### 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

*distribution*or

*generalized function*[13,44]. (It is still commonly called a ``delta function'', however, despite the misnomer.)

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Poles and Zeros

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Impulse Response