### 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