Prove that the function in the domain of Z is a high pass filter

Question

I need prove that the function \begin{equation} L(z) = 1-z^{-1} \end{equation} is a high pass filter, but I have not much understanding of the $z$ transform and what really the $z$ domain is. So how could I prove it?

Answer 1

In order to go from a (discrete time) signal's $z$-transform to its DTFT, substitute $z = e^{jw}$. So, we have $$ L(e^{jw}) = 1 - e^{-jw} $$ We want to check how $|L(e^{jw})|$ for different values of $w$. We have $$ \begin{align} \left| L(e^{jw}) \right| &= \left| (1 - \cos(w)) + j\sin(w) \right| = \sqrt{ (1 - \cos(w))^2 + \sin^2(w) }\\ &= \sqrt{\sin^2(w) + \cos^2(w) - 2 \cos(w) + 1} = \sqrt{2(1-\cos(w))} \end{align} $$ Why does this magnitude spectrum correspond to that of a high-pass filter? Try graphing it from $-\pi$ to $\pi$.