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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online