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2

They are pretty much the same thing and there is no material difference. The difference in scale factor just comes from the initial approach. The DFT starts with "what's the frequency content of a discrete periodic time signal?" whereas the DTFS starts with "how do I make a discrete periodic time signal out of complex exponentials?" If ...

1

Okay, this function $$x(t)=\sum\limits_{n=-\infty}^{\infty}\operatorname{rect}(t-n)$$ is identically equal to one: $x(t) = 1 \quad \forall t\in\mathbb{R}$ because $$\operatorname{rect}(x) \triangleq \begin{cases} 1,\quad & |x|<\tfrac12 \\ \tfrac12, \quad & |x|=\tfrac12 \\ 0, \quad & |x|>\tfrac12 \\ \end{cases}$$ Now, if these rectangular ...

3

Why does repeating periodically a signal means removing some frequency intervals from the single pulse spectrum? Because your signal is periodic. A periodic signal only contains frequencies that are integer multiples of the signal's repetition frequency. In other words, if the signal is periodic all of it's constituent parts must be periodic as well. In ...

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