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Fourier Transform (FT) and Inverse

The Fourier transform of a signal $ x(t)\in{\bf C}$, $ t\in(-\infty,\infty)$, is defined as

$\displaystyle X(\omega) \isdef \int_{-\infty}^\infty x(t) e^{-j\omega t} dt, \protect$ (B.1)

and its inverse is given by

$\displaystyle x(t) = \frac{1}{2\pi}\int_{-\infty}^\infty X(\omega) e^{j\omega t} d\omega. \protect$ (B.2)

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 $ x(t)$ to be absolutely integrable, i.e.,

$\displaystyle \left\Vert\,x\,\right\Vert _1 \isdef \int_{-\infty}^\infty \left\vert x(t)\right\vert dt < \infty . $

This requirement can be stated as $ x\in L1$, meaning that $ x$ belongs to the set of all signals having a finite $ L1$ norm ( $ \left\Vert\,x\,\right\Vert _1<\infty$). It is similarly sufficient for $ x(t)$ to be square integrable, i.e.,

$\displaystyle \left\Vert\,x\,\right\Vert _2^2\isdef \int_{-\infty}^\infty \left\vert x(t)\right\vert^2 dt < \infty, $

or, $ x\in L2$. More generally, it suffices to show $ x\in Lp$ for $ 1\leq p\leq 2$ [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

$\displaystyle \delta(t) \isdef \lim_{\Delta \to 0} \left\{\begin{array}{ll} \fr... ...eq t\leq \Delta \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array} \right. \protect$ (B.3)

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,

$\displaystyle \int_{-\infty}^\infty f(t) \, \delta(t)\, dt = f(0), \protect$ (B.4)

provided $ f(t)$ is continuous at $ t=0$. 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

$\displaystyle \delta(t) \isdef \lim_{\Omega\to\infty}\frac{\sin(\Omega t)}{\pi t}. $

(Note, incidentally, that $ \sin(\Omega t)/\pi t$ is in $ L2$ but not $ L1$.) 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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