Zeroes and poles for a system

Question

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?

Answer 1

The poles of an FIR filter get many people tripped. The poles are the roots of the pole polynomial which you already have identified as

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

1. How many roots does this polynomial have ?
2. Where are the roots all located ?