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An interactive guide to the Fourier transform (2012)

betterexplained.com

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Re: An interactive guide to the Fourier transform (2012)

#11

Post by duped »

earlymodernlvr wrote: post #5

Request for someone to make an intuitive explanation of why the Fourier Transform is (almost) it's own inverse. I know the math proof from taking analysis, but the formula is too pretty and symmetrical for the explanation to be so technical. Same for why it preserves L2 norm.

Same answer for both. It's an orthogonal transform, aka a change of basis. You're conceptually rotating the function/series to an equivalent one that's orthogonal to the original. The magnitude/energy hasn't changed (Parseval's theorem is a more succinct definition). And to perform the inverse transform you need to conceptually rotate it back to the original, which should mirror the original transform very nicely.

If it didn't preserve energy then there wouldn't necessarily be an inverse transform, since that implies information was lost.

Re: An interactive guide to the Fourier transform (2012)

#12

Post by sampo »

sorenjan wrote: post #10

The short explanation of the DFT that I like the most is that it projects the signal vector on to the vector room with the exponential functions as basis vectors. Then you can see how much of the signal that each basis vector, corresponding to a frequency, can explain. It's intuitive to see if you start with a 2D vector space (the regular euclidian plane) and a 2D vector, and then you can expand the definition of vec…

> vector room

Is Swedish, "rum" means both space, and a room. In English, "vector space" is used.

Re: An interactive guide to the Fourier transform (2012)

#13

Post by [deleted] »

segfaultbuserr wrote: post #9
chasil wrote: post #7

It would also be nice to have an understanding of its relation to the Laplace transform, something more than saying that the real component goes to zero.

+1. Pop-sci explanations of the frequency domain and Fourier transform are too many already, but does anyone know if there's a similar physical interpretation or visualization of the Laplace transform and the S domain?

[deleted]

Re: An interactive guide to the Fourier transform (2012)

#14

Post by kkapelon »

It is super to easy to find resources on HOW the FT works. But I find it very difficult to find resources on WHY we need it and WHERE it is useful to have.

Does anybody have some good sources that explain the practical applications and how it is useful on real world usage?

Re: An interactive guide to the Fourier transform (2012)

#15

Post by amelius »

sorenjan wrote: post #10

The short explanation of the DFT that I like the most is that it projects the signal vector on to the vector room with the exponential functions as basis vectors. Then you can see how much of the signal that each basis vector, corresponding to a frequency, can explain. It's intuitive to see if you start with a 2D vector space (the regular euclidian plane) and a 2D vector, and then you can expand the definition of vec…

Minor point, it's not a projection because you don't lose dimensions.

Re: An interactive guide to the Fourier transform (2012)

#16

Post by CaptainFever »

kkapelon wrote: post #14

It is super to easy to find resources on HOW the FT works. But I find it very difficult to find resources on WHY we need it and WHERE it is useful to have. Does anybody have some good sources that explain the practical applications and how it is useful on real world usage?

From TFA:

Stop. Here's where most tutorials excitedly throw engineering applications at your face. Don't get scared; think of the examples as "Wow, we're finally seeing the source code (DNA) behind previously confusing ideas".

If earthquake vibrations can be separated into "ingredients" (vibrations of different speeds & amplitudes), buildings can be designed to avoid interacting with the strongest ones.

If sound waves can be separated into ingredients (bass and treble frequencies), we can boost the parts we care about, and hide the ones we don't. The crackle of random noise can be removed. Maybe similar "sound recipes" can be compared (music recognition services compare recipes, not the raw audio clips).

If computer data can be represented with oscillating patterns, perhaps the least-important ones can be ignored. This "lossy compression" can drastically shrink file sizes (and why JPEG and MP3 files are much smaller than raw .bmp or .wav files).

If a radio wave is our signal, we can use filters to listen to a particular channel. In the smoothie world, imagine each person paid attention to a different ingredient: Adam looks for apples, Bob looks for bananas, and Charlie gets cauliflower (sorry bud).

The Fourier Transform is useful in engineering, sure, but it's a metaphor about finding the root causes behind an observed effect.

Re: An interactive guide to the Fourier transform (2012)

#18

Post by sizzzzlerz »

Just a nit. The pair of equations the author showed at the beginning of the article are not the equations of the Fourier Transform and its inverse. The Transform is a continuous function operating on an infinite input. The equation for the Transform involves the use of the integral taken over +/- infinity. What is shown, using the summation operator, is a discrete form on the Transform where the input is a limited time series.

A good alternative explanation can be found on Grant Sanderson's 3 Blue 1 Brown channel https://youtu.be/spUNpyF58BY?si=uqq2OOSATYcWmaG8

Re: An interactive guide to the Fourier transform (2012)

#19

Post by sizzzzlerz »

kkapelon wrote: post #14

It is super to easy to find resources on HOW the FT works. But I find it very difficult to find resources on WHY we need it and WHERE it is useful to have. Does anybody have some good sources that explain the practical applications and how it is useful on real world usage?

Your cell phone wouldn't exist without the Fourier Transform, or the discrete fourier transform, to be correct. Image compression is another application, albeit, a 2-dimensional version. Software defined radios, or SDR, are completely dependent on the DFT. Radar processing. Earthquake analysis. The list goes on and on. Basically, our technological society would not exist in its current form without the fourier transform. To me, it is one of the key mathematical algorithms of the 20th century.

Re: An interactive guide to the Fourier transform (2012)

#20

Post by sheepshear »

chasil wrote: post #7

It would also be nice to have an understanding of its relation to the Laplace transform, something more than saying that the real component goes to zero.

Maybe this? https://youtu.be/iP4fckfDNK8

It shows how the FT is a 2D slice of the 3D LT in the s-domain.

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