### 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 . 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 G 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 §G.10.

**Next Section:**

Repeated Poles

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FIR Part of a PFE