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Introduction to Digital Filters
    Implementation Structures for Recursive Digital Filters
       Series and Parallel Filter Sections
          Parallel First and/or Second-Order Sections
             Real Second-Order Sections

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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

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

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

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.

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Next: Implementation of Repeated Poles

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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