Spectra of Nonintegrable Functions
Pp. 81-102 (22)
Khamies M. A. El-Shennawy
The main evidence of the Fourier transform concludes that when the integral, Eq.(3.2), is absolutely integrable (finite), the Fourier transform exists. More sufficient evidence for the existence of the Fourier transform is, since the magnitude of the exponential e−j2πft, equals unity, then the integral
|x(t)| dt < ∞ (3.3a)
, (Dirichlet`s conditions), Eq.(3.3a) must be finite (finite area), that is x(t) must be unperiodic signal (energy signal) which has finite energy, zero average power, and infinite periodic time To . But, is the condition of the absolutely integrable is always necessary, the answer is No, because there are some special functions are not absolutely integrable (do not satisfy Dirichlet`s conditions) and have Fourier transforms in the limit such as the Dirac delta function (unit impulse) δ(t), the unit step function u(t) and the signum function sgn(t), Fig.1.4. It is necessary to evaluate the Fourier transform of these non-integrable functions to obtain their continuous frequency spectrum and study how these functions allow the easy solution of many communication problems. Also some of these non-integrable functions are related to each other by mathematical differentiation and integration formulas.
Dirac delta function, Signum function, Unit step function, Error probability function.
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