Z Transform 1P/1Z AllPass
Cuthbert Nyack
If a system has a pole at p inside the unit circle and a zero at
1/p outside the unit circle, both on the real axis then the transfer function
can be written as the following equation. Along the unit circle
it can readily be shown that the modulus of this transfer function
is constant and equal to 1/p. This means that the gain is constant
at all frequencies up to half the sampling frequency but the phase delay
will be frequency dependent. The result is a first order allpass
filter.
In 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. 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.
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Copyright 2000 © Cuthbert A. Nyack.