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

We extend t he fourier transform by generalizing t he

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Unformatted text preview: ncyclopaedia B ritannica, M icropaedia I V, 1 5th ed., Chicago, IL. 1982. 8. 6 .2-1 +2 1 ( g) (s ( d) S2(s5+ 2) ( e) s sis + 1)2 3 s+5 8 Bell, E .T., M en o f M athematics, Simon a nd Schuster, New York, 1937. 6. 2 + 4s + 13 (s + 1)2 ( e) Doetsch, G ., I ntroduction t o the T heory a nd A pplications o f t he Laplace T ransformation w ith a Table o f L aplace T ransformations, Springer Verlag, New York, 1974. LePage, W .R., C omplex Variables a nd t he Laplace T ransforms f or E ngineers, M cGraw-Hill, New York, 1961. D urant, W ill, a nd Ariel, T he A ge o f Napoleon, T he S tory o f C ivilization S eries, P art XI, S imon a nd Schuster, New York, 1975. (e) B : d irect.integration find t he Laplace transforms of t he signals shown in Fig. P6.1-2. s2 3. 1- F ig. P 6.1-2 6 .1-2 References 2. 1_ (b) ( b) 1. 1! (a) ( d)e -3 s2 - 23 +e + 38 +1 +2 T he Laplace transform of a causal periodic signal c an b e determined from t he kn I edge of t he L aplace transform o f its first cycle (period). ow (a) I f t he Laplace transform of f it) in Fig. P 6.2-4a is F (s) t hen sho h , w t a t G (s), t he Laplace transform of get) [Fig. P6.2-4b], is G (s) = F (s) 1 - e-sTo R es> 0 462 6 1(1)1 Continuous-Time System Analysis Using the Laplace Transform Problems 463 f\ ~ ''')~ t =0 + ( a) ( b) 4 0V IF DD[ 8 16 10 18 24 ( e) F ig. P 6.3-4 t- 6 .3-4 Fig. P 6.2-4 ( b) Using t his r esult, find t he L aplace t ransform o f t he signal pet) i llustrated i n Fig. P6.2-4c. Hint: 1 + x + x 2 + x 3 + ... = l~X for Ixl < 1. . 2-5 . 2-6 . 3-1 ( a) F ind t he L aplace t ransform o f t he p ulses in Fig. 6.3 in t he t ext by using only t he t ime-different i ation p roperty, time-shifting property, a nd t he f act t hat 0( t) { :=> 1. ( b) I n E xample 6.7, t he L aplace t ransform o f I (t) is found by finding t he L aplace t ransform o f d 2J / dt 2 F ind t he L aplace t ransform o f J (t) in t hat e xample b y finding t he L aplace t ransform o f dJ / dt. H int for p art ( b): dJ / dt c an b e e xpressed a s a s um o f s tep f unctions (delayed by various a mounts) whose t ransforms c an b e d etermined readily. ( b) (D2 ( e) (D2 . 3-2 . 3 -3 + 4 D + 4) + 6 D + 25) yet) = (D + 2 )J(t) yet) if y (O-) if y (O-) = 1 a nd J (t) = e -'u(t) 2 dy dy ( b) d3y dt3 + 6 d t2 - 11 dt 4 dy dy ( e) dt 4 + 4dt 6 .3-6 6 .3-7 dJ = 5 di d 2J dJ dt dt dJ = 3 di+2J(t) Fo;. ~aeh o f .the systems specified by t he following t ransfer f unctions find t he differ ' en I equatIOn r elating t he o utput yet) t o t he i nput J(t): 8 2 +5 s + 38 + 8 2 ( e) H (s) = 5s + 78 + 2 2 8 - 28 + 5 2 8 +3s+5 ( b)H(8)= F or a s ystem w ith t ransfer f unction 8 8 +5 + 58 + 6 ( a) F ind t he ( zero-state) response i f t he i nput J (t) is: 3 4 ~ju;;(t~~IS s ystem w rite t he d ifferential e quation r elating t he o utput yet) t o t he R epeat P rob. 6.3-7 if H (8)= 6 .3-9 a nd t he i nput J (t) is: ( a) lOu(t) R epeat P rob. 6.3-7 i f a nd t he i nput J (t) = (1 - 2 8+3 + 28 + 5 ( b) u (t _ 5). 8 D etermine t he t ransfer f unctions r elating o utputs Y l(t) a nd Y2(t) t o t he i nput J (...
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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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