# lead high pass filter and lag low pass filter

I have a question regarding lead high pass filter and lag low pass filter.

I do not quite understand the reasoning behind these two special case (when z=0) ? Thanks!

• You haven't really asked a question but rather said what you don't understand and I cannot fathom out what it is you are having problems with. – Andy aka Apr 1 '16 at 9:08

Lag low-pass and lead high-pass are in fact the "standard" low-pass and high-pass filters, in the sense that an ideal low-pass filter should have a gain of zero for $\omega\rightarrow\infty$, and an ideal high-pass filter should have a gain of zero at $\omega=0$. These conditions are satisfied by the lag low-pass filter (with a zero at $s\rightarrow\infty$), and by the lead high-pass filter (with a zero at $s=0$).
The phase of the lead low-pass filter is greater (i.e., less negative) than the phase of the lag low-pass filter ($\Rightarrow$ "lead"), but the magnitude is worse because its gain only decays from $z_1/p_1>1$ to $1$ for $\omega\rightarrow \infty$. A similar thing is true for the lag high-pass filter. Its gain is not zero at $\omega=0$ but it equals $z_1/p_1<1$. Its phase is smaller (i.e., less positive) than the phase of the lead high-pass filter ($\Rightarrow$ "lag").