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What Is the Fourier Transform?

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474 points rbanffy | 2 comments | 04 Sep 25 22:11 UTC | HN request time: 0.001s | 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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anvuong ◴[04 Sep 25 23:31 UTC] No.45133418[source]▶
You are being confused with #samples needed for perfect reconstruction, i.e. Nyquist sampling frequency. Fourier series/transforms work regardless of the bandwidth of the signal, as long as the integral exists, i.e. it must vanish at infinity.

Essentially it's just projection in infinite-dimensional vector spaces.

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1. esafak ◴[04 Sep 25 23:52 UTC] No.45133565[source]▶
That's what is commonly understood by reconstruction: perfect reconstruction. And for that you need a band-limited signal. Otherwise he would have said approximate- or lossy reconstruction.
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2. nomel ◴[05 Sep 25 22:37 UTC] No.45144491[source]▶
> And for that you need a band-limited signal.

Luckily, we live in a physical universe, where such mathematical oddities, like infinite bandwidth signals, cannot exist, so this isn't an actual issue. Any signal that that contains infinite bandwidths only exists because it has sampling artifacts. You would, necessarily, be attempting to reconstruct errors. There are many "tricks" around dealing with such flawed signals. But yes, you can't fully reconstruct impossible signals with FFT.