Is it possible to represent an aperiodic signal using an array of N samples? I am confused about this because you obviously have to window a function in the time domain to sample it.
Now what happens with aperiodic signals?
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$\begingroup$ is this really a question about the DFT and the inherent periodicity that comes from that? $\endgroup$– robert bristow-johnsonCommented Apr 30, 2019 at 22:08
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$\begingroup$ Hi, no. Its just a question about an aperiodic signal of infinite duration, and how we can sample it N times. $\endgroup$– Nash BrewerCommented Apr 30, 2019 at 22:10
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1 Answer
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Whether the signal is periodic or not is largely irrelevant to sampling. What matters is the bandwidth. See this answer: https://dsp.stackexchange.com/a/10339/11256
There are two main cases:
If the signal is aperiodic and of infinite duration (for example, Gaussian noise), then $N$ samples will always be insufficient for reconstruction.
If the signal is aperiodic and of finite duration, then in theory its bandwidth is infinite and it cannot be reconstructed from any set of samples. However, many practical signals tend to zero as $t \rightarrow \pm\infty$, and their spectrum also tend to zero as $f \rightarrow \pm\infty$. In this case, you can obtain a reconstructed signal that is very close to the original, and for engineering applications this is more than enough.
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$\begingroup$ thank you. I am interested in the proof that aperiodic, finite duration signal have an infinite bandwith, if you have the time, can you suggest where I can find a proof for this? $\endgroup$ Commented Apr 30, 2019 at 22:09
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$\begingroup$ this is continuous time signal that is finite in duration? or is the signal already sampled but the number of non-zero samples is finite? $\endgroup$ Commented Apr 30, 2019 at 22:12
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$\begingroup$ Hi, this is just a continuous signal that is aperiodic and infinite in time. From what I learned we have to take a finite window of this signal to sample it, so my question was how can we sample such a signal. The answer provided above was that aperiodic, N samples, will always be insufficient for reconstruction. In the case of a aperiodic energy signal, the bandwidth will be infinite, but it can be reconstructed as an approximation. This is what I understand now. $\endgroup$ Commented Apr 30, 2019 at 22:19
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$\begingroup$ @ArmaArmedAssualt Your understanding is correct. For more information about why finite duration signals have infinite bandwidth, see this answer: dsp.stackexchange.com/a/40624/11256 $\endgroup$– MBazCommented Apr 30, 2019 at 23:40
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$\begingroup$ @ArmaArmedAssualt Here's an intuitive explanation why you can't reconstruct a finite-duration signal from its samples: Say you sample from $t=-\infty$ to $\infty$ at sampling rate $f_s$. The signal starts at an unknown time. When the signal starts, you'll have a sample that is zero followed by another that is not zero. However, you don't know when the signal started; it could have been at any time between the two samples. Without that knowledge, exact reconstruction is obviously impossible. $\endgroup$– MBazCommented May 1, 2019 at 1:28