Summary of the Partial Fraction Expansion

In summary, the partial fraction expansion can be used to expand any rational z transform

$$H(z) = \frac{B(z)}{A(z)} = \frac{b_0 + b_1 z^{-1}+ \cdots + b_M z^{-M}}{1 + a_1 z^{-1}+ \cdots + a_N z^{-N}}$$

as a sum of first-order terms

$$H(z) \isdef \frac{B(z)}{A(z)} = \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$$ (7.17)

for $M<N$, and

$$H(z) = F(z) + z^{-(K+1)}\sum_{i=1}^{N}\frac{r_i}{1-p_iz^{-1}} \protect$$ (7.18)

for $M\geq N$, where the term $z^{-(K+1)}$ is optional, but often preferred. For real filters, the complex one-pole terms may be paired up to obtain second-order terms with real coefficients. The PFE procedure occurs in two or three steps:

1. When $M\geq N$, perform a step of long division to obtain an FIR part $F(z)$ and a strictly proper IIR part $B^\prime(z)/A(z)$.
2. Find the $N$ poles $p_i$, $i=1,\ldots,N$ (roots of $A(z)$).
3. If the poles are distinct, find the $N$ residues $r_i$, $i=1,\ldots,N$ from
   $$r_i = \left.(1-p_iz^{-1})\frac{B(z)}{A(z)}\right\vert _{z=p_i}$$

4. If there are repeated poles, find the additional residues via the method of §6.8.5, and the general form of the PFE is

   $$H(z) = F(z) + z^{-(K+1)}\sum_{i=1}^{N_p}\sum_{k=1}^{m_i}\frac{r_{i,k}}{(1-p_iz^{-1})^k} \protect$$ (7.19)

   
   
   where $N_p$ denotes the number of distinct poles, and $m_i\ge 1$ denotes the multiplicity of the $i$th pole.

In step 2, the poles are typically found by factoring the denominator polynomial $A(z)$. This is a dangerous step numerically which may fail when there are many poles, especially when many poles are clustered close together in the $z$ plane.

The following matlab code illustrates factoring $A(z) = 1 - z^{-3}$ to obtain the three roots, $p_k=e^{jk2\pi/3}$, $k=0,1,2$:

A = [1 0 0 -1];  % Filter denominator polynomial
poles = roots(A) % Filter poles

See Chapter 9 for additional discussion regarding digital filters implemented as parallel sections (especially §9.2.2).