# FIR or IIR filter from poles and zeros

Is it possible to differentiate between FIR and IIR Filter just by knowing the poles and zeros of the system?

• If yes, how?
• If not, why?

Thank you.

Axel Mancino's answer is correct for causal filters. In general, FIR filters have poles at either $$z=0$$ or $$|z|\rightarrow\infty$$, or both.

Take as an example a fourth-order causal FIR filter:

$$H_1(z)=a+bz^{-1}+cz^{-2}+dz^{-3}+ez^{-4}\tag{1}$$

Clearly, $$H_1(z)$$ has all its poles at $$z=0$$. An anti-causal FIR filter such as

$$H_2(z)=az^4+bz^3+cz^2+dz+e\tag{2}$$

has all its poles at $$|z|\to\infty$$. And, finally, a general non-causal FIR filter has poles at $$z=0$$ as well as at $$|z|\rightarrow\infty$$:

$$H_3(z)=az^2+bz+c+dz^{-1}+ez^{-2}\tag{3}$$

In sum, a filter with all its poles at $$z=0$$ or $$|z|\to\infty$$ (or both) is an FIR filter. Filters with poles in the region $$0<|z|<\infty$$ must be IIR.

Yes, if all the poles are in $$z=0$$ then it is FIR. If not, it is IIR.