The Continuous-Time Impulse
The continuous-time impulse response was derived above as the inverse-Laplace transform of the transfer function. In this section, we look at how the impulse itself must be defined in the continuous-time case.
An impulse in continuous time may be loosely defined as any ``generalized function'' having ``zero width'' and unit area under it. A simple valid definition is
More generally, an impulse can be defined as the limit of any pulse shape which maintains unit area and approaches zero width at time 0. As a result, the impulse under every definition has the so-called sifting property under integration,
provided is continuous at . This is often taken as the defining property of an impulse, allowing it to be defined in terms of non-vanishing function limits such as
Poles and Zeros