Real Second-Order Sections

In practice, however, signals are typically real-valued functions of time. As a result, for real filters (§5.1), it is typically more efficient computationally to combine complex-conjugate one-pole sections together to form real second-order sections (two poles and one zero each, in general). This process was discussed in §6.8.1, and the resulting transfer function of each second-order section becomes

$$\frac{r}{1-pz^{-1}} + \frac{\overline{r}}{1-\overline{p}z^{-1}} = \frac{r-r\overline{p}z^{-1}+\overline{r}-\overline{r}pz^{-1}}{(1-pz^{-1})(1-\overline{p}z^{-1})}$$

$$= \frac{2\mbox{re}\left\{r\right\}-2\mbox{re}\left\{r\overline{p}\right\}z^{-1}}{1-2\mbox{re}\left\{p\right\}z^{-1} + \left\vert p\right\vert^2 z^{-2}}, \protect$$ (10.3)

where $p$ is one of the poles, and $r$ is its corresponding residue. This is a special case of the biquad section discussed in §B.1.6.

When the two poles of a real second-order section are complex, they form a complex-conjugate pair, i.e., they are located at $z=R\exp(\pm j\theta)$ in the $z$ plane, where $R=\vert p\vert$ is the modulus of either pole, and $\theta$ is the angle of either pole. In this case, the ``resonance-tuning coefficient'' in Eq.(9.3) can be expressed as

$$2$$$$\mbox{re\ensuremath{\left\{p\right\}}}$$$$= 2R\cos(\theta)$$

which is often more convenient for real-time control of resonance tuning and/or bandwidth. A more detailed derivation appears in §B.1.3.

Figures 3.25 and 3.26 (p. ) illustrate filter realizations consisting of one first-order and two second-order filter sections in parallel.