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

T his is because a s s hown in sec 33 t he e nergy of

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Unformatted text preview: rary a mplitudes a nd phases. This i nstrument t herefore h ad t he a bility of self-checking its o peration by analyzing a signal f (t) a nd t hen a dding t he r esulting 80 c omponents t o see w hether t he s um yielded a close approximation of f (t). Michelson found t hat t he i nstrument checked very well w ith m ost of signals analyzed. However, when he t ried a d iscontinuous function, such as a square wave,t a curious behavior was observed. T he s um of 80 c omponents showed oscillatory behavior (ringing) w ith a n o vershoot of 9% i n t he v icinity of t he p oints of discontinuity. Moreover, this behavior was a c onstant f eature regardless of t he n umber of t erms t Actually i t was a periodic s awtooth signal ----------.------------------ ..."~.------ 206 3 Signal R epresentation by Orthogonal Sets 3.5 Exponential Fourier Series 207 Moreover, this set is a complete set. 6 ,7 From Eqs. (3.44) a nd (3.45), it follows t hat a signal j (t) c an be expressed over an interval of duration To seconds as a n e xponential Fourier series 00 L j (t) = D nejnwot (3.70) j (t)e-jnwot d t (3.71) n =-oo where [see E q. (3.45)] Dn = r ~ iTo To T he e xponential Fourier series is basically another form of t he t rigonometric Fourier series. Each sinusoid of frequency w c an be expressed as a sum of two exponentials jwt e a nd e - jwt . T his results in t he e xponential Fourier series consisting of components of t he form e inwot w ith n varying from - 00 t o 0 0. T he e xponential Fourier series in Eq. (3.70) is periodic with period To. I n order to see its close connection with t he t rigonometric series, we shall rederive t he e xponential Fourier series from t he t rigonometric Fourier series. A sinusoid in t he t rigonometric series can be expressed as a sum of two exponentials using Euler's formula: A lbert M ichelson (left) and W illard J . G ibbs (right). C n cos ( nwot a dded. Larger number of terms made t he oscillations proportionately faster, b ut regardless of t he n umber of terms added t he overshoot remained 9%. T his puzzling behavior caused Michelson t o s uspect some mechanical defect in his synthesizer. He wrote a bout h is observation in a l etter t o N ature (December 1898). Josiah Willard Gibbs a n e minent m athematical physicist (inventor of vector analysis), a nd a professor 'at Yale investigated a nd clarified this behavior for a s awtooth periodic signal in a letter t o N ature. 9 L ater, in 1906, B acher generalized the result for any function with discontinuity.lO I t was Bacher who gave t he n ame Gibbs p henomenon t o t his behavior. G ibbs showed t hat t he peculiar behavior in the synthesis of a square wave was inherent in t he b ehavior of the Fourier series because of nonuniform convergence a t t he p oints o f discontinuity. This, however, is not t he e nd o f t he story. B oth Bacher a nd Gibbs were under t he impression t hat t his property had remained undiscovered until Gibbs'~ l etter in 1899. I t is n ow known t hat t he Gibbs phenomenon had been observed I II 1848 by Wilbraham of Trinit...
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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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