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

37b a signal therefore h as a dual i dentity the time

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Unformatted text preview: ients ao, a n, and bn in Eq. (3.51) must be 1 = - 17r/2 cos n t dt = -2 11' -7r /2 r If(t)1 dt < 00 n = 1, 5, 9, 13, . .. (3.60b) n = 3, 7, l 1, 15, . .. (3.59) tThis behavior o f the Fourier series is dictated by its error energy minimization property, discussed in Sec. 3.3. 2 n even finite. From E qs. (3.51a), (3.51b), a nd (3.51c), it follows t hat t he existence of these coefficients is g uaranteed if f (t) is absolutely integrable over one period; t hat is, lTo sin ( 11' n) n1l' 117r/2 sin n t dt = -7r/2 bn = ; 0 (3.60c) 196 3 Signal Representation b y O rthogonal Sets I -21t -31t It It 2: - It "2 It 21t 31t I~ (a) 2 11 0.5 3.4 Trigonometric Fourier Series 197 Using t he a bove values we could p lot a mplitude a nd p hase spectra. However, t o simplify o ur t ask i n t his s pecial case, we will allow a mplitude en t o t ake on negative values so t hat we d o n ot n eed a phase o f - 7r t o a ccount for t he sign. I n o ther words, phases of allcomponents a re zero, so we can discard t he p hase s pectrum a nd m anage with only t he a mplitude s pectrum, as shown in Fig. 3.8b. Observe t hat t his simpler procedure involves no loss o f i nformation a nd t hat t he a mplitude s pectrum i n Fig. 3.8b has t he c omplete information a bout t he F ourier series in (3.61). T herefore, w henever all s ine t erms v anish (b n = 0), i t is convenient t o a llow en t o t ake o n n egative values. T his p rocedure p ermits t he s pectral i nformation t o b e conveyed by a single s pectrum-the a mplitude s pectrum. B ecause en c an b e p ositive as well as negative, t he s pectrum is called t he a mplitude s pectrum r ather t han t he m agnitude s pectrum. • • E xample 3 .5 F ind t he c ompact t rigonometric Fourier series for t he t riangular p eriodic signal f (t) i llustrated in Fig. 3.9a, a nd s ketch t he a mplitude a nd p hase s pectra for f (t). I n t his case t he p eriod To = 2. Hence 2 7r (b) o 10 WO a nd -2 f (t) ':lit = ao + L = 2" =7r an cos n7l:t + bn sin n7rt n =l F ig. 3 .8 A s quare pulse periodic signal a nd i ts Fourier s pectra. T herefore 2(cos t - 1 f (t) = - + 2 71: 1 1 1 - cos 3 t + - cos 5 t - - cos 7t + . .. ) 3 5 7 w here f (t) = (3.61) Observe t hat bn = 0 a nd all t he sine t erms a re zero. O nly t he cosine t erms a ppear i n t he t rigonometric series. T he series is therefore a lready in t he comp~~t form ex.cept t hat t he a mplitudes of a lternating h armonics a re n egative. Now, by defimtlOn, a mphtudes en. a re p ositive [see Eq. (3.53a)J. T he n egative sign c an b e a ccommodated by a phase o f 71: r adIans as seen from t he t rigonometric i dentityt - cos x = cos ( x - ~+ ~ [ cos t + 21 32 1 1/2 = 1 3/2 2 Atcos n7l:tdt + 2 A(1- t)eos n7l:tdt - 1/2 i cos 9 t+ . .. J T his is precisely t he F ourier series in t he c ompact t rigonometric form, T he a mplitudes a re eo = ~ < t ::; ~ / f(t) cos n 7rtdt an = 2 - 1/2 71:) ~COS(3t -71:) + ~cos 5 t+ ~COS(7t -71:) + ~ Here i t will be advantageous t o choose t he i nterval o f i ntegration from - ~ t o ~ r ather t han 0 t o 2. A glance a...
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