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 .
This gives a linear system of equations in unknowns ,
.

Yet another method for finding residues is by means of Taylor series
expansions of the numerator and denominator about each
pole , 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.