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.