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If I consider a generic aperiodic signal $x(t)$, how can I prove that rapid changes in signal correspond to high frequencies?

Thank you for your time.

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    $\begingroup$ Exactly how are you defining "rapid changes"? $\endgroup$ – hotpaw2 Sep 1 '17 at 17:36
  • $\begingroup$ Hi @hotpaw2, rapid change=high amplitude of derivative. $\endgroup$ – Gennaro Arguzzi Sep 1 '17 at 18:32
  • $\begingroup$ Here's a rough handwavy explanation: rapid changes in a signal imply periods where its time derivative is large relative to the "steady state" value. Higher-frequency components have larger first derivatives, so they are needed for the signal to change quickly. $\endgroup$ – Jason R Sep 1 '17 at 22:08
  • $\begingroup$ Hi @JasonR can you explain me your last sentence please? $\endgroup$ – Gennaro Arguzzi Sep 1 '17 at 22:25
  • $\begingroup$ The amplitude of the first derivative of a sinusoid of frequency $f$ Hz is proportional to $f$. This makes sense; if the period repeats more quickly, it has to change at a faster rate. $\endgroup$ – Jason R Sep 1 '17 at 22:32
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One might argue that the Dirac delta $\delta(t)$ , is a signal that exhibits maximum change because all frequencies are uniformly increased, not just the high frequencies

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