## Pole-Zero Analysis

Since our example transfer function

*zeros*of the filter), its denominator roots (filter

*poles*), and a constant gain factor:

Figure 3.12 gives the pole-zero diagram of the specific example filter
. There are three zeros,
marked by ``O`' in the figure, and five poles, marked by
``X`'. Because of the simple form of digital comb filters, the
zeros (roots of ) are located at 0.5 times the three cube
roots of -1 (
), and similarly the poles (roots
of ) are located at 0.9 times the five 5th roots of -1
(
). (Technically, there are also two more
zeros at .) The matlab code for producing this figure is simply

[zeros, poles, gain] = tf2zp(B,A); % Matlab or Octave zplane(zeros,poles); % Matlab Signal Processing Toolbox % or Octave Forgewhere

`B`and

`A`are as given in Fig.3.11. The pole-zero plot utility

`zplane`is contained in the Matlab Signal Processing Toolbox, and in the Octave Forge collection. A similar plot is produced by

sys = tf2sys(B,A,1); pzmap(sys);where these functions are both in the Matlab Control Toolbox and in Octave. (Octave includes its own control-systems tool-box functions in the base Octave distribution.)

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Alternative Realizations

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Phase Response