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Parallel Complex Resonators

Implementing the partial fraction expansion6.8.7) directly results in a set of parallel complex resonators for the IIR part. The same result is obtained by implementing a diagonalized state-space model (Eq. (E.22)). In practice, however, signals are typically real-valued functions of time. As a result, for real filters,10.5it is typically more economical to combine complex-conjugate one-pole sections together to form real second-order sections (two poles, one zero, in general). This process was discussed in §6.8.1, and the resulting transfer function of each second-order section becomes

\frac{r}{1-pz^{-1}} + \frac{\overline{r}}{1-\pc z^{-1}}
&=& \f...
+ \left\vert p\right\vert^2 z^{-2}},

where $ p$ is one of the poles, and $ r$ is its corresponding residue. Figure 3.21 and Fig.3.22 illustrates a filter realization consisting of one first-order, and two second-order parallel filter sections.

Finally, 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

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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