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.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.