## Fourier Transform (FT) and Inverse

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

and its inverse is given by

Thus, the Fourier transform is defined for continuous time and continuous frequency, both unbounded. As a result, mathematical questions such as existence and invertibility are most difficult for this case. In fact, such questions fueled decades of confusion in the history of harmonic analysis (see Appendix G).

### Existence of the Fourier Transform

Conditions for the *existence* of the Fourier transform are
complicated to state in general [36], but it is *sufficient*
for
to be *absolutely integrable*, *i.e.*,

(3.6) |

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.*,

(3.7) |

or, . More generally, it suffices to show for [36, 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 loosely called the
*delta function*), discussed in §B.10.

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Fourier Theorems for the DTFT

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Discrete Time Fourier Transform (DTFT)