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The Continuous-Time Impulse
An impulse in continuous time must have ``zero width''
and unit area under it. One definition is
![$\displaystyle \delta(t) \isdef \lim_{\Delta \to 0} \left\{\begin{array}{ll} \frac{1}{\Delta}, & 0\leq t\leq \Delta \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array} \right. \protect$](img2450.png) |
(B.26) |
An impulse can be similarly defined as the limit of any pulse
shape which maintains unit area and approaches zero width at time 0
[150]. As a result, the impulse under every definition has
the so-called
sifting property under integration,
![$\displaystyle \int_{-\infty}^\infty f(t) \delta(t) dt = f(0), \protect$](img2451.png) |
(B.27) |
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
![$\displaystyle \delta(t) \isdef \lim_{\Omega\to\infty}\frac{\sin(\Omega t)}{\pi t}.$](img2452.png) |
(B.28) |
(Note, incidentally, that
is in
but not
.)
An impulse is not a function in the usual sense, so it is called
instead a distribution or generalized function
[36,150]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)
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