Alternate PFE Methods

Another method for finding the pole residues is to write down the general form of the PFE, obtain a common denominator, expand the numerator terms to obtain a single polynomial, and equate like powers of $z^{-1}$. This gives a linear system of $N$ equations in $N$ unknowns $r_i$, $i=1,\ldots,N$.

Yet another method for finding residues is by means of Taylor series expansions of the numerator $B(z)$ and denominator $A(z)$ about each pole $p_i$, using l'Hôpital's rule.

Finally, one can alternatively construct a state space realization of a strictly proper transfer function (using, e.g., tf2ss in matlab) and then diagonalize it via a similarity transformation. (See Appendix E for an introduction to state-space models and diagonalizing them via similarity transformations.) The transfer function of the diagonalized state-space model is trivially obtained as a sum of one-pole terms--i.e., the PFE. In other words, diagonalizing a state-space filter realization implicitly performs a partial fraction expansion of the filter's transfer function. When the poles are distinct, the state-space model can be diagonalized; when there are repeated poles, it can be block-diagonalized instead, as discussed further in §E.10.