# Design a filter which passes all frequencies except $\omega=\pm\frac{\pi}{2}$ and plot its pole-zero diagram

Also draw its normalized frequency response.

What is the ROC?

This has to be done in z-plane so there must be two poles at $+i$ and $-i$ since they cannot be included in region of convergence. Is my assumption correct?

And what is meant by normalized frequency?

The kind of filter you are looking for is a notch filter. Using filter design toolbox in Matlab you can get it as I'm showing you in the following picture:

This has to be done in z-plane so there must be two poles at +i and −i since they cannot be included in region of convergence. Is my assumption correct?

Yes! This is correct.

Assuming you want to create a causal filter (a right sided sequence with non-zero samples only for n > 0) then your Region of Convergence ROC is :

If you want to understand this better I would recommend you take a look at example 3.1 of Alan V. Oppenheim Discrete-Time Signal Processing 3rd edition.

Normalized frequency response means that instead of dealing with a sampling frequency you make a design independent of it in which you only care about digital frequency. For instance if I say that $w = \pi/2$ then if sampling frequency is 48 kHz (this is $2*\pi$), the maximum frequency that you may get is $\pi$ or 24 kHz and the notch will be at 12 KHz which is $\pi/2$. But on that very same filter if sampling frequency is later 96 kHz then the notch will be at 24 kHz.

The way in which you can do this filter in Matlab is this:

If you select a broader bandwidth (for instance 0.1 instead of 0.001) you will notice how the zero and the pole maintaining the same angle start to separate from each other in the diagram

with sampling frequency 48 kHz:

with sampling frequency 96 kHz: