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Is there any signal which is periodic in Continuous Time but not in Discrete Time?

I have this doubt prevailing in me for a long time.

Are all CT periodic signals periodic in DT?

If so, how is sin wave a periodic signal? It is periodic only at intervals of pi radians, which is an irrational number and not an integer.

But for a DT signal to be periodic shouldn't the signal be periodic at discrete intervals of time?

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  • $\begingroup$ Please clarify: do you want to know if a continuous-time periodic signal, when sampled at frequency $f_s$, can possibly become a non-period discrete-time signal? Any constraints on $f_s$? $\endgroup$
    – MBaz
    Nov 2 '20 at 17:28
  • $\begingroup$ @MBaz Yes. Will it be periodic if it is sampled at frequency fs? No constraints $\endgroup$ Nov 2 '20 at 18:01
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A periodic continuous-time signal satisfies $x(t)=x(t+T_0)$ for all $t$. The period $T_0$ doesn't need to be a rational number. A periodic discrete-time signal satisfies $x[n]=x[n+N]$ for all integers $n$. The period $N$ is an integer.

If you sample a periodic continuous-time signal, you don't necessarily get a periodic sequence. E.g., sampling the periodic signal $x(t)=\sin(\omega_0t)$ results in the sequence $x_d[n]=x(nT)=\sin(\omega_0Tn)$, which is only periodic if $\omega_0T=2\pi k/l$ with integers $k$ and $l$.

In general, sampling a $T_0$-periodic continuous-time function with sampling interval $T$ results in a periodic sequence only if

$$kT_0=lT,\qquad k,l\in\mathbb{Z}$$

is satisfied.

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Matt L's answer is correct but it's worth keeping in mind that discrete periodic sequences do not exist in our real world. Based on the traditional definition of periodicity, for a discrete sequence to be periodic it must be infinite in length. And we can't physically have infinite-length sequences in our real world. Periodic sequences are like a perfect circle, one of Euclid's lines having infinite length but zero thickness, and unicorns. They're interesting things to think about but they do not exist in reality.

Rather than using the phrase "periodic sequences" we should discuss "discrete sequences whose sample values repeat over a finite number of samples". I don't have a good (or clever) name for such finite-length sequences.

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  • $\begingroup$ I think you meant "...but they do NOT exist in reality" in the last line of your first paragraph? $\endgroup$
    – Gilles
    Nov 3 '20 at 12:10
  • $\begingroup$ @Giles. You are correct. (Thanks.) I've inserted the missing word "not". $\endgroup$ Nov 3 '20 at 17:42
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    $\begingroup$ Well, yes, but continuous periodic sequences don't exist in the real world, for the same reason -- time is not infinite. Since you brought up unicorns -- a pixellated unicorn is no more or less real than a continuous one. $\endgroup$
    – TimWescott
    Nov 3 '20 at 17:50
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    $\begingroup$ @TimWescott. OK. So you and I agree that periodic discrete sequences do not exist. How confusing that notion must be for DSP beginners who are regularly told that "N-point sequences applied to the DFT are periodic". $\endgroup$ Nov 4 '20 at 7:02

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