Real Second-Order Sections

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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

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

where

is one of the poles, and

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 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