I'm going through a Signal Processing lecture where the professor mentions this fact and the argument given is: Suppose you have a sinusoidal signal: $Acos(\omega t)$
Now if you change the phase of the signal: $Acos(\omega t + p)$ then if it would correspond to a time shift, then $Acos(\omega t + \omega t_0) = Acos(\omega t + p)$
So, $t_0 = p/\omega $
And this can't always have integral values, which is contradictory since we're considering the discrete case.
My argument is that, since we're considering the discrete case, the value of 'p' will be restricted to a certain set of values, right? And these values will only be those for which p/w is an integer.
According to the definition here the phase ($p$ here) gives you how far the signal is in it's cycle. So by default the value of $p$ would only be those values that allow p/w to be integral, since our signal is discrete right? What am I missing here?