## Fourier Transform (FT) and Inverse

The *Fourier transform* of a signal
,
, is defined as

and its inverse is given by

### Existence of the Fourier Transform

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.*,

*square integrable*,

*i.e.*,

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

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,

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

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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Fourier Series (FS) and Relation to DFT

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Discrete Time Fourier Transform (DTFT)