please help me find the error in the following counter example.
Consider we take sinus with period of $2\pi$. We sample it many time, and more than 3. We make convolution with rectangle of height 1 and width $2\pi$.
We’ve got sine wave sampled many times without periodicity.
Now we make fourier transform of it. We shall get delta at omega 2pi and as we know, since we transformed a discrete signal, we shall get this delta every 2pi, alas at 4pi, 6pi, and so on. Meaning by that, that each multiplication by two of this signal in the time domain shall fit the discrete signal’s samples.
But it does not work for me.
Say we have 999 samples on the original sine on single period. Signal which is shortened by two will simply remain long enough not to fit between each of two samples in each of the halves of the signal because it works only if we have 3 samples (on the 0s). Only then the $4\pi$ sinus will lay down on on the samples, which are the zeroes of the sine wave.
What am I missing? Please help.