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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#151

Post by Sharlin »

zwnow wrote: post #126
segfault99 wrote: post #69

Earlier quoted context omitted.

When I did EE, didn't have access to any kind of computer algebra system. Have 'fond' memories of taking Laplace transform transfer functions and converting to z-transform form. Expand and then re-group and factor. Used a lot of pencil, eraser and line printer fanfold paper for doing the very basic but very tedious algebra. Youngsters today don't know how lucky.. (ties onion to belt, etc., etc.)

No worries, as a self proclaimed youngster I didn't manage to understand Fourier in 2 days and never bothered again. Also had no other prior knowledge to algebra so maybe that's why I struggled. Never perceived algebra as useful in anything programming related, will continue to do so as most problems are solvable without it. I'll let the degree havers do all that stuff.

> Never perceived algebra as useful in anything programming related

Image, video, and audio processing and synthesis, compression algorithms, 2D and 3D graphics and geometry, physics simulation, not to mention the entire current AI boom that's nothing but linear algebra… yeah, definitely algebra isn't useful in anything programming related.

Re: What Is the Fourier Transform?

#152

Post by kragen »

jcgrillo wrote: post #143
kragen wrote: post #103

Earlier quoted context omitted.

Differential equations aren't limited to describing stable systems, though, and there are chaotic systems that are also in some sense stable. Ordinary differential equations can describe any system with a finite number of state variables that change continuously (as opposed to instantaneously jumping from one state to another without going through states in between) and as a function of the system's current state (as…

One amazing application of spectral theory I always harp on when this topic comes up is Chebfun[1]. Trefethen's Spectral Methods in Matlab is also wonderful. [1] http://www.chebfun.org/

I haven't read it! Thanks for the recommendation!

Re: What Is the Fourier Transform?

#153

Post by acjohnson55 »

lutusp wrote: post #77

Such a shame. In an otherwise well-written article, the author mentions Cooley and Tukey's discovery of the FFT, but without mentioning that Gauss discovered it first, among others, each of whom approached the same idea from different directions. The Wikipedia FFT article ( https://en.wikipedia.org/wiki/Fast_Fourier_transform ) credits Gauss with originating the FFT idea later expanded on by others, and correctly des…

As they say, everything in modern math is named for the 2nd person to discover it after Gauss.

Re: What Is the Fourier Transform?

#154

Post by dsego »

zwnow wrote: post #126
segfault99 wrote: post #69

Earlier quoted context omitted.

When I did EE, didn't have access to any kind of computer algebra system. Have 'fond' memories of taking Laplace transform transfer functions and converting to z-transform form. Expand and then re-group and factor. Used a lot of pencil, eraser and line printer fanfold paper for doing the very basic but very tedious algebra. Youngsters today don't know how lucky.. (ties onion to belt, etc., etc.)

No worries, as a self proclaimed youngster I didn't manage to understand Fourier in 2 days and never bothered again. Also had no other prior knowledge to algebra so maybe that's why I struggled. Never perceived algebra as useful in anything programming related, will continue to do so as most problems are solvable without it. I'll let the degree havers do all that stuff.

Have I got a video for you.

gingerBill – Tools of the Trade – BSC 2025 https://youtu.be/YNtoDGS4uak

Re: What Is the Fourier Transform?

#155

Post by analog31 »

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.

My control theory professor (who was also my physics advisor -- it was a small college) explained it like this: Physicists like Fourier transforms because they go from minus to plus infinity, like the universe. Control engineers like Laplace transforms because they start at zero, and a control system also has a starting point.

Re: What Is the Fourier Transform?

#156

Post by drmpeg »

cycomanic wrote: post #99

While we are talking about fun facts of the Fourier transform, the transfer function of the DFT is a sinc. That's how OFDM works, it performs an FFT on parallel input symbol streams. If you look at the spectrum of your signal it's sinc (approximately because they are truncated) channels spaced at 1/symbol rate which don't exhibit (theoretically if they weren't truncated) any interchannel interference.

It's going from the frequency domain (parallel symbol streams) to the time domain (multi-carrier RF), so it's using the inverse FFT.

Re: What Is the Fourier Transform?

#158

Post by HarHarVeryFunny »

astrange wrote: post #64

> A compression algorithm can then remove high-frequency information, which corresponds to small details, without drastically changing how the image looks to the human eye. I slightly object to this. Removing small details = blurring the image, which is actually quite noticeable. For some reason everyone really wants to assume this is true, so for the longest time people would invent new codecs that were prone to thi…

Well, there are use cases for lossy compression as well as non-lossy, and nobody is saying they are the same. If you really need to heavily compress to reduce file size or transmission bandwidth then you'll likely need to use a lossy CODEC, so the question then becomes how can you minimize the reduction in perceived quality of whatever you are compressing (photos, video, audio), which comes down to how these various human sensory/perceptual systems work.

For vision we are much more sensitive to large scale detail (corresponding to low frequency FFT components) than fine scale detail (corresponding to high frequency components), so given the goal of minimizing reduction in perceived quality this is an obvious place to start - throw away some of that fine detail (highest frequency FFT components), and it may not even be noticeable at all if you are throwing away detail at a higher level of resolution than we are able to perceive.

It turns out that human vision is more sensitive to brightness than color (due to numbers of retinal rods vs cones, etc), so compression can also take advantage of that to minimize perceptual degradation, which is what JPEG does - first convert the image from RGB to YUV color space, where the Y component corresponds to brightness and the U,V components carry the color information, then more heavily compress the color information than brightness by separately applying FFT (actually DCT) to each of the Y,U,V components and throwing away more high frequency (fine detail) color information than brightness.

But, yeah, there is no magic and lossy compression is certainly going to be increasingly noticeable the more heavily you compress.

Re: What Is the Fourier Transform?

#159

Post by arethuza »

arethuza wrote: post #117
kmarc wrote: post #115

Earlier quoted context omitted.

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…

To be fair (and because I've just remembered - it was ~40 years ago) we did get some practical stuff covered in the maths part of my CS degree in the application of group theory (groups, rings & fields) to coding theory.

Re: What Is the Fourier Transform?

#160

Post by zwnow »

Sharlin wrote: post #151
zwnow wrote: post #126

Earlier quoted context omitted.

No worries, as a self proclaimed youngster I didn't manage to understand Fourier in 2 days and never bothered again. Also had no other prior knowledge to algebra so maybe that's why I struggled. Never perceived algebra as useful in anything programming related, will continue to do so as most problems are solvable without it. I'll let the degree havers do all that stuff.

> Never perceived algebra as useful in anything programming related Image, video, and audio processing and synthesis, compression algorithms, 2D and 3D graphics and geometry, physics simulation, not to mention the entire current AI boom that's nothing but linear algebra… yeah, definitely algebra isn't useful in anything programming related.

Yea that's not what I work with my guy
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