# Z Transform 2P/2Z Frequency Response

Cuthbert Nyack
The Z transform of a system with 2 poles and 2 zeros is given below.
The frequency response is obtained by substituting ejwT = ej2pw/ws for z.
The variation of the frequency response of a system with the above transfer function is shown by the applet below. 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. Poles are shown by an "x" and zeros 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. The location of the poles therefore determine the frequency of the peak response. Having zeros in the transfer function means that the response can be set to zero at certain frequencies by placing zeros at those frequencies.
Digital filters can be designed using poles, zeros or a combination of both.
Parameters (0.8, 15.0, 1.0, 90.0, 375, NA, 10.0, 1.5, 1, 0.14) show a low pass configuration.

Parameters (0.8, 165.0, 1.0, 0.0, 188, NA, 10.0, 1.5, 1, 0.07) show a high pass configuration.