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

are integral multiples is t he r atio of t he g cf

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Unformatted text preview: an explain this role by considering a signal f (t) t hat has rapid changes such as j ump discontinuities. To synthesize a n i nstantaneous change a t a j ump discontinuity, t he phases of t he various sinusoidal components in its s pectrum m ust be such t hat all (or most) o f t he h armonic amplitudes will have one sign before t he discontinuity and t he o pposite sign after t he discontinuity. This will result in a s harp change in f {t) a t t he p oint of discontinuity. We can verify 204 3 Signal R epresentation by Orthogonal S ets 3.4 Trigonometric Fourier Series 205 t his a ssertion in a ny waveform with j ump discontinuity. Consider, for example, t he s awtooth w aveform in Fig. 3.10b. T his waveform has a discontinuity a t t = 1. T he F ourier series for t his waveform as given in Exercise E3.6b is f (t) = 2: + ~ cos (211"t + 90°) + ~ cos (311"t + ~ cos (411"t + 90°) + ...J [cos (1I"t - 90°) 90°) (3.68) F igure 3.12 shows t he first t hree c omponents of this series. T he p hases of all t he (infinite) c omponents a re such t hat t heir a mplitudes a re positive j ust before t = 1 a nd t urn n egative j ust a fter t = 1, t he p oint o f discontinuity. T he s ame b ehavior is also observed a t t = - 1, w here similar discontinuity occurs. T his sign change in all t he h armonics a dds u p t o p roduce t he j ump discontinuity. I n a chieving a s harp c hange in t he waveform, t he role of t he p hase s pectrum is crucial. I f we t ry t o r econstruct t his s ignal while ignoring t he p hase spectrum, t he r esult will b e a s meared a nd s pread-out waveform. I n g eneral t he p hase s pectrum is j ust a s crucial in d etermining t he waveform as is t he a mplitude s pectrum. T he s ynthesis o f a ny signal f (t) i s achieved by using a p roper c ombination o f a mplitudes a nd p hases o f various sinusoids. T his u nique combination is t he F ourier s pectrum o f f (t). t -+ F ig. 3 .13 Fourier Synthesis of a continuous signal using first 19 harmonics. Gibbs phenomenon is present only when t here is a j ump d iscontinuity in f (t). W hen a c ontinuous function f (t) is synthesized using t he first n t erms of t he F ourier series, t he s ynthesized function approaches f (t) for all t a s n --+ 0 0. No G ibbs phenomenon appears. Note t he a bsence of t he G ibbs phenomenon in Fig. 3.13, where a continuous signal is synthesized using first 19 h armonics. C ompare t he s imilar s ituation for a discontinuous signal in Fig. 3.11. 6 Fourier Synthesis of Discontinuous Functions: T he Gibbs Phenomenon Fig. 3.11 shows t he s quare function f (t) a nd i ts a pproximation by a t runcated t rigonometric F ourier series t hat i ncludes only t he first n h armonics for n =l, 3, 5, a nd 19. T he p lot of t he t runcated series approximates closely t he f unction f (t) as n increases, a nd w e e xpect t hat t he series will converge exactly t o f (t) a s n --+ 0 0. T his is because, a s s hown in Sec. 3.3, t he e nergy of t he difference between f (t) a nd i ts Fourier series o ver one period ( the e rror energy) --+ 0 a...
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