Example 2

For the filter

$$H(z) \isdef \frac{2+6z^{-1}+6z^{-2}+2z^{-3}}{1-2z^{-1}+z^{-2}}$$ (7.20)

$$= (2+10z^{-1}) + z^{-2}\left[\frac{8}{1-z^{-1}} + \frac{16}{(1-z^{-1})^2}\right] \protect$$ (7.21)

we obtain the output of residued (§H.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

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

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

See Sections H.5 and H.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