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

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

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Unformatted text preview: cted in Fig. P1.4-2a. ( d) ( h) b (t - 2) sin 1rtdt e (x-l) cos 5)]8(x - 3) dx 1 . 7 -2 CD (a) (b) 2 f (t) ~~ + (sin t )y(t) = ~ + 2 f(t) ( g) 2 t f (t) + 2y(t) = ~~ + 2y(t) = f(t)~ ( h) y (t) = 2 ( a) y(t) -1 -1 ~ED = f (t - ( b) y(t) 1 = f ( - t) ( e) y (t) = = f lat) ( f) y (t) F ig. P l.4-7 1 .4-7 Using t he g eneralized function definition, show t hat 8(t) is a n even function of t. f (r) d r 2) ( d) y (t) = t f(t - 2) ISs f(r)dr = (~) 2 F ind a nd s ketch J~oo f (x) dx for t he signal f (t) i llustrated in Fig. P l.4-7. 1 .4-8 loo 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 are time-invariant p arameter s ystems a nd which are time-varying p arameter s ystems. ( c) y (t) o + 2 = f (t) ( f) ~~ + y2(t) = f (t) ( d) 1 .4-6 ( c) 3 y(t) = f (t) Hint: S tart w ith Eq. (1.24a) as t he definition of 6(t). Now change variable t show t hat I: 1 .4-9 P rove t hat = - x to q ,(t)6(-t)dt = q,(O) 1 8(at) = ~6(t) Hint: Show t hat 1 1 00 - 00 . . 4 -10 Show t hat q,(t)8(at) dt I: = ~q,(0) 8(t)q,(t) dt = -4>(0) 1 . 7 -3 F or a certain LTI system with t he i nput f (t), t he o utput y (t) a nd t he two initial conditions Xl(O) a nd X2(O), following observations were made: f (t) o o u (t) 1 e2t cos 3t ( d) e - 2t ( e) e2t ( f) 5. e -tu(t) 2 e - t (3t -1 -1 + 2)u(t) 2u(t) D etermine y (t) when b oth t he i nitial conditions are zero a nd t he i nput f (t) is as shown in Fig. P l.7-3. Hint: T here a re t hree causes: t he i nput a nd each of t he two initial conditions. Because of linearity property, if a cause is increased b y a f actor k, t he r esponse t o t hat cause also increases by t he s ame factor k. Moreover, if causes are added, t he c orresponding responses add. w here q,(t) a nd 4>(t) a re continuous a t t = 0, a nd q,(t) - > 0 a s t - > ± oo. T his integral defines 8( t) a s a generalized function. Hint: Use i ntegration by p arts . .. 4 -11 A sinusoid eat cos wt c an be expressed as a s um o f exponentials est a nd e -· t (Eq. ( l.30c) w ith complex frequencies s = a + j w a nd s = a - jw. L ocate in t he complex plane t he frequencies of t he following sinusoids: ( a) cos 3t ( b) e - 3t cos 3t ( c) y (t) Xl(O) -5 1 F ig. P 1.7-3 f {t) t __ ---~--------- 102 1 .1-4 1 I ntroduction t o S ignals a nd S ystems 1 .7-8 A system is specified by its i nput-output relationship as = loo ( c) yet) J(T)dT = ret) n , integer ( d) yet) = cos [J(t)] ( b) yet) = J(3t - 6) Show t hat t he circuit in Fig. P1.7-5 is zero-state linear b ut is not zero-input linear. Assume all diodes to have identical (matched) characteristics. Hint: In zero s tate (when the initial capacitor voltage vc(O) = 0), t he circuit is linear. If t he i nput J (t) = 0, a nd vc(O) is nonzero, t he c urrent yet) does not exhibit linearity with respect to its cause vc(O). y (t) For t he systems described by t he e quations below, with the i nput J(t) a nd o utput yet), d etermine which of t he s...
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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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