Part 1
The function was chosen at random from HW1: HW1.4 Hang Zhang - Periodic vs Non-period Functions_ECE301Fall2008mboutin
$ \,x(t)=2cos(2\pi t)\, $
Periodic Signal in DT:
If $ x(t) $ is sampled at $ period=0.1 $, the function
$ \,y[n]=x[0.1n]=2cos(\frac{2\pi n}{10})\, $
would be periodic, since
$ \,y[n]=y[n+10N], \forall N\in Z\, $
This can be seen in the following plot (notice how the values lines up horizontally):
Non-Periodic Signal in DT:
However, if $ x(t) $ is sampled at $ period=1/2\pi $, the function
$ \,z[n]=x[\frac{n}{2\pi }]=2cos(t)\, $
Is not periodic in DT, since
This can be seen in the following plot (notice how the values do not line up horizontally):
