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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#41

Post by yshklarov »

As everyone in this thread is sharing links, I'm gonna pitch in, too.

This lecture by Dennis Freeman from MIT 6.003 "Signals and Systems" gives an intuitive explanation of the connections between the four popular Fourier transforms (the Fourier transform, the discrete Fourier transform, the Fourier series, and the discrete-time Fourier transform):

https://ocw.mit.edu/courses/6-003-signals-and-systems-fall-2...

Re: What Is the Fourier Transform?

#42

Post by alphazard »

There's a lot of great Fourier visualizations out there (3Blue1Brown has a series on it as well). A great intuition pump would be a game where the player has to recreate a waveform given to them, by toggling on/off switches for each fundamental frequency (and phase).

Has anyone ever seen something like that?

Re: What Is the Fourier Transform?

#43

Post by sizzzzlerz »

hbarka wrote: post #39

Isn’t the Fourier series fundamentals generally a required course in undergraduate college EE field?

Of course. It's taught in both mathematics courses as well as engineering. The Fourier transform and it's digital domain cousin, the discrete Fourier transform play such a fundamental role across nearly every engineering discipline as well as physics and many other scientific areas, you cannot get through school without learning about them.

Re: What Is the Fourier Transform?

#44

Post by CamperBob2 »

femto wrote: post #21
CamperBob2 wrote: post #17

Earlier quoted context omitted.

And of infinite duration, if you want to split hairs.

Same thing! :-) In the purest sense, finite bandwidth requires infinite duration and finite duration requires infinite duration. The real world is somewhere in between. It must involve quantum mechanics (in a way I don't really understand), as maximum bandwidth/minimum wavelength bump up against limits such as the Planck length and virtual particles in a vacuum.

Related, but not quite the same thing. Band-limiting is needed to avoid aliasing. The infinite-duration part (or perfect phase continuity at the boundaries, or a window function...) is needed to avoid Gibbs ringing.

An interesting anecdote from Lanczos[1] claims that Michelson (of interferometer fame) observed Gibbs ringing when he tried to reconstruct a square wave on what amounted to a steampunk Fourier analyzer [2]. He reportedly blamed the hardware for lacking the necessary precision.

1: https://math.univ-lyon1.fr/wikis/rouge/lib/exe/fetch.php?med...

2: https://engineerguy.com/fourier/pdfs/albert-michelsons-harmo...

Re: What Is the Fourier Transform?

#46

Post by antimora »

I think the best explanation and understanding I received about Fourier Transform was from studying linear algebra.

The Fourier Transform equation essentially maps a signal from the time domain onto orthogonal complex sinusoidal basis functions through projection.

And the article does not even mention this. =)

Re: What Is the Fourier Transform?

#47

Post by ThrowawayR2 »

Kind of a tangent but why are there so many articles and videos popularizing the Fourier transform and practically none for the Laplace transform? The first person to do a well done 3Blue1Brown style video that focuses on intuition and visualization would probably be an overnight sensation (well, among engineers at least).

Re: What Is the Fourier Transform?

#48

Post by esalman »

antimora wrote: post #46

I think the best explanation and understanding I received about Fourier Transform was from studying linear algebra. The Fourier Transform equation essentially maps a signal from the time domain onto orthogonal complex sinusoidal basis functions through projection. And the article does not even mention this. =)

Almost every paragraph of this article says the same thing but in different ways.

Re: What Is the Fourier Transform?

#50

Post by hovden »

laszlokorte wrote: post #26

Shameless plug: If you are interested in Fourier Transform and signal processing you might enjoy my somewhat artistic 3D visualisation of the fourier transform as well as the fractional fourier transform [1] (Fractional fourier transform on the top face of the cube) And for short time fourier transform showing how a filter kernel is shiftes across the signal. [2] [1]: https://static.laszlokorte.de/frft-cube/ [2]: htt…

If I might also plug ‘the Atlas of Fourier Transforms’. If your interested in understanding building intuition of symmetry and phase in fourier space, the book illustrates many structures.
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