Fourier Transform (FT) and Inverse

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

$\displaystyle X(\omega) \isdefs \int_{-\infty}^\infty x(t) e^{-j\omega t} dt \protect$ (3.4)

and its inverse is given by

$\displaystyle x(t) \eqsp \frac{1}{2\pi}\int_{-\infty}^\infty X(\omega) e^{j\omega t} d\omega \protect$ (3.5)

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

$\displaystyle \left\Vert\,x\,\right\Vert _1 \isdefs \int_{-\infty}^\infty \left\vert x(t)\right\vert dt \;<\; \infty .$ (3.6)

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.,

$\displaystyle \left\Vert\,x\,\right\Vert _2^2\isdefs \int_{-\infty}^\infty \left\vert x(t)\right\vert^2 dt \;<\; \infty,$ (3.7)

or, $ x\in L2$ . More generally, it suffices to show $ x\in Lp$ for $ 1\leq p\leq 2$ [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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Discrete Time Fourier Transform (DTFT)