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[Phys-l] some questions related to sampling

I'm ultimately going somewhere with this, but feel I need to get some things clear in my head first. While it may not be the ideal starting place, I want to begin with this article:

You will have to read/skim the article for context. I'm not sure whether some of my confusion is from being dense, or just reading an article that is not well-written/organized. In particular, I am interested in this quote from the article:

The DFT and the DTFT can be viewed as the logical result of applying the standard continuous Fourier transform to discrete data. From that perspective, we have the satisfying result that it's not the transform that varies, it's just the form of the input:
If it is discrete, the Fourier transform becomes a DTFT.
If it is periodic, the Fourier transform becomes a Fourier series.
If it is both, the Fourier transform becomes a DFT.

I'm not getting the *significance* of the difference between the DTFT and the DFT. There seems to be a lot of mathematical effort to distinguish the two. The statement above implies that the DFT is a particular case of the DTFT.

I have been applying the DFT to sampled data (specifically a single-pulse, hence non-periodic, waveform) without any issues AFAICT, but then I ran across this article and out of curiosity am trying to understand if it has any true significance. To put it another way, if I am fourier-transforming discrete data, who cares if it merely discrete, or discrete and periodic? Who cares about the semantics of DFT or DTFT?

Like I said, I'm going somewhere with this (other questions), but just want to get some other thoughts before I try to explain further.

Stefan Jeglinski