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

T he relationship between t he p hase s pectrum a nd

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Unformatted text preview: s n --+ 0 0. Yet t he c urious fact, as seen f rom F ig. 3.11, is t hat even for large n , t he t runcated series exhibits a n o scillatory b ehavior a nd a n o vershoot approaching a value of a bout 9% in t he v icinity of t he d iscontinuity a t t he first p eak o f oscillation. Regardless of t he value of n , t he o vershoot r emains a t a bout 9%. T his s trange b ehavior a ppears t o c ontradict t he m athematical r esult derived in Sec. 3.3-2 t hat t he e rror energy --+ 0 as n --+ 0 0. I n f act, t his a pparent c ontradiction puzzled m any p eople a t t he t urn of t he c entury. Josiah W illard G ibbs gave a m athematical e xplanation of this behavior (now called called t he G ibbs p henomenon). We can reconcile t he two confiicting notions by observing from F ig. 3.11 t hat t he frequency of oscillation of t he s ynthesized signal is n , so t he w idth o f t he spike w ith 9% o vershoot is a pproximately 1 /2n. As we i ncrease n , t he n umber of t erms i n t he series, t he frequency o f o scillation increases a nd t he spike w idth 1 /2n diminishes. As n --+ 0 0, t he e rror energy - -+ 0 b ecause t he e rror consists m ostly o f t he spikes, whose widths - > O. T herefore, as n --+ 0 0, t he c orresponding F ourier series differs from f (t) b y a bout 9% a t t he i mmediate left a nd r ight of t he p oints of discontinuity, a nd y et t he e rror energy --+ O. W hen we u se o nly t he first n t erms in t he F ourier series t o s ynthesize a signal, we a re a bruptly t erminating t he series, giving a u nit weight t o t he first n h armonics a nd zero weight t o all t he r emaining harmonics beyond n . T his a brupt t ermination o f t he series c auses t he G ibbs phenomenon in synthesis of discontinuous functions. More discussion o n t he G ibbs phenomenon, its ramifications, a nd c ure a ppear i n section 4.9. E xercise E 3.8 B y inspection of signals in Figs. 3.7b, 3 .l0a, a nd 3.lOb, determine t he a symptotic r ate of decay o f t heir a mplitude s pectra. Answer: l in, l /n 2 , a nd l in, respectively. 'V A Historical Note on the Gibbs Phenomenon N ormally speaking, troublesome functions w ith s trange b ehavior are invented by mathematicians, although we rarely see such oddities in practice. I n t he case of t he G ibbs phenomenon, however, t he t ables were t urned. A r ather p uzzling behavior was observed in such a m undane o bject as a mechanical wave synthesizer, a nd t hen well-known m athematicians of t he d ay were dispatched o n t he s cent of it t o discover its hide-out. A lbert Michelson (of Michelson-Morley fame) was an intense. practical m an who developed ingenious physical i nstruments o f e xtraordinary precision, mostly in t he field of optics. I n 1898 he developed a n i nstrument ( the h armonic analyzer) which could c ompute t he first 80 coefficients of t he F ourier series of a signal f (t) specified by any graphical description. T he h armonic analvzer could also be used as a harmonic synthesizer, which could plot a function f(t)" g enerated by s umming t he first 80 h armonics (Fourier components) of a rbit...
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