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

I n t his s et t he sinusoid of frequency wo called t

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Unformatted text preview: 1a)] an t an = ;:p .LO I + e- t2 / = 0.504 ( - - 2 2 ) 1 + 16n sin 2 ntdt = 0.504 0 (+ 16n ~) 1 T herefore '+TO I (t)sinnwotdt n = 1 ,2,3, . .. (3.51c) t, j (t) = 0.504 [ 00 2 1 + ~ 1 + 16n 2 (cos 2 nt . + 4 nsm 2] nt) Compact Trigonometric Fourier Series T he t rigonometric Fourier series in Eq. (3.48) contains sine and cosine terms of t he same frequency. We c an combine t he two terms t o o btain a single sinusoid o f t he s ame frequency using t he trigonometric identity an cos nwot + bn sin nwot = e n cos (nwot + (}n) (3.52) To find t he c ompact F ourier series, we c ompute i ts coefficients using Eq. (3.53) as Co = ao = 0.504 Cn = () n J a~ + b~ = 0.504 = t an - 1 (-b ) = ~ (1+1!n2)2 + (1:;:::2)2 = 0.504 ("h:16n t an - 1( - 4 n ) = - tan - 1 4 n 2) (3.55) 192 3 Signal R epresentation by O rthogonal Sets (a) ~ o - It ~ 193 Periodicity o f t he Trigonometric Fourier Series f {t) -21t 3.4 Trigonometric Fourier Series 21t It I ~ ~-112 ~ < p(t) 1- ~ (b) We have shown how a n a rbitrary signal f (t) may b e expressed as a t rigonometric Fourier series over any interval of To seconds. In E xample 3.3, for instance, w~ r epresented e - t / 2 over t he interval from 0 t o 7 r / 2 only. T he Fourier series found in Eq. (3.56) is e qual t o e - t / 2 over t his i nterval alone. O utside t his interval t he series is n ot necessarily equal t o e - t / 2 • I t would be interesting t o find o ut w hat h appens t o t he F ourier series outside this interval. We now show t hat t he t rigonometric Fourier series is a periodic function o f p eriod To ( the p eriod o f t he f undamental). Let us denote t he t rigonometric Fourier series o n t he r ight-hand side o f Eq. (3.54) by <.p(t). T herefore 00 - 2n 0 - It 21t It 1_ <.p(t) = Co + a nd r'w I o <.p(t + To) = Co + L Cn cos [nwo(t + To) + On] n =l 0.125 T , T 4 2 (c) 6 00 • = Co + = (d) f ~cos(2nt 1 + 16n 2 tan- 1 4n) 0 ~ t ~ 7r o~ t + 0.084 cos (6t - 85.24°) + 0.063 cos (8t - 86.42°) + . .. Remember t hat t he r ight-hand side represents e - t !2 over the interval 0 t o t his interval, t he two sides need not b e equal. Co +L 0.504 0 I I 2 0.244 0.125 - 75.961 - 82.87 I I 3 0.084 I I I - 85.24\ 4 I I 5 0.063 0.0504 8 6.421 87.14 I 6 I 0.042 87.61 I Cn cos (nwot + On) = (3.56a) 7r ~ 7r (3.56b) only. Outside <.p(t) I I 0.036 87.95 ao 7 I I I • for all t (3.57) This result shows t hat t he t rigonometric Fourier series is a periodic function o f p eriod To ( the p eriod o f i ts fundamental). For instance, 'P(t), t he Fourier series on t he r ight-hand side o f E q. (3.56), is a periodic function in which t he s egment o f f (t) in Fig. 3.7a over t he i nterval (0 ~ t ~ 7 r) r epeats periodically every 7r seconds, as illustrated in Fig. 3 ·7h·t T hus, when we represent a signal f (t) by t he t rigonometric Fourier series over a certain interval o f d uration To, t he f unction f (t) a nd its Fourier series <.p(t) need b e e qual only over t hat i nterval o f To seconds. O utsi...
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