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What is meant to be explained in the section between the 2nd paragraph and the folding section in the Sampling sinusoidal functions section shown in this link?
Why is $Nf_0$ written in $f+Nf_0$ instead of frequency in the given equation?
Also, what exactly do I need to understand from the gif on the right?
I think the main take away of this whole section is
If you sample a signal in time it becomes periodic in frequency
Let's look at an example: A sinusoidal signal with a frequency of $f = 1Hz$ sampled at a rate of $f_s = 100Hz$. The spectrum (continuous Fourier Transform) of the original has only two lines: one at 1Hz and -1Hz.
The continuous spectrum of the sampled signal repeats this spectrum every 100Hz. You still have a line spectrum but it's an infinite number of lines at ..., -201Hz, -199Hz, -101Hz, -99Hz, -1Hz, 1Hz, 99Hz, 101Hz, 199Hz, 201Hz, ...
So the line frequencies of the samples are images of the original frequencies repeated at the sampling rate. This can be written as
$$f_N = f+N\cdot f_s$$
The GIF is trying to illustrates what happens when you increase the signal frequency so it approaches the sample rate. When you exceed the Nyquist frequencies the image cross each other. That's aliasing.
Let's increase the frequency to 40Hz: In the interval $[0,f_s]$ we get the original line at 40Hz. The -40Hz line images to 60Hz: so we see 40Hz and 60 Hz.
Let's increase further to 60Hz. The original line is at 60Hz and -60Hz images to 40 Hz. That means we ALSO get a line and 40Hz and 60Hz. In fact the spectra are identical and there no way to tell what the original frequency was!
This would be much easier to see, if the picture would show a larger frequency interval, and not just $[0,f_s]$
