Difference between revisions of "HW2.A Jeff Kubascik ECE301Fall2008mboutin" - Rhea

Revision as of 14:08, 11 September 2008

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})\, $

is periodic, since

$ \,\exists N\in \mathbb{Z}\, $ such that $ \,y[n]=y[n+N], \forall n\in \mathbb{Z}\, $

$ \,2cos(\frac{2\pi n}{10})=2cos(\frac{2\pi n}{10}+\frac{2\pi N}{10})\, $

This is true when

$ \,\frac{2\pi N}{10}=2\pi \, $

$ \,N=10\, $

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(n)\, $

Is not periodic in DT, since is no integer $ N\in \mathbb{Z} $ such that

$ \,z[n]=z[n+N], \forall n\in \mathbb{Z}\, $

$ \,2cos(n)=2cos(n+N)\, $

This would be true when

$ \,N=2\pi k, k\in \mathbb{Z}\, $

This can be seen in the following plot (notice how the values do not line up horizontally):

Part 2