# How to determine if a filter is bandpass/stopband from its pole-zero diagram in z-domain

How can we determine if a filter is bandpass or stopband, just by looking at its pole-zero diagram in z-domain?

For exmaple, if we have a system with third-order pole at the origin and a zero on the real axis at 0.5, what is the type of this filter?

In general, what is the effect of zeros outside the unit circle on frequency response magnitude?

For strongly selective filters that have well defined pass and stop bands and narrow transition bands and which, therefore, can approximate ideal brickwall filters to some desired degree, poles and zeros are generally distributed around and closer to the unit circle. Normally you would like to have all of them inside the unit circle but due to other reasons, such as linear phase FIR impulse response, some zeros can be put outside of unit circle too.

Effective frequency of a single pole/zero is the angle of its location from real axis counter clockwise. If that's a zero, it will stop frequencies nearby. If it's a pole it will amplify freqencies nearby.

By distributing those poles and zeros in groups you will be defining pass bands (that include band of poles) and stop bands (that include band of zeros), or bandpass filters as mixture.

When the magnitude of a single zero is away from unity, its selectivity decreases;i.e., pass-stop band distinction dissapears. When the magnitude is larger than one, it will also have an overall gain despite being a zero. If the magnitude is close to zero (zero at origin) then the gain approaches one.

Similarly, when the magnitude of a single pole is away from unity, its selectivity decreases;i.e., pass-stop band distinction dissapears. When the magnitude is larger than one, it will also have an overall attenuation despite being a pole. If the magnitude is close to zero (zero at origin) then the attenuation approaches one.

I highly recommend this video - https://www.youtube.com/watch?v=m5TP2uG_O2M

There David Dorran explains this topic very well, around 6 mins in he shows how the frequency response becomes more selective as the Poles reach the unit circle. 11 mins he shows how it's the same for Zeros.

So parts of the frequency can be supressed and parts can be placing appropriate zeros and poles respectively. This is because, if the frequecy is approaching a pole, the H(z) term goes to infinity due to the zero in the denominator. While H(z) goes to 0 as the frequency approaches the zero since it's in the numerator. Hence use poles to amplify and zeros to diminish.

Ok so a zero on the real axis at 0.5, this means that you would be supressing DC values (as said in here), not much, 0.5 is not very selective.

If you had kept 2 zeros at +j and -j and 2 poles at origin (like this) then since it's is at angle $$\frac{\pi}{2}$$, we have highly selected the 0.25 normalized frequency. Which is $$\frac{f_s}{4}$$ where $$f_s$$ is the sampling frequency. Since it's a zero, it means that we have a very selective stop band around that frequency.