# Poles and zeros map

Given the following map of poles and zeros for some $H^L(s)$:

How can I understand from the map that the given Transfer function is LPF? Hence the poles are located at the hight frequencies it should implement HPF right? But according to the solution that was given to the question in original, it's implementing a LPF.

• A lowpass has an "unwritten" zero somewhere off the chart. A highpass should also have a zero somewhere, too, I forgot where. – a concerned citizen Jul 17 '18 at 6:48
• I don't really get the point. We know that the frequencies at the right side are the hight ones right? So that means if I had a pole at the right side it's on the hight frequencies, and since we know that the poles boost the frequency that says that they boost the hight frequencies, unlike what was written in the solution. – Sama Assi Jul 17 '18 at 6:59

Note that the pole locations would be the same for all $4$ classic types of frequency-selective filters (low pass, high pass, band pass, band stop). It's the location of the zeros that determines which filter type it is. Since we don't see any zeros, we can assume that they are at infinity, hence, the filter is a low pass filter. For a high pass filter you would get a double zero at $s=0$ (DC). A band pass filter would have one zero at $s=0$ and the other one at infinity. And, finally, a band stop filter would have two zeros on the frequency axis (the vertical axis in the plot), close to the imaginary part of the poles. Of course, all of the above is valid for second-order filters, as the one shown in the plot.