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Impulse Response

In the same way that the impulse response of a digital filter is given by the inverse z transform of its transfer function, the impulse response of an analog filter is given by the inverse Laplace transform of its transfer function, viz.,

$\displaystyle h(t) = {\cal L}_t^{-1}\{H(s)\} = \tau e^{-t/\tau} u(t)
$

where $ u(t)$ denotes the Heaviside unit step function

$\displaystyle u(t) \isdef \left\{\begin{array}{ll}
1, & t\geq 0 \\ [5pt]
0, & t<0. \\
\end{array} \right.
$

This result is most easily checked by taking the Laplace transform of an exponential decay with time-constant $ \tau>0$ :

\begin{eqnarray*}
{\cal L}_s\{e^{-t/\tau}\}
&\isdef & \int_0^{\infty}e^{-t/\tau} e^{-st} dt
=\int_0^{\infty}e^{-(s+1/\tau)t} dt\\
&=&\left. \frac{-1}{s+1/\tau} e^{-(s+1/\tau)t} \right\vert _0^\infty\\
&=& \frac{1}{s+1/\tau} = \frac{RC}{RCs+1}.
\end{eqnarray*}

In more complicated situations, any rational $ H(s)$ (ratio of polynomials in $ s$ ) may be expanded into first-order terms by means of a partial fraction expansion (see §6.8) and each term in the expansion inverted by inspection as above.


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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