The first order discrete system is described by

The impulse response can be found by solving the following

to be

Alternatively, we can take z-transform of the DE and get

and the transfer function of the system (assumed causal)

has a zero at and a pole and its ROC is the region outside the pole. If , then the unit circle can be included in the ROC, the Fourier transform exists and the system is stable. The impulse response (unit sample response) of the system is

Although seems to have a form different from the typical impulse response in continuous case , they are essentially the same as can be rewritten as

where

Letting in , we get the frequency response function of the system

where and are two 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

which can be evaluated graphically in the z-plane as the frequency changes in the range . If we assume , then when , the denominator reaches its minimum of , and is maximized to be ; and when , the denominator reaches its maximum of , and is minimized to be . The phase angle of is zero when or , and is negative for and positive for .