Parallel Complex Resonators

Implementing the partial fraction expansion (§6.8.7)
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,^{10.5}it 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

where is one of the poles, and 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

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