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An interesting thing about a few unique DT signals is that although they may be periodic in the DT domain, they are aperiodic in the CT domain.

One example of such a signal is: $$x[n]=\cos\left(\frac\pi{10}n^2\right)$$ (whose fundamental period is sought out for in this video, at 10:18: https://www.youtube.com/watch?v=AhoeYb6Qq2c).

What are some other examples of periodic DT signals that are aperiodic in CT?

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That might happen to any signal that is sampled without respecting Nyquist's sampling theorem.

The effect of that theorem is that

If your signal has a certain bandwidth, then a real-valued sampling must have at least twice the sampling rate; otherwise, you cannot fully represent your continuous-time signal in discrete-time samples.

\$\cos(\text{const.} \cdot n^2)\$ has infinite bandwidth (simply because the higher you set \$n\$, i.e. the longer you look, the faster the cosine oscillates), and hence, there's no way you can properly represent this continuous signal in discrete time.

So, your signal class is basically "a lot of signals for which you don't respect Nyquist"; that's not a very useful class.

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