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
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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