I want to understand spectral leakage.
I understand that whenever we feed $N$ time-samples of a periodic, continuous, signal into a FFT algorithm we are multiplying in time-domain the true periodic, continuous, signal with a rectangular window, this resulting in a convolution in the frequency domain of the true signal's FT with the rectangular window's FT. I also know that the FT of a square function is a sinc function.
I do not understand the idea of having an integer number of cycles within the time period covered by those $N$ time-samples of the continuous signal we feed into the FFT. I want to find out why if we have a (windowed by a square due to the finite value of $N$) pure sinusoid (one frequency only) as input to the FFT, and if this length-$N$-signal contains an integer number of sinusoid's cycles within the time range it covers, then there is no spectral leakage.
What I think is that in the frequency-domain we shall get a sinc function extending to $\pm \infty$, with its main lobe centered at the frequency of the sinusoid. So there is going to be non-zero counts in all the bins across the (horizontal) frequency axis. This is just how life is due to the finite $N$.
Does each bin from across the returned-from-the-FFT frequency axis ''summons'' / ''displays'' a sinc function with its main lobe centered on that specific bin? That would be, if we gather $N=10$ bins from the FFT algo because we fed in a length $N=10$ signal, we will have a superposition of $N=10$ sinc functions in a spectrum plot?
If somehow I manage to make the returned-from-the-FFT frequency 1D array to contain in its components the frequency of my sinusoid, will I obtain ONLY 1 sinc function appearing on the spectrum plot? That sinc function will have its main lobe centered on the frequency of interest.
It seems from the above two points that there is going to be spectral leakage to other-than-the-center-lobes of that only sinc function appearing in the spectogram, just by the nature of things. Is the explanation for not seeing this leakage in the spectrum plot the fact that the sinc function's side lobes which still contain a reasonable count-value (so which doesn't tend to 0, but it's still visible with the eye) are at such frequencies that they don't reach the next nearby bin? In other words, the very next bins (left and right to the sinusoid's frequency bin) are so far away from the main (important) bin that the amount of counts they get is negligible and this is why when we plot the spectrogram we see a sharp peak on the important beam and "nothing" at other bins? Or is it actually a precise cancellation which is happening and by magic all the other bins get exactly 0 counts?
If I don't manage to make the returned-from-the-FFT frequency 1D array to contain in its components the frequency of my sinusoid, then the sinusoid's frequency will for sure be between 2 bins. Then 2 bins will each summon a sinc function and the spectogram will show the superposition of 2 sinc function now. Is this correct?
I would greatly appreciate (material from anywhere) with equations and or graphs, in addition to a "wordy explanation". I don't mind reading another answer already posted, if that responds to my question, however I kind of searched with a filter through all DSP's questions and read the potentially useful ones, however I still don't understand my questions above ...