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
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.