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I understand that a time shift in the time domain produces a corresponding phase change / phase shift in the frequency domain. But I don't understand why the magnitude is unchanged (I am referring to the time shift property here). I am thinking of the following example: Consider An Aperiodic signal, now we observe this signal over the time interval delta t. (and it is my understanding that the longer this interval is , the more accurate the peak of the obtained Fourier transform). But since the signal is aperiodic, this means by definition that it does not repeat itself. So , if we introduce a time shift , then within that time interval delta t that we are observing , a different part of the signal will appear. So how come the magnitude of the Fourier transform is still the same?

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So how come the magnitude of the Fourier transform is still the same?

Because the Fourier Transform integrates from $-\infty$ to $+\infty$. There is no finite "observation window" and you always integrate over the entire signal. The time shift doesn't change that.

Once you consider a finite interval, you are windowing the signal and that does indeed change both magnitude and phase.

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  • $\begingroup$ okay, I understand the argument for infinity, buy practically speaking it isn't possible to observe the signal for infinity and therefore we use the time window in reality. So can I say that in reality the time shift property won't hold? $\endgroup$
    – Hyako Issei
    Jan 17, 2022 at 15:01
  • $\begingroup$ Not really. The time shift properties of the Fourier Transform holds just fine. You can apply Short Term Fourier Transform to chop a long signal into manageable pieces, but that's a different transform with different properties. Once you put numbers in a computer you have discretized the signal and now you are dealing with the Discrete Fourier Transform, which is a completely different animal again. They are all different transforms, with different properties and for different applications. $\endgroup$
    – Hilmar
    Jan 17, 2022 at 17:02
  • $\begingroup$ oh I see, thank you for the explanation. $\endgroup$
    – Hyako Issei
    Jan 17, 2022 at 17:49

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