# Z Transform Transfer Functions 1P 1Z Frequency Response

Cuthbert Nyack
The Z transform of a system with a single pole at p1 and a single zero at z1 along the real Z axis is given by:- The frequency response is obtained by substituting ejwT = ej2pw/ws for z.
The applet below shows the frequency response of the system which is also the Z transform along the unit circle in the Z plane. The gray curve above the magenta lines show the response between 0 and ½ ws while the gray curve above the green lines show the response between ½ ws and ws. Orange lines show the phase variation. White disc shows the unit circle, black line is the real z axis and orange line is the imaginary z axis. Pole is shown by the "x" and the zero by an "o". When the "Theta" parameter is zero the view is along the imaginary axis from the top. -90° corresponds to a view along the real z axis from the left and +90° represents a view along the real z axis from the right. Note that having a pole near the unit circle produces a peak in the response while having a zero on the unit circle produces a zero in the response.

Setting Fn = 1 shows the unfolded spectrum between 0 and 2ws. Numerical values can be seen by changing wL.