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Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

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Existence of the Fourier Transform

Conditions for the existence of the Fourier transform are complicated to state in general [35], but it is sufficient for $ x(t)$ to be absolutely integrable, i.e.,

$\displaystyle \left\Vert\,x\,\right\Vert _1 \isdefs \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\isdefs \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$ [35, 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), discussed in §B.1.10.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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