# Z Transform
Comb Filter

Cuthbert Nyack

A comb filter is one which has poles linearly spaced around a circle
with radius close to but less than 1. The transfer function is shown below.
Here the poles are on a circle of radius r.
The transfer function also has zeros at the origin as seen from the
second form of the equation above.

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.

The filter does have a "comb" shaped response with several peaks
linearly spaced around a circle with radius slightly less than 1.
This filter and its variants can be used for audio and video applications.

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Copyright 2000 © Cuthbert A. Nyack.