>one man’s mathematical obsession gave way to a calculation that now underpins much of mathematics and physics
Underpins much of mathematics, science and engineering
What Is the Fourier Transform?
51–60 of 214 posts
Re: What Is the Fourier Transform?
#52Earlier quoted context omitted.
I love the visualization! Thanks for sharing. How do you compute the fractional FT? My guess is by interpolating the DFT matrix (via matrix logarithm & exponential) -- is that right, or do you use some other method?
I am glad you like it! Yes the simplest way to think of it is to exponentiate the dft matrix to an exponent between 0 and 1 (1 being the classic dft). But then the runtime complexity is O(n^2) (vector multiplied with precomputed matrix) or O(n^3) opposed to the O(n log n) of fast fourier transform. There are tricks to do a fast fractional fourier transform by multiplying and convolving with a chirp signal. My impleme…
I made a cool rust fft tui a long time ago too
Re: What Is the Fourier Transform?
#53>one man’s mathematical obsession gave way to a calculation that now underpins much of mathematics and physics Underpins much of mathematics, science and engineering
That is a very true comment. Electrical Engineering for example would be nothing without Laplace (which is more or less an even more general Fourier).
Re: What Is the Fourier Transform?
#54Earlier quoted context omitted.
Same thing! :-) In the purest sense, finite bandwidth requires infinite duration and finite duration requires infinite duration. The real world is somewhere in between. It must involve quantum mechanics (in a way I don't really understand), as maximum bandwidth/minimum wavelength bump up against limits such as the Planck length and virtual particles in a vacuum.
Related, but not quite the same thing. Band-limiting is needed to avoid aliasing. The infinite-duration part (or perfect phase continuity at the boundaries, or a window function...) is needed to avoid Gibbs ringing. An interesting anecdote from Lanczos[1] claims that Michelson (of interferometer fame) observed Gibbs ringing when he tried to reconstruct a square wave on what amounted to a steampunk Fourier analyzer [2…
Can we agree that a fun thing about the Fourier Transform is the many different ways such a simple idea (a liner combination of basis functions) can be viewed and the many subtle implications it has?
For example, one viewpoint is that "Gibbs ringing" is always present if the bandwidth is limited, just that in the "non-aliased" case the sampling points have been chosen to coincide with the zero-crossings of the Gibbs ringing.
I find that my brain explodes each time I pick up the Fourier Transform, and it takes a few days of exposure to simultaneously get all the subtle details back into my head.
Re: What Is the Fourier Transform?
#55Fourier transforms are useful in applications like avoiding moire when downscaling manga and reducing rainbow artifacts in color eink manga. All in kindle comic converter.
Re: What Is the Fourier Transform?
#56Isn’t the Fourier series fundamentals generally a required course in undergraduate college EE field?
Yes, but how many software engineers remember any of that? Most aren't using it.
Re: What Is the Fourier Transform?
#57My favorite algorithm (just for the name) is [Lomb-Scargle Periodogram](https://docs.astropy.org/en/latest/timeseries/lombscargle.ht...) which is a method for performing a Fourier transform on irregularly sampled data.
Re: What Is the Fourier Transform?
#58Earlier quoted context omitted.
Related, but not quite the same thing. Band-limiting is needed to avoid aliasing. The infinite-duration part (or perfect phase continuity at the boundaries, or a window function...) is needed to avoid Gibbs ringing. An interesting anecdote from Lanczos[1] claims that Michelson (of interferometer fame) observed Gibbs ringing when he tried to reconstruct a square wave on what amounted to a steampunk Fourier analyzer [2…
Can we agree that a fun thing about the Fourier Transform is the many different ways such a simple idea (a liner combination of basis functions) can be viewed and the many subtle implications it has? For example, one viewpoint is that "Gibbs ringing" is always present if the bandwidth is limited, just that in the "non-aliased" case the sampling points have been chosen to coincide with the zero-crossings of the Gibbs…
For sure. As I understand it, though, the Gibbs phenomenon arises due to the sinc kernel's infinite support (sinc in Fourier domain = rectangular window in time domain, equivalent to no window at all.)
No amount of precision, no number of coefficients, no degree of lowpass filtering can get around the fact that sin(x)/x never decays all the way to zero. So if you don't have an infinitely-long (or seamlessly repeating) input signal, you must apply something besides a rectangular window to it or you will get Gibbs ringing.
There is always more than one way to look at these phenomena, of course. But I don't think the case can be made that bandlimiting has anything to do with Gibbs.
Re: What Is the Fourier Transform?
#59Reading the comments, it seems there are very few electrical engineers in hacker news.
Maybe they are EE’s, who just happen to still find some joy in the stuff from their intro classes. Sometimes little things can still make us happy, after all.
Re: What Is the Fourier Transform?
#60Earlier quoted context omitted.
Same thing! :-) In the purest sense, finite bandwidth requires infinite duration and finite duration requires infinite duration. The real world is somewhere in between. It must involve quantum mechanics (in a way I don't really understand), as maximum bandwidth/minimum wavelength bump up against limits such as the Planck length and virtual particles in a vacuum.
The Heisenberg uncertainty principle in quantum mechanics comes about precisely because position and momentum are a Fourier transform pair.
In that vein, I've always wondered whether the fact that we live in a fundamentally quantum universe is just a mathematically consistent side effect of the Fourier transform and the Universe's limited extent in space-time. Is the Planck time just the point at which we can't determine the difference in frequency of two signals because the wavelength of their beat frequency has to fit inside the Universe? It's fun to think about.