either monad). Or a computation that can interacti with inputs and outputs (IO monad). So these are computations i…

Let me rephrase it in words that I have more intuitive grasp of and see if the translation holds. A monad is a way to pick out the specific constraints of interest and define a class of computations that satisfy those constraints, and how computations within this class compose.

Are the constraints limited to return values or can there be other kinds of constraints?

Re: The Fourier Transform, explained in one sentence (2014)

#108

Post by [deleted] »

TedDoesntTalk wrote: post #60
johnday wrote: post #42

Earlier 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?

[deleted]

Re: The Fourier Transform, explained in one sentence (2014)

#109

Post by chongli »

posix86 wrote: post #73
avip wrote: post #71

Earlier quoted context omitted.

I think you’re confusing proving with understanding . The two are almost orthogonal.

Disagree. If you truly understand something, you must have prooven it to yourself, everything is just a mental help ("donkey bridge" in german) that allows you to remember the statement better - weather it's right or not, you can only know once you've prooven it; and before you did that, it can often happen that your intuition on what's right is actually wrong.

I think a lot of people can understand simple statements in math such as Fermat’s Last Theorem or the Collatz Conjecture but proving them is an entirely different matter. While it may be the case that proving some statements is sufficient to understanding them, I would say it’s never necessary.

Re: The Fourier Transform, explained in one sentence (2014)

#110

Post by antegamisou »

prof-dr-ir wrote: post #69

Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…

At last someone said it, thanks.

It's bothersome how popularized the notion

    I'm one,two,three,...,n blog posts/YouTube videos away from fully grasping this complex mathematical concept which not only requires fundamental knowledge I may not be familiar with, but also solving problems/writing code where at first I'll have no idea how is it supposed to be of any use but over time it will eventually 'click'.
has become.

Beautiful animations and witty narrations are nice, but you're in for a (bad) surprise if you think they're adequate. Also nothing beats good old book -> pencil/paper.

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