Timeline for answer to Why does the frequency spectrum "fill in" under an existing envelope when you increase the period to go from Fourier Series to Fourier Transform? by Marcus Müller
Current License: CC BY-SA 4.0
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| 4 hours ago | comment | added | TimWescott | You don't have to just make the claim that $\mathcal F \left \{ e^{j 2 \pi f t} \right \} = \delta(\omega - 2 \pi f)$ and then backfill by showing it makes sense. You can time-limit the exponential to a rectangular pulse of duration $T$, or you can multiply it by a $\mathrm{sinc} \frac t T$, then find that in the limit as $T \to \infty$ you end up with something that acts like $\delta (\omega - 2 \pi f)$. It's a good exercise if you're feeling like the Fourier transform of a continuous sine wave isn't philosophically valid. | |
| 10 hours ago | history | answered | Marcus Müller | CC BY-SA 4.0 |