A second order discrete system is described by

The z-transform of this DE is

and the transfer function is

which has two repeated zeros and a pair of complex conjugate poles

For convenience, we define

and the transfer function becomes

and the poles become

For the system to be causal and stable, we have to have . When , can be expanded

where

The impulse response of the system can be found by inverse z-transform

When , has two repeated poles and becomes

and the impulse response becomes

When , has two repeated poles and becomes

and the impulse response becomes

To graphically evaluate the behavior of the system as a function of frequency in the z-plane, we let in and get the frequency response function of the system

where , and are three vectors in z-plane defined as

For any frequency represented by a point on the unit circle, the magnitude and phase angle of the frequency response function can be represented in the z-plane as

and

When , and are maximized and thereby is minimized. In particular, when is close to 1, is minimized when , i.e., has a peak.