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

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

In the series case, we get

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

Thus, two identical one-pole filters in parallel are equivalent to a one-pole filter, while the same two filters in series give a two-pole filter having a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms

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

corresponding to a pole $p$ repeated twice.