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

Hint use eqs 366 a nd 367 for symmetry l 34

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Unformatted text preview: s et (~, ~, 2), t he G CF of t he n umerator s et ( 2,6,2) is 2; t he L CM o f t he d enominator s et ( 3,7,1) is 21. T herefore, is t he l argest number o f which ~,~, a nd 2 a re integral multiples. fr. 3 Signal R epresentation by Orthogonal Sets 202 ------.1 I' - --l---_-n--_-n,....'2--+ I 1 1 --(a) n 1- (b) 3.4 Trigonometric Fourier Series 203 t inuous function with j ump discontinuities, a nd therefore its amplitude spectrum decays r ather slowly, as l in [see Eq. (3.61)]. O n t he o ther hand, t he t riangular pulse periodic signal in Fig. 3.9a is s moother because it is a continuous function ( no· j ump discontinuities). Its s pectrum decays rapidly with frequency as I /n 2 [see Eq. (3.63)]. We can shows t hat if t he first k - 1 derivatives of a periodic signal f {t) a re continuous a nd t he k th derivative is discontinuous, t hen i ts amplitude spectrum en decays with frequency a t least as rapidly as l ink+!. T his result provides a simple a nd useful means for predicting t he a symptotic r ate of convergence of t he Fourier series. In t he case of t he square-wave signal (Fig. 3.8a), the zero-th derivative of t he signal (the signal itself) is discontinuous, so t hat k = o. For t he t riangular periodic signal in Fig. 3.9a, t he first derivative is discontinuous; t hat is, k = 1. For this reason, t he s pectra of these signals decay as l in a nd I /n2, respectively. (e) (d) , .' . ~. """ •• I ..\ . ".,' (e) J"v Fig. 3.11 monics. 1- Synthesis of a square pulse periodic signal by successive addition of its har- t he n ineteenth harmonic), t he edges of t he pulses become sharper a nd t he signal resembles f {t) more closely. Asymptotic Rate o f Amplitude Spectrum Decay T he a mplitude s pectrum indicates t he a mounts (amplitudes) of various frequency components of f {t). I f f {t) is a s mooth function, its variations are less rapid. Synthesis of such a function requires predominantly lower-frequency sinusoids a nd r elatively small amounts of rapidly varying (higher frequency) sinusoids. T he a mplitude s pectrum of such a function would decay swiftly with frequency. To synthesize s uch a function we require fewer terms in the Fourier series for a good approximation. I n c ontrast, a signal with s harp changes, such as j ump discontinuities, contains r apid variations a nd its synthesis requires a relatively large amount of high-frequency components. T he a mplitude spectrum of such a signal would decay slowly w ith frequency, a nd t o synthesize such a function, we require many terms in its Fourier series for a good approximation. T he square wave f {t) is a discon- Fig. 3.12 Role of the phase spectrum in shaping a periodic signal. The Role o f t he Phase Spectrum T he r elationship between t he a mplitude s pectrum a nd t he waveform f (t) is reasonably clear. T he relationship between t he p hase s pectrum a nd t he periodic signal waveform is n ot so direct, however. Yet t he phase s pectrum plays a n equally i mportant role in waveshaping. We c...
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