Signal Processing and Linear Systems-B.P.Lathi copy

2 jcontrast this situation with the polynomial

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Unformatted text preview: rought b efore h im in chains. Napoleon scolded Fourier for his ungrateful behavior b ut r eappointed h im t he p refect of R hone a t Lyons. W ithin four months Napoleon was defeated a t W aterloo, a nd was exiled t o St. Helena, never t o r eturn. 3.4 Trigonometric Fourier Series 189 Fourier once again was in disgrace as a B onapartist, a nd h ad t o p awn his effects t o keep himself alive. B ut t hrough t he i ntercession of a former s tudent, who was now a prefect of Paris, he was a ppointed d irector of t he s tatistical b ureau of t he Seine a p osition t hat allowed him ample t ime for scholarly pursuits. L ater, in 1827, h~ was elected t o t he powerful position of p erpetual s ecretary of t he P aris A cademy of Science, a section of t he i nstitute. 4 W hile serving as t he p refect of Grenoble, Fourier carried on his e laborate investigation of p ropagation of h eat in solid bodies, which led him t o t he F ourier series a nd t he F ourier integral. O n 21 D ecember 1807, h e announced these results in a prize p aper on t he t heory of heat. Fourier claimed t hat a n a rbitrary function (continuous or w ith d iscontinuities) defined in a finite interval by a n a rbitrarily c apricious g raph c an always b e e xpressed as a s um of sinusoids (Fourier series). T he judges, who included t he g reat F rench m athematicians L aplace, Lagrange, Monge, a nd L aCroix a dmitted t he novelty a nd i mportance of Fourier's work, b ut c riticized i t for lack o f m athematical rigor a nd generality. Lagrange t hought i t i ncredible that. a s um o f sines a nd cosines could a dd u p t o a nything b ut a n infinitely differentiable function. Moreover, one of t he p roperties of a n infinitely differentiable function is t hat if we know its behavior over a n a rbitrarily s mall interval, we c an d etermine its behavior over t he e ntire range ( the T aylor-Maclaurin series). Such a f unction is far from a n a rbitrary o r a capriciously d rawn g raph. 5 F ourier t hought t he c riticism u njustified b ut was unable t o prove his claim because t he t ools required for o perations w ith infinite series were n ot available a t t he t ime. However, p osterity h as proved Fourier t o b e closer t o t he t ruth t han his critics. T his is t he classic conflict between pure m athematicians a nd p hysicists or engineers, as we shall see again in t he life of Oliver Heaviside (p. 381). I n 1829 Dirichlet proved Fourier's claim concerning capriciously drawn functions w ith a few restrictions (Dirichlet conditions). Although t hree o f t he four judges were in favor of publication, this p aper was rejected because o f v ehement opposition by Lagrange. Fifteen years later, after several a ttempts a nd d isappointments, Fourier published t he r esults in e xpanded form as a t ext, Theorie analytique de la chaleur, which is now a classic. 3.4 Trigonometric Fourier Series C onsider a signal set: { I, cos wot, cos 2wot, . .. , cos nwot, . .. ; sin wot, sin 2wot, . .. , s in nwot, . .. } (...
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