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

S ketch cpt for all values of t 3 4 3 for each o f t

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Unformatted text preview: erms w ith Wo = 2, we c onstruct a p ulse f (t) = It I over - 1 ::; t ::; 1, a nd r epeat i t e very rr seconds (Fig. P3.4-8b). T he r esulting signal is a n even function with period 7r. Hence, its Fourier series will have only cosine t erms w ith Wo = 2. T he r esulting Fourier series represents f (t) = t over 0 ::; t ::; 1 a s desired. We do n ot c are w hat i t r epresents outside t his interval. Represent f (t) = t over 0 ::; t ::; 1 by a Fourier series t hat h as ( a) Wo = ~ a nd c ontains all harmonics, b ut cosine t erms only. ( b) wo = 2 a nd c ontains all harmonics, b ut s ine t erms only. ( c) wo = ~ a nd c ontains all harmonics, which a re n either exclusively sine nor cosine ( d) Wo = 1 a nd c ontains only o dd h armonics a nd cosine terms. ( e) Wo = ~ a nd c ontains only o dd h armonics a nd s ine terms. ( f) Wo = 1 a nd c ontains only o dd h armonics, which a re n either exclusively sine nor cosine. Hint: For p arts d, e, a nd f, y ou need t o use half wave s ymmetry discussed in Prob. 3.4-7. Cosine t erms i mply possible de component. 3 .4-9 S tate, w ith reasons, w hether t he following signals a re p eriodic o r a periodic. For periodic signals, find t he p eriod a nd s tate w hich o f t he h armonics a re p resent in t he series. TO/2 =- P roblems f(t) s in nWot dt 0 ¥ + 3 cos ~ + 3sin (~+300) ( a) 3 s in t + 2 sin 3 t -4""'3 ( h) ( 3sin 2 t + sin 5t)2 ( d ) 7 cos r rt + 5 s in 2rrt t . .. 10 ( g) s in 3 t + cos ¥ t ( c ) 2 sin 3t + 7 cos 7rt 8 ( f) s in ( b) 2 + 5 sin 4 t + 4 cos 7 t Using t hese r esults, find t he F ourier series for t he periodic signals in Fig. P3.4-7. (a) ( e) 3 cos ht + ( i) (5 sin 2t)3 5 cos 2t (b) F ig. P3.4-1O. F ig. P 3.4-1. ;1 ;1 - Jt Jt ( a) M ~Nd M (b) F ig. P3.4-B. 1_ 3 .4-10 F ind t he t rigonometric Fourier series for f (t) s hown in Fig. P3.4-10 over t he i nterval [0, IJ. Use Wo = 2rr. S ketch t he F ourier series ",(t) for all t. C ompute t he e nergy o f t he e rror signal e(t) if t he n umber of t erms in t he F ourier series a re N for N = 1 ,2,3 a nd 4. H int: Use Eq. (3.40) t o c ompute e rror energy. 3 .4-11 W alsh functions, which c an t ake o n only two a mplitude values, form a complete set of o rthonormal f unctions a nd a re o f g reat p ractical i mportance i n digital applications because t hey c an b e easily g enerated b y logic circuitry a nd b ecause multiplication with these functions c an b e i mplemented by simply using a p olarity r eversing switch. 232 3 S ignal R epresentation b y O rthogonal S ets P roblems 233 I f (/) (a) <,(" (b) It o · It 0.5 - 11-------1 I~ F ig. P 3.4-12. L egendre polynomials a re o rthogonal. R eader m ay verify t hat I~" 1 1 Pm(t)Pn(t) dt = tJJ l I 0.75 FkFL 2 m+l m =n 0 -1 0.25 {_2 ( a) R epresent f (t) i n Fig. P 3.4-12a by t he L egendre Fourier series over t he i nterval - 1 < t < 1. C ompute o nly first two nonzero coefficients of t he series. C ompute t he e nergy o f e (t), t he e rror for t he o ne a nd two (non-zero) t erm a pproximations. ( b) R epresent f (t) i n Fig. P 3.4-12b u sing t he L egendre Fourier series. C ompute t he coefficients o f t he s eries for t he first two non-zero t erms. H int: A lthough t he series r epresentation is valid only over - 1 < t < 1, i t c an b e e xtended t o a ny r egion b y s uitable t ime scaling. I 0" 3 .5-1 For each o f t he p eriodic signals in Fig. P...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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