81–90 of 214 posts
They do a really good job at breaking down the fundamental knowledge needed to build an understanding.
I have a pet theory that the reason why the FT, and other transforms (generating functions, Mellin/Laplace/Legendre/Haar), are so useful is because many real world functions are sparse and lend themselves to compressed sensing. The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the…
And there's another good reason why so many real-world signals are sparse (as you say) in the FT domain in particular: because so many real-world systems involve periodic motion (rotating motors, fly's wings as you noted, etc). When the system is periodic, the FT will compress the signals very effectively because every signal has to be harmonic of the fundamental frequency.
https://www.youtube.com/watch?v=spUNpyF58BY&list=PL4VT47y1w7...
I have a pet theory that the reason why the FT, and other transforms (generating functions, Mellin/Laplace/Legendre/Haar), are so useful is because many real world functions are sparse and lend themselves to compressed sensing. The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the…
I have a pet theory that the reason why the FT, and other transforms (generating functions, Mellin/Laplace/Legendre/Haar), are so useful is because many real world functions are sparse and lend themselves to compressed sensing. The FT, as are many other transforms, are 1-1, so, in theory, there's no information lost or gained. In many real world conditions, looking at a function in frequency space greatly reduces the…
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…
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 stability to function. Ergo, by this anthropic argument there must be many oscillatory systems.
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…
I would say that the it's very difficult to imagine a world that would not be governed by differential equations. So it's not just that life wouldn't exist it's that there wouldn't be anything like the laws of physics.
As a side note chaotic systems are often better analysed in the FT domain, so even in a world of chaotic systems (and there are many in our world, and I'd argue that if there wasn't life would not exist either) the FT remains a powerful tool