Software for Partial Fraction Expansion

Figure 6.3 illustrates the use of residuez (§H.5) for performing a partial fraction expansion on the transfer function

$$H(z) = \frac{1 + 0.5^3 z^{-3}}{1 + 0.9^5z^{-5}}$$

The complex-conjugate terms can be combined to obtain two real second-order sections, giving a total of one real first-order section in parallel with two real second-order sections, as discussed and depicted in §3.11.

Figure 6.3: Use of residuez to perform a partial fraction expansion of an IIR filter transfer function $H(z)=B(z)/A(z)$.

 

B = [1 0 0 0.125];
A = [1 0 0 0 0 0.9^5];
[r,p,f] = residuez(B,A)
% r =
%   0.16571 
%   0.22774 - 0.02016i
%   0.22774 + 0.02016i
%   0.18940 + 0.03262i
%   0.18940 - 0.03262i
% 
% p =
%   -0.90000 
%   -0.27812 - 0.85595i
%   -0.27812 + 0.85595i
%    0.72812 - 0.52901i
%    0.72812 + 0.52901i
% 
% f = [](0x0)