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

339 t he a dvantage of t his a pproximation of a

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Unformatted text preview: ed new schools to replenish their ranks. T he 26-year old Fourier was appointed chair of mathematics a t t he newly created school Ecole Normale in 1794. 2 :j:Contrast this situation with the polynomial approximation of f (t). Suppose we wish to approximate f (t) by a polynomial in t such that the polynomial is equal to f (t) at two points t I and t2. This can be done by choosing a first order polynomial ao + a l t with f (tl) = ao + a liI and f (t2) = ao + a lt2 Solution of these equations yields the desired values of ao and a l. For a three-point approximation, we must choose the polynomial ao + a l t + a2t2 with f (t;) = ao + a lti + a2ti2 i = 1,2, and 3 The approximation improves with larger number of points (higher-order polynomial), but the coefficients ao, a i, a2, . .. do not have the finality property. As we increase the number of terms in the polynomial, we need to recalculate the coefficients. 188 3 Signal R epresentation by Orthogonal Sets J oseph F ourier (left) and N apoleon (right). Napoleon was t he first m odern r uler with a scientific education, a nd h e was one of t he r are p ersons w ho was equally comfortable w ith soldiers a nd s cientist. T he age of Napoleon w as o ne of t he m ost f ruitful in t he h istory of science. Napoleon liked t o sign himself as a ' member o f i nstitut d e F rance'(a f raternity o f s cientists), a nd h e once e xpressed t o L aplace his regret t hat "force of circumstances h as led me so far from t he c areer of a scientist.,,3 Many g reat figures in science a nd m athematics, including F ourier a nd Laplace, were honored a nd p romoted by Napoleon. I n 1798, he t ook a g roup o f s cientists, artists, a nd s cholars-Fourier a mong t hem-on his Egyptian e xpedition, w ith t he p romise of a n e xciting a nd historic union of adventure a nd r esearch. F ourier proved t o b e a capable a dministrator o f t he newly formed I nstitut d 'Egypte, which, incidentally, was responsible for t he discovery of t he R osetta S tone. T he i nscription on t his s tone in two languages a nd t hree scripts (hieroglyphic, d emotic, a nd Greek) enabled T homas Young and Jean-Francois C hampollion, a p rotege of Fourier, t o i nvent a m ethod o f t ranslating h ieroglyphic writings of ancient E gypt-the o nly significant result o f N apoleon's E gyptian e xpedition. Back i n F rance in 1801, F ourier briefly served in his former position as professor of m athematics a t t he Ecole P olytechnique in Paris. I n 1802 N apoleon appointed h im t he p refect of Isere ( with its h eadquarters in Grenoble), a position in which Fourier served w ith d istinction. Fourier was created B aron o f t he E mpire by Napoleon in 1809. L ater, when Napoleon was exiled t o E lba, his r oute was t o t ake h im t hrough G renoble. Fourier h ad t he r oute c hanged t o avoid meeting Napoleon, which would h ave displeased Fourier's new master, t he B ourbon K ing Louis XVIII. W ithin a y ear, N apoleon escaped from Elba. O n his way home, a t G renoble, Fourier was b...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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