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

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 Forgewhere

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University