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,^10.5it 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

$$\begin{eqnarray*} \frac{r}{1-pz^{-1}} + \frac{\overline{r}}{1-\pc z^{-1}} &=& \f... ...{re}\left\{p\right\}z^{-1} + \left\vert p\right\vert^2 z^{-2}}, \end{eqnarray*}$$

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

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

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.