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Parallel, Real, Second-Order Sections

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 ($ f_s=1$).

Figure 3.21: Impulse response of real two-pole section $ H^r_2(z)$ of the real partial-fraction-expansion of the example filter.
\includegraphics[width=\twidth]{eps/arir2}

Figure 3.22: Frequency response of real two-pole section $ H^r_2(z)$.
\includegraphics[width=\twidth]{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 $ H^r_3(z)$ of the real partial-fraction-expansion of the example filter.
\includegraphics[width=\twidth]{eps/arir3}

Figure 3.24: Frequency response of real two-pole section $ H^r_3(z)$.
\includegraphics[width=\twidth]{eps/arfr3}


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Parallel Second-Order Signal Flow Graph
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First-Order Parallel Sections