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What Is the Fourier Transform?

(www.quantamagazine.org)
474 points rbanffy | 3 comments | 04 Sep 25 22:11 UTC | HN request time: 0s | source
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Salgat ◴[04 Sep 25 22:35 UTC] No.45133006[source]▶
Always blew my mind that every signal can be recreated simply by adding different sine waves together.
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esafak ◴[04 Sep 25 22:42 UTC] No.45133064[source]▶
Only if it is band-limited.
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CamperBob2 ◴[04 Sep 25 23:08 UTC] No.45133266[source]▶
And of infinite duration, if you want to split hairs.
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femto ◴[04 Sep 25 23:17 UTC] No.45133325[source]▶
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.

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1. ndriscoll ◴[05 Sep 25 00:14 UTC] No.45133692[source]▶
The Heisenberg uncertainty principle in quantum mechanics comes about precisely because position and momentum are a Fourier transform pair.
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2. femto ◴[05 Sep 25 01:38 UTC] No.45134213[source]▶
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.
3. cycomanic ◴[05 Sep 25 05:32 UTC] No.45135260[source]▶
And you can essentially "observe" the Heisenberg principle when looking at a moving object. If you are observing a plane for example to more accurately know its velocity you will need to observe over a longer time, but if you do this you loose accuracy about its position. This does affect radar systems, who can either send short pulses to accurately pin down the position or long pulses to measure the speed.