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What Is the Fourier Transform?
101–110 of 214 posts
Re: What Is the Fourier Transform?
#102If 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…
While this is a good idea, I think it's unrelated to the Laplace transform except that they're named after the same dude?
Re: What Is the Fourier Transform?
#103Earlier 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…
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 opposed to nondeterministically or under the influence of the past or future or some kind of supernatural entity).
Partial differential equations extend this to systems with infinite numbers of variables as long as the variables are organized in the form of continuous "fields" whose behavior is locally determined in a certain sense—things like the temperature that Fourier was investigating, which has an infinite number of different values along the length of an iron rod, or density, or pressure, or voltage.
It turns out that a pretty large fraction of the phenomena we experience do behave this way. It might be tempting to claim that it's obvious that the universe works this way, but that's only because you've grown up with the idea and never seriously questioned it. Consider that it isn't obvious to anyone who believes in an afterlife, or to Stephen Wolfram (who thinks continuity may be an illusion), or to anyone who bets on the lottery or believes in astrology.
But it is at least an excellent approximation that covers all phenomena that can be predicted by classical physics and most of quantum mechanics as well.
As a result, the Fourier and Laplace transforms are extremely broadly applicable, at least with respect to the physical world. In an engineering curriculum, the class that focuses most intensively on these applications is usually given the grandiose title "Signals and Systems".
Re: What Is the Fourier Transform?
#104As 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.
Really, do you think they've somehow fallen out of favor? If so, that's a surprise to me.
In any case, they are a bit more advanced, and out of scope for the undergraduate course I linked to.
Re: What Is the Fourier Transform?
#105Earlier 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…
> Well, stable systems are can either be stationary or oscillatory.
In practice this is probably true, but I can see another possibility. The system could follow a trajectory that bounces around endlessly in some box without ever repeating or escaping the box.
Re: What Is the Fourier Transform?
#106Such 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…
Yes, bad article to omit that. It is such a cool fun fact. Gauss was unreal.
Re: What Is the Fourier Transform?
#107This is maybe a good first thing to read if you've never heard of the Fourier Transform before, but it makes it sound a great deal more arbitrary and random than it actually is. It might set your understanding back by giving you the illusion that you understand things you don't actually understand, and that would be a shame, because some of those things are more beautiful than a sunrise or a hummingbird.
It would be very sad to lose those treasures by passing them by because you thought you already had them. As the song says:
> When you're young, you're always looking
> On the far side of the hill;
> You might miss the fairest flower
> Standing by your side very still.
And the flowers of Fourier analysis are as fair as the fairest flowers in this universe, or any universe.
https://news.ycombinator.com/item?id=45134843 may be a clue to the hidden beauty, for those who are puzzled as to what I might be talking about.
Re: What Is the Fourier Transform?
#108IANAMathematician, but I've tried to come up with a metaphor which I hope isn't too wrong: 1. You've start with a signal fluctuating going up and down, and it's on a strip of little LEDs labeled from -1 to +1. 2. You mount that strip to a motor, and spin it at a certain rate. After a while the afterimages make a blob-shape. 3. For each rotation rate, measure how much the shape appears off-center. In this way you can…
Sounds like a variation on the mechanical strobe tuner.
Re: What Is the Fourier Transform?
#109This is maybe a good first thing to read if you've never heard of the Fourier Transform before, but it makes it sound a great deal more arbitrary and random than it actually is. It might set your understanding back by giving you the illusion that you understand things you don't actually understand, and that would be a shame, because some of those things are more beautiful than a sunrise or a hummingbird. It would be…
As usual, 3 Blue 1 Brown delivers: https://youtu.be/spUNpyF58BY?si=nSqHf_3zbhyu9YGd
Re: What Is the Fourier Transform?
#110This is maybe a good first thing to read if you've never heard of the Fourier Transform before, but it makes it sound a great deal more arbitrary and random than it actually is. It might set your understanding back by giving you the illusion that you understand things you don't actually understand, and that would be a shame, because some of those things are more beautiful than a sunrise or a hummingbird. It would be…
As usual, 3 Blue 1 Brown delivers: https://youtu.be/spUNpyF58BY?si=nSqHf_3zbhyu9YGd
This is a much better introduction than the article, and it's only 20 minutes.