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