# Determining type of analog filter given its pole zero plot

How can i classify an analog filter given its pole zero map.

For example:
I've got my zero's located at $$±2j$$ and my poles located at $$-1$$ & $$-2$$ , then what is the nature of the filter?
$$i.e,$$ It is L.P.F or H.P.F or B.P.F or B.S.F or A.P.F ?

My Approach:
Consider the following fig.: Let P be a point on positive $$j \Omega$$ axis ,
$$i.e,$$ we are considering positive range of frequencies
Let pole at '$$-1$$' be denoted by $$X_1$$,
Let pole at '$$-2$$' be denoted by $$X_2$$,
Let zero at $$2j$$ be denoted as $$Z_1$$ and
Let zero at $$-2j$$ be denoted as $$Z_2$$

Now, at frequency $$=0$$ , P is at the origin (O) ,
Distance of the poles from P at this frequency is $$PX_1 =1$$ & $$PX_2=2$$
and distance of zero is $$PZ_1=2$$ $$[\because \text{ we are considering +ve freq } , \therefore \text{ neglected } Z_2]$$
So, Magnitude Response of the given filter at $$\Omega=0$$ is $$M= \frac{2}{1.2}=1 \quad \dots (i)$$ $$\implies$$ Pass Band at $$\Omega=0$$

Now, from the fig, we recognise as $$\Omega$$ increases from $$0 \to 2$$ , $$M$$ decreases ;
and at $$\Omega=2$$ , $$M=0 \implies$$ Stop Band at $$\Omega=2$$;
Now again as $$\Omega$$ increases from $$2 \to \infty$$ , $$M$$ increases ;
So, the nature of filter should be B.S.F or Notch filter whose notch frequency is at $$\Omega_0=2 \quad \dots (ii)$$

But, if we invoke its magnitude at $$\Omega \to \infty$$, then we get: $$M= \frac{PZ_1}{(PX_1).(PX_2)}=0 \text{ as } \Omega \to \infty$$ which contradicts $$(ii)$$ , so how to solve this?
I know it's the two zeroes that cancelling the effect of two poles at $$\Omega \to \infty$$ , but if we take into consideration of two zeroes then how will we approach $$(i)$$ ? Any help please...