## Pole-Zero Analysis

Since our example transfer function

(from Eq.(3.4)) is a ratio of polynomials in , and since every polynomial can be characterized by its roots plus a scale factor, we may characterize any transfer function by its numerator roots (called the zeros of the filter), its denominator roots (filter poles), and a constant gain factor:

The poles and zeros for this simple example are easy to work out by hand. The zeros are located in the plane at

where we assume , and the poles are similarly given by

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 Forge

where 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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