# Z Transform
Second Order AllPass Filter

Cuthbert Nyack

A transfer function with 2 poles and 2 zeros is shown below. The
situation is similar to a first order allpass filter with the same
pole zero relationship. In this case a pole at p has a corresponding
zero at 1/p at the same phase angle. This produces a second order
allpass filter and has a constant gain of 1/p^{2}.
Allpass filters can be used to modify phases without
affecting the amplitudes at the different frequencies.

In the applet below
the gray curve above the magenta lines show the response between 0 and
½ w_{s} while the
gray curve above the green lines
show the response between ½ w_{s}
and w_{s}. 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.