## 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)