Pole-Zero Analysis
Since our example transfer function
![$\displaystyle H(z) = \frac{1 + g_1 z^{-M_1}}{1 + g_2 z^{-M_2}}
$](http://www.dsprelated.com/josimages_new/filters/img343.png)
![$ \,$](http://www.dsprelated.com/josimages_new/filters/img94.png)
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
![$\displaystyle H(z) = g \frac{(1-q_1z^{-1})(1-q_2z^{-1})\cdots(1-q_{M_1}z^{-1})}{(1-p_1z^{-1})(1-p_2z^{-1})\cdots(1-p_{M_2}z^{-1})}
$](http://www.dsprelated.com/josimages_new/filters/img344.png)
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
![$\displaystyle q_k = - g_1^{\frac{1}{M_1}} e^{j2\pi\frac{k}{M_1}}, \quad
k=0,2,\dots,M_1-1
$](http://www.dsprelated.com/josimages_new/filters/img345.png)
![$ g_1>0$](http://www.dsprelated.com/josimages_new/filters/img346.png)
![$\displaystyle p_k = - g_2^{\frac{1}{M_2}} e^{j2\pi\frac{k}{M_2}}, \quad
k=0,2,\dots,M_2-1.
$](http://www.dsprelated.com/josimages_new/filters/img347.png)
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