Z Transform Transfer Functions 2 CC Poles Frequency Response

Cuthbert Nyack
The Z transform of a 2 pole system is shown below:-
The frequency response is obtained by substituting ejwT = ej2pw/ws for z.
The applet below shows the frequency response of a 2 pole system. 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 the "x"s. 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. As the angle varies from 0 to 180º the peak response moves from 0 to half of the sampling frequency. i.e. response changes from a lowpass to a bandpass to a highpass one.

eg parameters (0.99, 60.0, NA, NA, 62, 0, 10.0, 1.5, 1, 0.1)
show a sharp resonant peak of ~44.679 at ~ 0.165ws = (60/360)ws. At this frequency the phase goes through zero.

eg parameters (0.8, 60.0, NA, NA, 62, 0, 10.0, 1.5, 1, 1.0)
show a reduced Q resonant peak of ~3.2063 at ~ 0.165ws = (60/360)ws. The phase now goes through zero at 0.185ws.



Return to main page
Return to page index
Copyright 2000 © Cuthbert A. Nyack.