Implementation of Repeated Poles

Fig.9.5 illustrates an efficient implementation of terms due to a repeated pole with multiplicity three, contributing the additive terms

$$\frac{r_1}{1-pz^{-1}} + \frac{r_2}{(1-pz^{-1})^2} + \frac{r_3}{(1-pz^{-1})^3}$$

to the transfer function. Note that, using this approach, the total number of poles implemented equals the total number of poles of the system. For clarity, a single real (or complex) pole is shown. Implementing a repeated complex-conjugate pair as a repeated real second-order section is analogous.

Figure 9.5: Implementation of a pole $p$ repeated three times.