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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Implementation Structures for Recursive Digital Filters
       Series and Parallel Filter Sections
          Parallel First and/or Second-Order Sections
             Implementation of Repeated Poles

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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 \end{figure}

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Next: Formant Filtering Example

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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