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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 squarewave signal (Fig. 3.8a), the zeroth 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 lowerfrequency 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
highfrequency 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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 Spring '13
 Bayliss
 Signal Processing, The Land

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