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Impulse Response

The impulse response is again the inverse Laplace transform of the transfer function. Expanding $ H(s)$ into a sum of complex one-pole sections,

$\displaystyle H(s) = 2\eta\cdot\frac{s}{s^2 + 2\eta s + \omega_0^2}
= \frac{r_1}{s-p_1} + \frac{r_2}{s-p_2}
= \frac{(r_1+r_2)s - (r_1p_2 + r_2p_1)}{s^2-(p_1 + p_2) + p_1p_2},
$

where $ p_{1,2}\isdef -\eta \pm \sqrt{\eta^2 - \omega_0^2}$ . Equating numerator coefficients gives

\begin{eqnarray*}
r_1+r_2 &=& 2\eta \;\mathrel{=}\; \frac{1}{RC}\\
r_1p_2 + r_2p_1 &=& 0.
\end{eqnarray*}

This pair of equations in two unknowns may be solved for $ r_1$ and $ r_2$ . The impulse response is then

$\displaystyle h(t) = r_1 e^{p_1 t} u(t) + r_2 e^{p_2 t} u(t).
$


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA