Z Transform 2P/2Z Frequency Response
The Z transform of a system with 2 poles and 2 zeros is
The frequency response is obtained by substituting
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.
Return to main page
Return to page index
Copyright 2000 © Cuthbert A. Nyack.