Fourier Transform (FT) and Inverse
The
Fourier transform of a
signal

,

, is defined as

 |
(B.1) |
and its inverse is given by
 |
(B.2) |
Conditions for the
existence of the Fourier transform are
complicated to state in general [
12], but it is
sufficient
for

to be
absolutely integrable,
i.e.,
This requirement can be stated as

, meaning that

belongs to the set of all
signals having a finite
norm
(

). It is similarly sufficient for

to be
square integrable,
i.e.,
or,

. More generally, it suffices to show

for

[
12, p. 47].
There is never a question of existence, of course, for Fourier
transforms of real-world signals encountered in practice. However,
idealized signals, such as
sinusoids that go on forever in
time, do pose normalization difficulties. In practical engineering
analysis, these difficulties are resolved using Dirac's ``
generalized
functions'' such as the
impulse (also called the
delta function) [
38].
The Continuous-Time Impulse
An
impulse in continuous time must have
``zero width''
and
unit area under it. One definition is
![$\displaystyle \delta(t) \isdef \lim_{\Delta \to 0} \left\{\begin{array}{ll} \fr...
...eq t\leq \Delta \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array} \right. \protect$](http://www.dsprelated.com/josimages_new/mdft/img1700.png) |
(B.3) |
An impulse can be similarly defined as the limit of
any
integrable 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,
 |
(B.4) |
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
(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
[
12,
38]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)
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