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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)

#21

Post by benreesman »

amuresan wrote: post #9
benreesman wrote: post #7

IMHO an under-explored bit of intuition is crops in the frequency domain. If you want to try this for yourself grab a megabyte of photos of “conventionally attractive” selfies and a megabyte of selfies chosen at random and gzip them. I’m personally not all that symmetrical so I don’t compress all that well. But I suspect you’ll find that “pretty” people compress better. Same experiment with pop music vs serious jazz,…

What do you mean by crop in frequency domain? Low pass filtering i.e. removing higher frequency components?

I hope I adequately advertised this as personal metaphysics, certainly I don’t have figures that I can share.

Asymmetry, blemish, teeth a different color than the corpus all consume entropy in the Shannon sense.

In a very real sense, movie stars are the most boring people to take still photos of.

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

#22

Post by bobowzki »

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.

Haha! That's a great joke! It can definitely be applied to a lot of situations.

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

#23

Post by linhns »

thrdbndndn wrote: post #19
munchler wrote: post #17

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.)

I'm currently in uni, learnt about both but has already forgotten what the Laplace transform is about, but still remember Fourier vividly as I got to play with it a lot.

If you read more into it's history, the DFT actually helped a lot during the nuclear arms race as it can detect any nuclear test that happened anywhere except underground.

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

#24

Post by Jensson »

thrdbndndn wrote: post #19
munchler wrote: post #17

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.)

Laplace transform is for exponentials, fourier is for the sine function (ie a complex exponential). So they do the same thing just with different weight functions.

All of this is just linear algebra, with different linear transforms. A fourier transform is a linear transform that transforms point space basis to harmonic oscillation basis.

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

#25

Post by wholinator2 »

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.

What do you mean? A Monad is just a Monoid in the category of Endofunctors.

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

#26

Post by cousin_it »

hgsgm wrote: post #11
onos wrote: post #4

I see this as a confusing explanation of the algorithm for calculating coefficients, not as a line that explains what the Fourier transform is. As for a single line on intuition, can you beat: “method of decomposing a function into a sun of sines and cosines?”

Then you have to get into why you are talking about sines and cosines, adding unnecessary complexity to the explanation. The OP is better. It's the frequency that matters for intuiook, the shape of the periodic function is a technical detail.

Sines and cosines are the best choice here because they're the "elementary periodic function" for a given frequency: the sum of two (possibly scaled and shifted) sinusoids at some frequency is also a (scaled and shifted) sinusoid at the same frequency. For other shapes of periodic function it's not true: for example, if you sum two triangle waves of the same frequency but offset from each other, you'll get a shape that's more complex than a triangle wave. Same for two square waves etc. So the most natural way to decompose a signal is into a sum of sinusoids rather than triangles or squares etc.

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

#27

Post by shreyshnaccount »

I find this kind of math translated to natural language very useful to visualise things. It adds tremendous value for someone like me. The comments saying that if the sentence was enough you wouldn't need the article are in a very "ha gotchu" way and it's counter to both the people who want to understand it and the spirit of HN

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

#28

Post by meindnoch »

Or just say that the (complex) amplitude at a particular frequency is the dot product of the signal and a (complex) sinusoid of that frequency.

Then you’ll immediately understand how the Fourier transform can be just a change of basis (a matrix multiplication in the discrete case).

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

#29

Post by voidhorse »

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.

Unfortunately I can't remember where I saw this explanation. It might've been in the Scientist's and Engineers Guide to DSP book.

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

#30

Post by dlandau »

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.

That was my experience studying statistical physics in Uni. For "some reason" the fourth book I read was the "only one" written in a way that made sense. No relation to the order I read them in, just written better.
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