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I am performing spectral analysis of a finite length signal that saturates to a non-zero value. The signal ($s(t)$) can, practically, be write as $s(t) = f(t) \big(1-H(t-t_0)\big) $, where $t_0$ is the time at which the signal ends and $H(t)$ is the Heaviside step function. The sudden step from a non-zero value to zero at $t = t_0$ induces artifacts in the analysis.

I am looking for advice/literature on how to deal with signals of this type.

Question: How does one deal with artifacts cause by signals terminating at a non-zero value when performing spectral analysis.

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  • $\begingroup$ What you're looking for is quite literally windowing. I'm not sure how saturation comes into play here? Could you explain what saturation has to do with your rectangularly windowed signal? $\endgroup$ Commented Dec 21, 2020 at 12:04
  • $\begingroup$ In some sense it is equivalent to windowing I guess. I am not exactly sure what you want me to explain. The saturation comes from the fact that the signal is supposed to represent a signal of infinite length which does not return to zero after some time. It is not a signal which passes and when it ends the signal is zero. However, due to computational constraints, I do not have the infinite length signal. The signal s looks like it does without being windowed. $\endgroup$ Commented Dec 21, 2020 at 12:12
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    $\begingroup$ I think saturation means something else than you think it does, sorry. What you mean is windowed. $\endgroup$ Commented Dec 21, 2020 at 12:27
  • $\begingroup$ Maybe the word saturation is not the right one to use. Let me put it this way. How to deal with a signal that has an abrupt end? A signal that does not tend to zero. The end goal is to understand which part of the spectrum is caused by this sudden termination. $\endgroup$ Commented Dec 21, 2020 at 12:42
  • $\begingroup$ as said, you apply a window, like the Nuttal, Blackman-harris, Hann, or Tschebychev windows. en.wikipedia.org/wiki/Window_function That window doesn't have to be rectangular (i.e. just cutting off), but can be (and should be) tapered at the ends $\endgroup$ Commented Dec 21, 2020 at 12:44

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You can & should window your signal. Choice of window depends on the specific requirements of your application. Wikipedia has a really good overview on properties and trade-offs: https://en.wikipedia.org/wiki/Window_function

Keep in mind that you already ARE windowing: your finite length window is simply an infinite length signal multiplied with a rectangular window.

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  • $\begingroup$ Ok, so basically there is no fixed way of fully determining/isolating the artifacts. You have to try different things and make a judgment call. Thanks to you and to Marcus. $\endgroup$ Commented Dec 21, 2020 at 12:58
  • $\begingroup$ well, that "judgement call" can be based on math; see the wikipedia page, and make sure you understand the artifact suppression vs spectral resolution tradeoff to be made. The time–bandwidth product of anything is always limited; that is literally the underlying math of the Heisenberg uncertainty principle. $\endgroup$ Commented Dec 21, 2020 at 13:07

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