Partial Fraction Expansion

An important tool for inverting the z transform and converting among digital filter implementation structures is the partial fraction expansion (PFE). The term ``partial fraction expansion'' refers to the expansion of a rational transfer function into a sum of first and/or second-order terms. The case of first-order terms is the simplest and most fundamental:

where

and . (The case is addressed in the next section.) The denominator coefficients are called the poles of the transfer function, and each numerator is called the residue of pole . Equation (6.7) is general only if the poles are distinct. (Repeated poles are addressed in §6.8.5 below.) Both the poles and their residues may be complex. The poles may be found by factoring the polynomial into first-order terms,^7.4e.g., using the roots function in matlab. The residue corresponding to pole may be found analytically as

when the poles are distinct. The matlab function residuez^7.5 will find poles and residues computationally, given the difference-equation (transfer-function) coefficients.

Note that in Eq.(6.8), there is always a pole-zero cancellation at . That is, the term is always cancelled by an identical term in the denominator of , which must exist because has a pole at . The residue is simply the coefficient of the one-pole term in the partial fraction expansion of at . The transfer function is , in the limit, as .

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Subsections

• Example
• Complex Example
• PFE to Real, Second-Order Sections
• Inverting the Z Transform
• FIR Part of a PFE
  • Example: The General Biquad PFE
  
  
• Alternate PFE Methods
• Repeated Poles
  • Dealing with Repeated Poles Analytically
  • Example
  • Impulse Response of Repeated Poles
  • So What's Up with Repeated Poles?
  
  
• Alternate Stability Criterion
• Summary of the Partial Fraction Expansion
• Software for Partial Fraction Expansion
  • Example 2
  • Polynomial Multiplication in Matlab
  • Polynomial Division in Matlab

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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.