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I am reading "Digital Signal Processing" - Proakis and often read that sampling guarantees periodicity (not exact as read)

But I wonder how sampling aperiodic signal will result in periodic repetitions of the aperiodic signal?

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sampling guarantees periodicity

Not exactly. Sampling in one domain guarantees periodicity in the other domain. So sampling in time creates a periodic spectrum and sampling in frequency creates a periodic time signal. Or the other one around: a periodic time signal has a discrete spectrum and a periodic spectrum has a discrete time signal.

That's why there are four different "types" of Fourier Transform. One for each permutations of continuous vs discrete in each domain. See for example http://fourier.eng.hmc.edu/e101/lectures/handout4/node3.html or https://www.dspguide.com/ch8/1.htm

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Aperiodic signal will also have certain band of interest which you actually try to sample. This band of interest repeats itself, which can be observed in frequency domain. Thus, your aperiodicity has now converted into periodicity.

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