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I have been studying signals and systems lately and I have came across the following claim:

The uniform sampling of a periodic continuous-time signal may not be periodic!

Can someone please explain why this statement is true?

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If the ratio between your sampling frequency and the frequency of your signal is irrational, you will not have a periodic discrete signal.

Assuming you have a 1-kHz sine wave and you sample at 3000*sqrt(2) Hz. You will have approximately 4.2 samples per period. However you will not be able to sample the sine wave exactly at the same place. Hence your digital signal will not be periodic.

However, if you sampled the same 1-kHz signal at 4 kHz, you would get a periodic discrete signal. The period would be 4 samples.

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    $\begingroup$ And quite interestingly (correct me if I am wrong), since the measure of the rationals is zero, if you sample a continuous periodic signal discretely without knowing its frequency, the probability of getting a periodic discrete signal is zero (theoretically speaking, however in practice due to quantization things won't be so bad). $\endgroup$ – Apollys supports Monica May 21 at 22:18
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    $\begingroup$ @Apollys On the other hand, the rationals are dense in the reals and the lifetime of the Universe is perhaps and ours is certainly bounded, hence getting something close-enough to periodic (though perhaps with a long period) is more than likely - in particular, when the signal and sample are not generated by controlled processes in zero-gravity and near the absolute zero temperatur and whatnot ... $\endgroup$ – Hagen von Eitzen May 22 at 4:26
  • $\begingroup$ Correct me if i am wrong: But when the input singal is 1kHz and you sample with 3.5kHz, you get a periodic signal with a period time of 2ms. To get a periodic signal, f_s does not need to be n*f_in but can be n*f_in/m $\endgroup$ – 12431234123412341234123 May 22 at 10:44
  • $\begingroup$ Yes, the ratio between 3.5 kHz and 1 kHz is rational number, 2/7 i.e not irrational. $\endgroup$ – Ben May 22 at 11:59
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    $\begingroup$ @Apollys : Yes but in some systems they implement a control loop to adjust the sampling frequency to a multiple of the signal of interest frequency. For example in power systems, where the sampling frequency is tracking the grid frequency. This makes some calculations easier, calculating the mean, RMS and harmonics for example. $\endgroup$ – Ben May 22 at 12:05

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