Parallel, Real, Second-Order Sections

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Figure 3.21 shows the impulse response of the real two-pole section

$\displaystyle H^r_2(z) \isdef H_2(z) + H_3(z) = \frac{0.3788 -0.2413z^{-1}}{1 - 1.4562z^{-1}+ 0.8100z^{-2}},$

and Fig.3.22 shows its frequency response. The frequency axis unnecessarily extends all the way to the sampling rate (

**Figure 3.21:** Impulse response of real two-pole section of the real partial-fraction-expansion of the example filter.
$\includegraphics[width=\textwidth]{eps/arir2}$

**Figure 3.22:** Frequency response of real two-pole section .
$\includegraphics[width=\textwidth]{eps/arfr2}$

Finally, Fig.3.23 gives the impulse response of the real two-pole section

$\displaystyle H^r_3(z) \isdef H_4(z) + H_5(z) = \frac{0.4555 + 0.0922z^{-1}}{1 + 0.5562z^{-1}+ 0.8100z^{-2}},$

and Fig.3.24 its frequency response.

**Figure 3.23:** Impulse response of real two-pole section of the real partial-fraction-expansion of the example filter.
$\includegraphics[width=\textwidth]{eps/arir3}$

**Figure 3.24:** Frequency response of real two-pole section .
$\includegraphics[width=\textwidth]{eps/arfr3}$

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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