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/p2. 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 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.