### 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 denotes the Heaviside unit step function

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.

Next Section:
The Continuous-Time Impulse
Previous Section:
Transfer Function