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)
$](http://www.dsprelated.com/josimages_new/filters/img1813.png)
![$ u(t)$](http://www.dsprelated.com/josimages_new/filters/img1770.png)
![$\displaystyle u(t) \isdef \left\{\begin{array}{ll}
1, & t\geq 0 \\ [5pt]
0, & t<0. \\
\end{array}\right.
$](http://www.dsprelated.com/josimages_new/filters/img1814.png)
![$ \tau>0$](http://www.dsprelated.com/josimages_new/filters/img1815.png)
![\begin{eqnarray*}
{\cal L}_s\{e^{-t/\tau}\}
&\isdef & \int_0^{\infty}e^{-t/\tau...
...ght\vert _0^\infty\\
&=& \frac{1}{s+1/\tau} = \frac{RC}{RCs+1}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1816.png)
In more complicated situations, any rational (ratio of
polynomials in
) 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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The Continuous-Time Impulse
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Transfer Function