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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#111

Post by dsego »

alphazard wrote: post #42

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?

I created a small piece trying to build intuition, my approach was to think of it as probing with test signals, so instead of recreating a waveform, it's more poking at it with sines and seeing how well the test sine matches the target signal.

https://dsego.github.io/demystifying-fourier/

Re: What Is the Fourier Transform?

#112

Post by energy123 »

RachelF wrote: post #98
yshklarov wrote: post #41

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...

I wonder what happened to Wavelet transforms? The were very popular years ago, and now one never hears about them.

The use-case is slightly different. Wavelets are suited for non-stationary signals, while Fourier Transform has no time localization so it's more for stationary signals. Although short-time Fourier transform exists, which can handle non-stationary signals under the assumption of local stationarity.

Also, a property of wavelets is they're non-parametric, which limits their utility in knowledge discovery applications.

For ML applications, my opinions is that they're somewhat superseded by deep learning methods that apply less restrictive inductive bias. As data grows, the restrictive prior assumptions of wavelets will hurt, sort of like how CNN is being abandoned for ViT, even though CNN can outperform in situations where data is scarce.

So overall, they have a pretty small set of usecases where they're more suited than other alternative tools.

Re: What Is the Fourier Transform?

#113

Post by arethuza »

anyfoo wrote: post #33

If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform). This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.

When I think of Laplace Transforms I always think of control theory - poles, zeros etc.

Re: What Is the Fourier Transform?

#114

Post by zkmon »

The most shocking thing for me was the realization that frequency and wavelength are kind of dual forms. Ofcourse, every concept and thing that comes into existence does so only in the context of it's opposite or dual form. But recognizing that form is genius. 3b1b video goes a bit deeper.

Also a related fact is that the rate of change of a sine wave is itself shifted by pi/2, while the exponential curves need no shifting. I'm not a professional in these matters, but I guess there is a deeper connection between the rate of change and the static world.

Re: What Is the Fourier Transform?

#115

Post by kmarc »

arethuza wrote: post #113
anyfoo wrote: post #33

If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform). This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.

When I think of Laplace Transforms I always think of control theory - poles, zeros etc.

Probably that's why we are learning about it in the "Control Theory" classes at university. :-)

Jokes aside, I graduated as "Computer Engineer" (BSc) and then also did a "Master in Computer Science"; I was (young and) angry at the universe why soooo many classical engineering classes and then theory I had to sit through (Control theory, Electrical engineering, Physics), and we never learned about the cool design patterns etc etc.

Today I see that those formative years helped me a lot with how I develop intuition when looking at large (software) systems, and I also understand that those ever changing best design patterns I can (could have) just look up, learn, and practice in my free time.

I wish a today-me would have told my yesterday-me all this.

Re: What Is the Fourier Transform?

#117

Post by arethuza »

kmarc wrote: post #115
arethuza wrote: post #113

Earlier quoted context omitted.

When I think of Laplace Transforms I always think of control theory - poles, zeros etc.

Probably that's why we are learning about it in the "Control Theory" classes at university. :-) Jokes aside, I graduated as "Computer Engineer" (BSc) and then also did a "Master in Computer Science"; I was (young and) angry at the universe why soooo many classical engineering classes and then theory I had to sit through (Control theory, Electrical engineering, Physics), and we never learned about the cool design patt…

I learned about it after I graduated with a CS degree - I mean in true university degree fashion we'd been taught about Laplace and Z transforms (and related things) but with no practical applications.

After graduating I joined an academic research team based mainly in a EE department who were mainly Control Engineers - we were mainly doing stuff around qualitative reasoning and using it for fault diagnosis, training etc.

Re: What Is the Fourier Transform?

#118

Post by jojobas »

anyfoo wrote: post #33

If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform). This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.

Then there's the whole mindfuck of fractional order Fourier (and other) transforms.

Re: What Is the Fourier Transform?

#120

Post by hackandthink »

abdullahkhalids wrote: post #88
mitthrowaway2 wrote: post #83

Earlier quoted context omitted.

As mentioned by other commenters, a reason for the FT's dominance in particular is because sine, cosine, and complex exponentials are the eigenfunctions of the derivative operator. Since so many real-world systems are governed by differential equations, the Fourier Transform becomes a natural lens to analyze these systems. Sound waves are one (of many) examples. And there's another good reason why so many real-world…

The question is why "so many real-world systems are governed by differential equations" and "so many real-world systems involve periodic motion". Well, stable systems are can either be stationary or oscillatory. If the world didn't contain so many stable systems, or equivalently if the laws of physics didn't allow so, then likely life would not have existed. All life is complex chemical structures, and they require s…

Natura non facit saltus.

https://en.wikipedia.org/wiki/Natura_non_facit_saltus

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