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We have a signal $s(t)$.

  1. If we do an upsampling, does the signal duration increase?
  2. What is the point of upsampling if the signal time increases?
  3. Can we do an upsampling if we don't use a shaping filter?
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No, if you upsample a signal, its duration (i.e. measured in real-world units like "seconds") does not change.

  • If your signal $s[n]$ is sampled at a rate $f_s$ Hz, and it is $N$ samples long, then the total duration of the signal is $\frac{N}{f_s}$ seconds.

  • Say you upsample the signal by a factor $M$, yielding a new signal $s_u[n]$. It will be $MN$ samples long, but it will be sampled at a new rate $Mf_s$. Its duration is therefore $\frac{MN}{Mf_s} = \frac{N}{f_s}$. You can see that the signal duration is the same as $s[n]$.

It's not clear what you mean by a "shaping filter" in your third question.

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  • $\begingroup$ In the third question, 'shaping filter' is for example a root raised cosine filter: Can we do an upsampling if we don't use a shaping filter (for example a root raised cosine filter)? $\endgroup$ – user39389 Dec 8 '18 at 19:25
  • $\begingroup$ upsampling has not necessarily something to do with pulse shaping, so yes. $\endgroup$ – Marcus Müller Dec 9 '18 at 0:06

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