## 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.*,

*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

*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)