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

# Lathi b p signals and systems berkeley cambridge press

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Unformatted text preview: 3-3 k =m k=m a ssuming a ll frequencies t o b e d istinct, t hat is, 1 .1-6 ( a) 10 cos ( lOOt + ( c) (10 i) + 2 s in 3t) cos ( e) 10 sin 5 t c os lOt 1 .3-1 Wi i ' W k for all i i ' k 4 D etermine t he p ower a nd t he r ms value for each o f t he following signals: ( b) 10 cos (lOOt lOt + i) + 16 sin ( 150t+ (a) i) 1- ( d) 10 cos 5 t cos lOt ( f) ejo. t cos wot -4 I n Fig. P 1.3-1, t he s ignal /1(t) = f ( - t). E xpress signals h (t), h (t), f4(t), a nd f5(t) in t erms o f s ignals f (t), /1 (t), a nd t heir t ime-shifted, time-scaled o r t ime-inverted versions. F or i nstance h (t) = f (t - T) + / 1(t - T). f it) F ig. P l.4-2 1 .4-1 S ketch t he s ignals ( a) u (t-5)-u(t-7) ( b) u (t-5)+u(t-7) ( c) t 2 [u(t-1)-u(t-2)] ( d) (t - 4)[u(t - 2) - u(t - 4)] 1 .4-2 E xpress each of t he s ignals in Fig. P1.4-2 by a single expression valid for all t. 1 .4-3 F or a n e nergy signal f (t) w ith e nergy E f, show t hat t he e nergy of a nyone o f t he s ignals - f(t), f ( - t) a nd f (t - T) is E f. Show also t hat t he e nergy o f f rat) as well as f rat - b) is E fla. T his s hows t hat t ime-inversion a nd t ime-shifting does n ot affect signal energy. O n t he o ther h and, t ime c ompression of a signal (a > 1) reduces t he energy, a nd t ime e xpansion o f a signal (a < 1) i ncreases t he energy. W hat is t he effect o n s ignal energy if t he s ignal is multiplied by a c onstant a? 1 .4-4 Simplify t he following expressions: o ( b) ( a) (c) -I o ( s in t ) 6(t) t2 + 2 [ e- t cos (3t - 60 )]6(t) 2 0 ( lW + 2 ) 6(w) w2 + 9 ( d) C in 1 .3-2 For t he s ignal f (t) d epicted in Fig. P 1.3-2, s ketch t he signals: ( a) f (-t) ( b) f (t+6) ( c) f (3t) ( d) f(~). 1 .3-3 F or t he s ignalf(t) i llustrated in Fig. P1.3-3, s ketch ( a) f (t - 4) ( b) f ( 1 \) ( c) f ( - t) ( d) f (2t - 4) ( e) f (2 - t). ( e) ( _._ 1_) 6(w JW + 2 + 3) ( f) (SinwkW) 6(w) [2 ~(t t F ig. P 1.3-1 H int: Use Eq. (1.23). For p art ( f) u se L 'H6pital's rule. +4 2)]) 6 (t-1) 100 1 .4-5 1 I ntroduction t o S ignals a nd S ystems E valuate t he following integrals: ( a) ( b) ( c) I: I: I: I: b (r)f(t - r)dr ( e) j (r)8(t - r)dr ( f) 6 (t)e- jwt ( g) dt I: I: I: I: 101 P roblems 1 .5-1 1 .6-1 (t 3 + 4)6(1 - t) dt W rite t he i nput-output r elationship for a n ideal integrator. Determine t he z ero-input a nd z ero-state components of t he response. 1 . 7 -1 t 8(t + 3 )e- dt F ind a nd s ketch t he o dd a nd t he even components of ( a) u (t) ( b) t u(t) ( c) sin wat u (t) ( d) cos w atu(t) ( e) s in wat ( f) cos wat. For t he s ystems described by t he e quations below, w ith t he i nput f (t) a nd o utput y (t), d etermine which of t he s ystems a re l inear a nd which are nonlinear. f (2 - t)8(3 - t) dt [~(x - dy ( a) dt + 2y(t) 2 ( b) ~~ + 3ty(t) = Hint: 8(x) i s l ocated a t x = O. For example, 8(1 - t) is located a t 1 - t = 0, a nd so on. ( e) (~~) ( a) F ind a nd sketch df / dt for t he signal f (t) shown in Fig. P1.3-3. ( b) F ind a nd sketch d2f / dt 2 for t he signal f (t) d epi...
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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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