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:

$$H(z) \isdef \frac{B(z)}{A(z)} = \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$$ (7.7)

where

$$\begin{eqnarray*} B(z) &=& b_0 + b_1 z^{-1}+ b_2z^{-2}+ \cdots + b_M z^{-M}\\ A(z) &=& 1 + a_1 z^{-1}+ a_2z^{-2}+ \cdots + a_N z^{-N} \end{eqnarray*}$$

and $M<N$. (The case $M\geq N$ is addressed in the next section.) The denominator coefficients $p_i$ are called the poles of the transfer function, and each numerators $r_i$ is called the residue of pole $p_i$. Equation (6.7) is general only if the poles $p_i$ 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 $A(z)$ into first-order terms,^7.2 e.g., using the roots function in matlab. The residue $r_i$ corresponding to pole $p_i$ may be found analytically as

$$r_i = \left.(1-p_iz^{-1})H(z)\right\vert _{z=p_i} \protect$$ (7.8)

when the poles $p_i$ are distinct. In the Matlab Signal Processing Tool Box, the function residuez will find poles and residues computationally, given the difference-equation (transfer-function) coefficients.^7.3

Note that in Eq. (6.8), there is always a pole-zero cancellation at $z=p_i$. That is, the term $(1-p_iz^{-1})$ is always cancelled by an identical term in the denominator of $H(z)$, which must exist because $H(z)$ has a pole at $z=p_i$. The residue $r_i$ is simply the coefficient of the one-pole term $1/(1-p_i z^{-1})$ in the partial fraction expansion of $H(z)$ at $z=p_i$. The transfer function is $r_i/(1-p_i z^{-1})$, in the limit, as $z\to p_i$.