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 , 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 [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.
Fourier Theorems for the DTFT
Discrete Time Fourier Transform (DTFT)