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Example 2

For the filter

$\displaystyle H(z)$ $\displaystyle \isdef$ $\displaystyle \frac{2+6z^{-1}+6z^{-2}+2z^{-3}}{1-2z^{-1}+z^{-2}}$ (7.20)
  $\displaystyle =$ $\displaystyle (2+10z^{-1}) + z^{-2}\left[\frac{8}{1-z^{-1}} + \frac{16}{(1-z^{-1})^2}\right] \protect$ (7.21)

we obtain the output of residued (§J.6) shown in Fig.6.4. In contrast to residuez, residued delays the IIR part until after the FIR part. In contrast to this result, residuez returns r=[-24;16] and f=[10;2], corresponding to the PFE

$\displaystyle H(z) = 10+2z^{-1}-\frac{24}{1-z^{-1}} + \frac{16}{(1-z^{-1})^2},$ (7.22)

in which the FIR and IIR parts have overlapping impulse responses.

See Sections J.5 and J.6 starting on page [*] for listings of residuez, residued and related discussion.

Figure 6.4: Use of residued to perform a partial fraction expansion of an IIR filter transfer function $ H(z)=B(z)/A(z)$ .
  
B=[2 6 6 2]; A=[1 -2 1];
[r,p,f,m] = residued(B,A)
% r =
%    8
%   16
% 
% p =
%   1
%   1
% 
% f =
%    2  10
% 
% m =
%   1
%   2

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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