Repeated Poles

When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have

$$H_1(z) \eqsp \frac{r_1}{1-pz^{-1}} + \frac{r_2}{1-pz^{-1}} \eqsp \frac{r_1+r_2}{1-pz^{-1}} \isdefs \frac{r_3}{1-pz^{-1}}.$$

In the series case, we get

$$H_2(z) \eqsp \frac{r_1}{1-pz^{-1}} \cdot \frac{r_2}{1-pz^{-1}} \eqsp \frac{r_1r_2}{(1-pz^{-1})^2} \isdefs \frac{r_3}{(1-pz^{-1})^2}.$$

Thus, two one-pole filters in parallel are equivalent to a new one-pole filter^7.8 (when the poles are identical), while the same two filters in series give a two-pole filter with a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms

$$\frac{r_{1,1}}{(1-pz^{-1})^2} + \frac{r_{1,2}}{(1-pz^{-1})}$$

for a pole $p$ having multiplicity 2.