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

Similarly t he component of f requency 3 is 6e j te

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Unformatted text preview: oreover, we m ay c ompute D n = a n /2 u sing E q. (3.66b), w hich r equires i ntegration o ver a h alf p eriod only. Similarly, w hen j (t) h as a n o dd s ymmetry, an = 0, a nd D n = - jb n /2 is i maginary ( positive o r n egative). H ence, L Dn c an o nly b e 0 o r ±11' / 2. M oreover, we m ay c ompute D n = - jb n /2 u sing E q. ( 3.67b), which r equires i ntegration o ver a h alf p eriod only. N ote, h owever, t hat i n t he e xponential c ase, we a re u sing t he s ymmetry p roperty i ndirectly b y f inding t he t rigonometric coefficients. We c annot a pply i t d irectly t o f ind D n f rom eq. (3.71). 6 E xercise E 3.9 The exponential Fourier spectra of a certain periodic signal f (t) are shown in Fig. 3.17. Determine and sketch the trigonometric Fourier spectra of f (t) by inspection of Fig. 3.17. Now write the (compact) trigonometric Fourier series for f (t). Answer: n wo = 211' 6 f (t) = 4 + 6 cos (3t - ~) + 2 cos (6t -:,f) + 4 cos (9t - ~) E xercise E 3.10 Find the exponential Fourier series and sketch the corresponding Fourier spectrum Dn vs. w for the full-wave rectified sine wave depicted in Fig. 3.18. Answer: f (t) = ~ ~ _ 1_e1 2nt 2 11' ~ 1 -4n 3 214 S ignal R epresentation b y O rthogonal S ets 3.5 215 E xponential F ourier S eries t L D, -9 -6 -1tI6 -1tI4 6 3 . .• ! -1tI2 ,- 9 .1 F ig. 3 .17 Fourier spectra for t he signal in Exercise E3.9. __ ___ A ............... 1+ Sint ~ Fig. 3 .19 (a) A clipped sinusoid cos wot (b) t he d istortion component fd(t) of t he signal in (a). ?t~ For a real f (t), ID-nl = IDnl. T herefore F ig. 3 .18 A full-wave rectified sine wave in Exercise E3.10. 00 P f = D02 + 2 L ID nl 2 ( 3.83b) n =l 3 .5-2 Parseval's Theorem • T he t rigonometric F ourier s eries o f a p eriodic s ignal f (t) is given b y 00 f (t) = Co + L C n cos (nwot + lin) n =l E very t erm o n t he r ight-hand s ide o f t his e quation is a power signal. M oreo:er, a ll t he F ourier c omponents o n t he r ight-hand s ide a re o rthogonal o ver o ne p eriod. H ence, t he p ower o f f (t) is e qual t o t he p ower o f t he s um ~f a ll t he s inusoidal c omponents o n t he r ight-hand side. We a lready d emonstrated I II E xample 1.2 t hat t he p ower o f t he s um o f sinusoids is e qual t o t he s um of t he p owers of all t he 2 s inusoids. M oreover, t he p ower of a sinusoid of a mplitude C n is C n / 2 r~gardless o f t he v alues o f i ts f requency p hase, a nd t he p ower o f a d c t erm Co is Co . T hus, t he p ower o f f (t) is given b y Pf 2 1 ~ = Co + 2 L .."Cn (3.82) 2 E xample 3 .9 T he i nput signal t o a n audio amplifier of gain 100 is given by x (t) = 0.1 cos wot. Hence, t he o utput is a sinusoid 10 cos wot. However, t he amplifier, being nonlinear a t higher amplitude levels, clips all amplitudes beyond ± 8 volts as shown in Fig. 3.19a. We shall determine t he harmonic distortion incurred in this operation. The o utput y(t) is t he clipped signal in Fig. 3.19a. The distortion signal Yd(t), shown in Fig. 3.19b, is t he difference between t he u ndistorted sinuso...
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