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What Is the Fourier Transform?

quantamagazine.org

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Re: What Is the Fourier Transform?

#131

Post by segfault99 »

zozbot234 wrote: post #127
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.

The so-called "Z transform" for discrete sequences is really just a misnomer for the actual method of generating functions (and formal power-series/Laurent-series). You just write a discrete sequence as a power series in z^(-1).

True dat. But you see there's this thing called 'Engineering Maths'. Apparently it's really bad for real mathematicians' blood pressure.

Re: What Is the Fourier Transform?

#132

Post by perching_aix »

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.

You might find LLMs to be a useful crutch for this to an extent, although it's very easy to take the wrong turn and go off into the deep end. But as long as you keep forcefully connecting it back to practical reality, you can get progress out of it. And of course, never actually make it calculate.

Re: What Is the Fourier Transform?

#134

Post by Shadowmist »

armanj wrote: post #75
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.

Years ago, I often struggled to choose between Amazon products with high ratings from a few reviews and those with slightly lower ratings but a large volume of reviews. I used the Laplace Rule of Succession to code a browser extension to calculate Laplacian scores for products, helping to make better decisions by balancing high ratings with low review counts. https://greasyfork.org/en/scripts/443773-amazon-ranking-la…

I always assume that all the ratings are fake when there is a low count of ratings since it is easy for the seller to place a bunch of game orders when they are starting out.

Re: What Is the Fourier Transform?

#135

Post by SillyUsername »

Amazing.

Until I read this article I didn't properly understand Fourier transforms (I didn't know how image compression bitmaps were derived), now it's opened a whole new world - toying with my own compression and anything that can be continuous represented as it's constituent parts.

I can use it for colour quantisation too possibly to determine main and averaged RGB constituents with respect to hue, allowing colour reduction akin to dithering, spreading the error over the wave instead and removing the less frequent elements.

It may not work but it'll be fun trying and learning!

Re: What Is the Fourier Transform?

#136

Post by eric-burel »

What always bothered me when trying to "feel" Fourier transforms is that to compute the oscillations, you need to wait some time. Mathematically, the transformation includes computing integrals. So it's tricky to understand how you compute the Fourier decomposition for a stream. Illustrations always show the whole signal over time but in real life you get the signal progressively. I'd be eager to read more on this.

Re: What Is the Fourier Transform?

#138

Post by CuriouslyC »

armanj wrote: post #75
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.

Years ago, I often struggled to choose between Amazon products with high ratings from a few reviews and those with slightly lower ratings but a large volume of reviews. I used the Laplace Rule of Succession to code a browser extension to calculate Laplacian scores for products, helping to make better decisions by balancing high ratings with low review counts. https://greasyfork.org/en/scripts/443773-amazon-ranking-la…

Just for reference, in case you find yourself in an optimization under uncertainty situation again: The decision-theoretic right way to do this is generate a bayesian posterior over true ranking given ranking count and a prior on true rankings, add a loss function (it can just be the difference between the true rating of the selected item and the true rating of the non-selected item for simplicity) then choose your option to minimize the expected loss. This produces exactly the correct answer.

Re: What Is the Fourier Transform?

#140

Post by yobbo »

eric-burel wrote: post #136

What always bothered me when trying to "feel" Fourier transforms is that to compute the oscillations, you need to wait some time. Mathematically, the transformation includes computing integrals. So it's tricky to understand how you compute the Fourier decomposition for a stream. Illustrations always show the whole signal over time but in real life you get the signal progressively. I'd be eager to read more on this.

As I remember intuitively, it's a convolution over a time window. The size of the time window limits the frequencies that can be detected.
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