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.*,

where the scaling by gives unity-gain in the passband, and denotes the

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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University