# Zeroes and poles for a system

I have run into some issues on an exercise for the course in signal analysis and systems I am currently studying.

We are to create an echo effect and are using the system below:

I am to find the systemes zeroes and poles for D=500.

I did find zeroes by first describing the system like this:

$$y(n)=x(n)+x(n-D)+x(n-2D)\\ y(z)=x(z)(1+z^{-D}+z^{-2D})\\ H(z)=1+z^{-D}+z^{-2D}=\frac{1+z^{D}+z^{2D}}{z^{2D}}\\ z^{D}+z^{2D}=-1$$

And after some thinking and tinkering found all the zeroes.

$$z=e^{\pm2\pi j\frac{1}{3D}+2\pi k\frac{1}{D}} k=0,1,2...D-1$$

I am quite confident this is correct. When it comes to the poles however, I'm not sure where or how to start.

Any suggestions?

• Because this is for a course, I've tagged it as homework and self-study. You've given your reasoning to date, so this is a good question for here... just don't expect us to "give" you the answer. Hilmar's response is good.
– Peter K.
Commented Jul 26, 2021 at 21:08

$$P(z) = z^{2D}$$