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

$\displaystyle \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.
\begin{figure}\input fig/repeatedpole.pstex_t

Next Section:
-Phase Filters
Previous Section:
Real Second-Order Sections