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.,
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 [12, 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 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
(Note, incidentally, that is in but not .) 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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Discrete Time Fourier Transform (DTFT)
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Convolution DSP tutorial